Modelling Materials Processing

A state of the art review and proposals for change:
New needs in training, data and software technology

Version 2.0

Synopsis

This report presents a recent (1992-93) review of materials process modelling technologies from the manufacturing engineer’s perspective, performed for the British Science and Engineering Research Council.

Several key issues are raised concerning organizational support for research in process modelling, training and educational needs in Britain from comparisons with America and Scandinavia. Some suggestions are made as to how these issues might be addressed.

The organizational and scientific developments needed to support effective materials process modelling are presented, based on wide-ranging discussions including several detailed case studies (in appendices). The Overview (section 1) deals with some of the generic issues in modelling while a more detailed discussion is provided in Models and Modelling (section 2). Section 3 summarizes with overall Conclusions and Recommendations.

Conclusions:

1. The current British approach to educating researchers and training process modellers leaves much to be desired; particularly with respect to the lack of interdisciplinary training and industrial perspective.
2. The perceived gulf between analytical "materials" modellers and numerical "industrial" modellers is unhelpful and unnecessary.
3. There are several ways of classifying modelling problems in terms of the physics and mathematics of the problem, the type of boundary conditions, and the type of data required.
4. It is important that this classification of materials process modelling is developed further, and used as a basis for training process modellers.
5. Computational power is not an issue except for a few, mature processes.
6. Much benefit could be derived from modern computer science and software engineering developments from the late 1980s which are not being used.
7. Modelling the physics of a process and building a useful process model are two different things. The latter requires major inputs from computer science and allied technologies.
8. Many long-term generic modelling and measurement problems require a new "aggregate physics" to be developed for turbulent or chaotic processes, such as dirty stick-slip friction, liquid/solid heat-transfer or grain-grain interactions.

This report is issued by the Department of Engineering, Cambridge University, as CUED/C-MATS/TR.206 and copies are available from The Librarian for a nominal charge.

This copy is version 2.0 dated 6 November 1993.

Converted to HTML 24 June 1997.
Converted from Microsoft HTML to standard HTML 30 April 2000.

How to read this report

It is intended that this report be read by managers responsible for modelling activities, policy formers advising government and academic funding agencies, industrial and academic modellers, experimentalists who measure data intended as input for models, materials scientists and manufacturing process engineers.

Our survey has gathered information from practising line managers, company development laboratories, software developers, materials data measurement laboratories, academic researchers, consultants and theoreticians. The subject matter is similarly wide, so any one reader is likely to find only some of the material familiar. We encourage readers to examine those areas which are new to them since many of the practical problems of materials process modelling are interdisciplinary to a high degree.

The report consists of a one-page synopsis, an overview, a discussion on models and modelling, and a conclusion (including recommendations for action). This is followed by a list of references and a number of case studies as appendices.

The synopsis, overview and conclusions largely stand on their own, but a full appreciation of the discussion section requires that the case studies in the appendices also be read. These case studies of various processing technologies are of varying length and depth of detail. No attempt is made at a uniform treatment except that each appendix ends with a list of lessons learned. These lessons use the same form of words as the Overview and Models and Modelling sections and are re-stated there in more general terms. All the technical points made in the main section of the report are backed up by lessons learned from at least one, and usually several, of the case studies or other processes discussed during interviews.

Acknowledgements

This review of modelling methods would never have been possible had not so many researchers and model developers given us so freely of their time and been so candid in their opinions. We are extremely grateful to all our colleagues and interviewees who are far too many to mention. We are particularly gratified that even in an area so close to industrial application as this is, our colleagues in industry have nevertheless felt able to be so supportive and informative. A partial list of our informants is given in Appendix F.

This work was supported by travel grant GR/H2412 from the Applications of Computers to Manufacturing Engineering (ACME) directorate of the British Science and Engineering Research Council (SERC).

[Cover illlustrations are not included with the HTML version of this report.]

Cover illustrations are all simulations and are, from top right, anti-clockwise, (1) a cross-section of a casting of a directionally-solidified turbine blade; (2) a longitudinal section of a directionally solidified block, both showing grain structure and courtesy of Michel Rappaz (EPFL, Lausanne); (3) a velocity plot of convection within a casting also showing shrinkage in the solid and gap formation modelled using an unstructured mesh, courtesy of Mark Cross (University of Greenwich, London) and (4) a thermal profile of a starting block and direct-chill continuous casting, courtesy of Ole Myhr (Hydro Aluminium, Sunndalsøra).

Philip Sargent^* and Hugh Shercliff

Cambridge University Engineering Dept.
Trumpington St., Cambridge, CB2 1PZ England
fax. +44 (1223) 332662
philip.sargents@bigfoot.com, hrs@eng.cam.ac.uk

Bob Wood

Manufacturing Engineering Dept.
Loughborough University of Technology, LE11 3TU, England
fax. +44 (1509) 267725

Contents

SECTION 1

Overview

1.1 Introduction

Manufacturing processes intrinsically involve the processing of materials. Understanding these processes at some specific level of detail is necessary if predictive models are to be used to optimize the process [Ashby, 1992]. However, while a substantial proportion of research funding is devoted to developing such models, too many projects are a black-hole of research effort which, while successful in their own narrow areas, never produce general, useful results. This study is based on the hypothesis that such non-generalizable projects should be, to a degree, identifiable in advance because of the nature of the materials information that they implicitly represent and manipulate (i.e. whether or not the data has characteristics typical of a wide range of materials and processes [Sargent, et al., 1993]).

A second hypothesis underlying this study is that inappropriate modelling techniques are too often used. Numerical methods are often used when really only qualitative results are required. Finite element modelling in particular is a notoriously all-absorbing activity. We have made case-studies which illustrate which types of process are most amenable to which types of modelling technique. There is clearly a need for a systematic means of classifying modelling problems in terms of the process physics and solution techniques. A start has been made at formulating such a system in section 2 of this report.

Conventional research projects in process modelling often intend to produce a general "methodology" for developing models of a particular type, but inevitably the development of a working, useful model for a particular process takes priority. The "methodology" is then abstracted from only a single example, and is asserted but not tested. We present a survey based on many different types of models.

The strength of this review is that it is focused directly on studying the current state of the development of useful models and is not sidetracked by attempting to produce or to update any specific model of a particular process or class of processes.

If enough resources are devoted to model sufficiently specific materials over a narrow range of properties, then predictive models can always be created. This is not the point. What we would like to be able to do is to know which types of process model are easily generalizable to a wide range of materials, and which models can be applied to a range of "similar" processes.

This section covers the problem under three headings: modelling, people and software.

1.2 Modelling: what is it for ?

This review covers only the modelling of materials processing from the manufacturing engineer’s perspective [Charles, 1992; Cross, et al., 1992; Dean, et al., 1990; Ion, et al., 1992; Jain, 1992; Lalli, 1992; Overfelt, 1992; Sellars, 1990; Spilling, 1992; Tjotta and Langsrud, 1992; Towers, 1989; Wills and McCartney, 1992], and not the modelling of fundamental physics aimed at predicting materials properties from an atomistic or other materials science perspective [Bhadeshia, 1992; Campbell, 1991; Choy, et al., 1990; Davis, et al., 1992; Funkenbusch, et al., 1986; Humphreys, 1992; Hunt, 1991; Mehta, 1992; Pettifor, 1992; Saunders, 1992; Shercliff and Ashby, 1991]. We are not surveying models as used as a tool in the course of scientific research. Thus all of the processing models we consider fall into one of the following problem classes as classified by a commercial need:

• modelling to develop a NEW or scaled-up process
• modelling to REDUCE ITERATION of trials (speed and cost)
• modelling to optimize an EXISTING process to reduce COST
• modelling to qualitatively understand an EXISTING process
• modelling to improve QUALITY by reducing product and process variability

Quite coarse models are appropriate for modelling entirely new processes: these can initially be simply based on equilibrium thermodynamics, supplemented later by gross kinetics. These simple models are also appropriate where an established process is to be scaled up. Simple models are needed for providing a first estimate of a trial (usually involving a new shape or geometry) which will then be tested empirically (e.g. new roll sections for shaped bar rolling, new multi-pass weld sequences for novel joint geometries, or moulds for new piston castings). If the shapes are simple then these models are simple, but they are highly constrained and usually model only some small part of the entire process. Such simple models can produce significant savings from just getting an improved first "guess". This does depend on the type of process, many are not as radically affected by variants for different geometries as are welds and castings, and so can be optimized generally at an early stage, e.g. some types of surface treatment. The models can then be "compiled down" into process diagrams such as forming limit diagrams [Chan, 1990] or laser surface treatment process plots [Ion, Shercliff and Ashby, 1992]. This requires that the process be understood, however, which will have required qualitative modelling.

Qualitatively understanding an existing process, as opposed to designing a new process or process variant, may require a more complex model if the issue that must be understood is the result of second order effects. Improving quality by reducing variability nearly always requires second order effects to be taken into consideration leading to a need for more complex models. This also requires a greater range and detail of experimental data to provide model parameters and validation tests.

1.2.1 The organization of modelling

This business issue is also important:

• Are we improving HIGH or MEDIUM-level technology with the modelling ?

The type of technology is important because "high" technology usually means that more resources are available to set up experiments to provide data for the models, that the materials are more expensive and as supplied have lower variability, and that more expensively trained and experienced modellers are available. Since greater benefits in terms of profits and reduced lead-times are likely, additional modelling costs can be justified. Deployment of the models either to production engineers or as part of packaged and simplified model-based control systems is not usually so much of a problem since both the process and the staff are already integrated with computer systems, money is potentially available and the procedures of software engineering [Brookes, 1982] and associated costs are more likely to be understood somewhere within the organization (though probably not in the processing division).

1.2.2 Types of prediction required

• Are we modelling to predict occurrence of major defects ?
• Are we modelling to predict microstructure and product performance?

Some types of defect, such as the size of dirt particles, occur at the tail-ends of probability distributions and these are not individually predictable. However, predicting the occurrence or non-occurrence of a systemic type of defect is possible if there is a "criterion", a continuous variable which predicts defects if it is well above some critical value. We can distinguish two types of systematic defect: those for which the modelled "field variables" (temperature, strain etc.) have an immediate, direct influence (they appear in the formula for the criterion) and those produced by the secondary effect of those fields on some internal microstructure (such as grain growth above a certain size). In reality there is a continuum between these extremes: simple criteria are easy to use but may not be accurate, complex criteria may be entire models in themselves and can require iterative coupling with the rest of the model.

Criteria do not have to have the same behavioural form as any "real" variable, they only have to be monotonic since the critical value for defect prediction can often be arbitrarily set or calibrated from experiment. Models for real variables which can (potentially) be measured, such as strain, crystal growth rate etc., have to evolve at least semi-quantitatively in accordance with measurement, a monotonic correlation is not good enough. (Processing "windows" are the result of several monotonic functions acting in opposite directions.)

Devising and calibrating such models is very difficult because microstructure tends to be affected by nearly every physical effect in the process. Thus, surprisingly, it is often much easier to predict the appearance of defects than to predict the numeric level of some microstructurally related feature such as grain size, or percentage recrystallization [Brown, et al., 1993]. Thus macroporosity in shaped castings is easily predicted for simple shapes by various "modulus" or "heat centre" ratios [Viswanathan, et al., 1992] and microporosity by the Niyama criterion [Overfelt, 1992]. An additional effect on model type is whether the model is to be used only in the process development phase or whether it will be used in conjunction with commercial production. If the latter, then the relationship between the model and control systems, which may be based on a reduced model, become important.

Thus the industrial context and purpose of the process modelling task already strongly constrains the appropriate types of modelling methods. The type of process of course determines the modelling methods within those constraints. Even taking these issues into account, there are at least 6 reasons why a good, new physically-based model of a process may not be implemented in an industrial tool:

1. It is so new that there has not yet been time to add it to commercial packaged software systems.
2. It models a highly specialized process for which the market is so small that there is no funding available to "productize" the model.
3. It makes unreasonable demands on either computational power (hardware) or software infrastructure with respect to the insight gained from its use: hence restricting it to niche use.
4. It makes unreasonable demands in terms of parameters and experimental data required as inputs.
5. It makes unreasonable demands on the expertise of the engineer who would be using the model in practice.
6. It produces results which are no better for a practising engineer than an existing (perhaps empirical) software package – despite being based on a sounder basis of real mechanisms.

However the major barriers to increasing use of models in industry are to do with how existing modelling technology is implemented, packaged and delivered.

1.3 Modelling: what is important ?

Given the constraints of the type of organization attempting the modelling and the types of results required, the major tasks are deciding what modelling methods are appropriate.

1.3.1 Analytic and mesh-structured models

One of the most important aspects is deciding when it is adequate to use analytic algebraic equations and when it is necessary to construct a discrete meshed model, e.g. for finite element or finite difference calculations. It is always necessary to do some non-meshed symbolic or analytical modelling in order to decide on appropriate boundary conditions for the meshed part of the problem. (It is the boundary conditions that "effectively model the physical problem and control the form of the final solution" [Edwards and Endean, 1990].)

There are only four reasons why a meshed method might be appropriate:

1. the modelled volume has some complex shape
2. the modelled volume contains numerous internal structures (e.g. grains)
3. the modelled volume contains discontinuous behaviour
4. the underlying process physics are very non-linear (typically material or boundary condition behaviour)

The first reason, complexity of shape, is very common in engineering application; the second is found more commonly in pure research; though it is used in some texture studies. Network models of grains, and cellular automata of grain-nuclei, are invariably found at the "pure" end of research [Humphreys, 1992; Rappaz and Gandin, 1993; Spittle and Brown, 1989a]. The third problem, discontinuities – such as a phase change in the material, can be handled easily if inelegantly with meshes using a volume-of-fluid (VOF) technique (e.g. [Wang, et al., 1993]).The fourth problem with boundary conditions often occurs in practice where, for example, the heat transfer coefficient is a non-linear function of temperature (often having a maximum at a particular temperature). This renders a closed-form analytical solution almost impossible – or requires many terms of a series expansion to be managed which approaches the complexity of a meshed formulation anyway – even for simple geometries.

Whether the shape is really complex enough to justify meshed models is a difficult decision because one does not usually know a priori whether some small geometrical feature is significant or not. In addition, the "shape" determining a materials processing operation is often not a physical shape at all, but the "shape" of a heat field. If a heat field can be treated as semi-infinite and highly symmetric (even if the physical material volume is not) then an analytic model can often be entirely adequate.

The "shape" can also depend on the material itself. When modelling heat affected zones from typical welding or laser surface treatments, the thermal conductivity of steel is sufficiently low that the heat source can be modelled as a line source and only a 2D section need be modelled as a semi-infinite plane, perpendicular to the plate and perpendicular to the direction in which the heat source travels [Ion, Shercliff and Ashby, 1992]. For aluminium alloys, for the same process, conduction is fast enough that metal ahead of the source "sees it coming" and also conduction through the thickness of the plate means that top and bottom surfaces have almost the same temperature. In this case a different 2D approximation to the thermal field is valid: infinite in all directions, but in the plane of the plate. The same distinction between aluminium and steel is found in continuous casting: heat flow is radial (outwards, from a long molten pipe) for steel, but longitudinal (along the strand) for aluminium.

Analytic models are always useful to distinguish between mechanisms that have to be modelled as a coupled set and mechanisms that can be modelled separately; thus heat diffusion problems often have characteristic time constants which are orders of magnitude smaller than solid state atomic diffusion problems (e.g. for carburizing or nitriding treatments for steels). Careful attention to length scales is useful in determining when a mechanism can be thought of as operating in a continuum, and thus being susceptible to analytic modelling, and when spatial distribution must be modelled explicitly.

Analytic models used to require closed-form algebraic descriptions so that the parameters of interest could be solved for directly; but that is no longer the case. Simple computer programs can track evolving solutions and iterate to find solutions to implicit formulations on very modest personal computers indeed [Ashby, 1992; Fleck, et al., 1992; Ion, Shercliff and Ashby, 1992; Shercliff and Ashby, 1991]. However, analytic models still require simple geometries and uniform boundary conditions even though they can cope with some complexity in the materials constitutive behaviour.

1.3.2 Boundary conditions and multi-physics models

Determining the boundary conditions of a novel processing modelling problem is a major part of the activity of process modelling [Edwards and Endean, 1990]. Boundaries are not just geometric shapes, they are also statements of symmetry, continuity and constancy of temperature, heat flux, strain, stress or humidity fields with respect to time or any other field value, e.g. heat transfer coefficients are often a function of temperature.

Analytic models require the modeller to make a very early commitment to specific boundary conditions because they control the entire form of the analytic description and do not appear in the model equations themselves. Meshed models, however, require explicit representation of boundary conditions separately from the model equations, and they are decoupled to some extent from the "continuum physics model" itself.

Thus meshed models can change boundary conditions at a much later stage than analytic models, and nearly all of the model can be re-used with different conditions. In practice this is vital because sensitivity analysis is required over different boundary conditions and not just over different parameter values; which for analytic models usually requires a complete new modelling effort for each trial.

Numerical meshed models are strongly affected by different boundary conditions but the effect takes place in the solving engine of the algorithms. This is also where different kinds of physics have their impact. Fundamentally, meshed models solve sets of partial differential equations (PDEs) and different types of physics: diffusion, flow, radiation etc., mean that different terms can be dropped from the PDEs. Thus, for example, the diffusion equation is a specialization of the lossy wave equation. Each type and sub-type of PDE formulation coupled with a particular type of boundary conditions and mesh-type can be "best" solved by a different algorithm; where "best" involves robustness, accuracy and speed.

Therefore intelligent use of numerical meshed models requires intelligent selection of numerical analysis techniques, which typically reduces to the selection of a particular software package (and solver algorithm), which is intellectually less difficult than constructing a solution from first principles. This has significant implications for education and training, and also for guiding the development of software appropriate for such people.

1.3.3 Modelling accuracy

Practical manufacturing problems often occur at the tail-ends of statistical distributions, at the 100s of ppm. level, from overlapping distinct processes (e.g. in solder reflow of surface-mount electronic devices on printed circuit boards where placement error and solder volume error combine). If there are 20 steps each with 96% success then overall productivity will only be 44% (0.96²⁰=0.44) but a model to study such a problem will have to be much better than 96% accurate. Being at 96% of the target property might not denote failure however, the target behaviour might be robust to the ±5% level. This kind of problem is especially significant in processes which are basically repetitive, such as lithography or soldering for electronics, multi-pass metal rolling or hammer forging.

Materials processors often assume that theirs is the only process relevant and forget packaging, painting, cleaning etc. These can initiate or aggravate materials mechanisms which, on their own, would cause no problems in the process. This can sometimes be used to great advantage, as is the case with a recently developed aluminium alloy for automotive body panels which is soft during pressing (reducing spring back and increasing formability) and which develops full strength only on passing through the paint curing oven.

1.3.4 Materials parameters

Any model of a materials processing operation requires numeric parameters describing the relevant properties of the material being modelled. Acquiring these data is almost always expensive and difficult [Charles, 1992; Jain, 1991] using the data appropriately is also not straightforward. Sensitivity analysis of any model, with respect to variation in the values of the materials parameters is always necessary because of the inevitable uncertainty in their values. In addition, because of the possible inaccuracy or slight inappropriateness of the model itself, the qualitative behaviour of the model with respect to parametric variation is often of much more use than the simple numeric "answer" produced by the model when "fed" the best estimates of the materials parameters.

There is only one exception to the requirement for sensitivity analysis and that is when separate models exist for both the upper and lower bounds of the materials behaviour [Cocks, 1993]. In this case, however, one must be careful to put the appropriate high or low estimates of the parameters to each bounding model. If there is only one parameter this is straightforward; but if there are many parameters, and the model is non-linear, it is often extremely difficult to predict whether increasing or decreasing the value of a particular parameter will have a positive or negative effect on the final modelled behaviour. In such cases, sensitivity analysis is again necessary.

In many cases, taking the complete set of upper and lower limiting estimates of a set of, say, 6 parameters, leads to such wide bounds on the modelled behaviour that the model appears to be of little use and there is a strong temptation to perform no sensitivity analysis at all but instead to merely use the "central" estimated parameter values. This is often due in reality to strong correlations between parameters, e.g. between the melting point and activation energy for diffusion, or between the Dorn "constant" and the power law index in the case of power-law creep. If these correlations are ignored when deciding on parameter variation for a sensitivity analysis, then the results will be unrealistically divergent.

Studying the sensitivity of a model to variations in one material parameter at a time is a common method for developing understanding and insight into a processing operation. However it must be appreciated that this is a very tentative form of understanding, and modelling in an industrial context which leads only to this type of insight is rarely cost effective [Herbertson and Austin, 1993]. The reason it is such a poor form of understanding is that even the qualitative effects of a single parameter are only valid for entirely linear, superimposable systems: an entire Taguchi-style matrix of sensitivity experiments is required for non-linear systems [Srinivasen and Chaudhary, 1990].

1.3.5 Generalizable models

It is the discrete differences in assumptions and types of approximations between different modelling problems that makes generalization difficult, not the differences in continuous parameter values. Thus the type of materials information used in a model only indirectly determines whether a model can be generalized. Thus a 2D approximate model of heat flow around a laser weld in a steel plate can be wrong for the same weld in an aluminium plate because the assumption about the relative speeds of thermal diffusion and weld velocity becomes invalid. It is this assumption which depends on the thermal conductivity values.

Although it appears that we can draw no completely general conclusions about the effect of material information type on the breadth of a model’s applicability, nevertheless certain correlations do occur fairly frequently. Thus surface properties: wear rates, friction coefficients or corrosion rates, are highly variable for otherwise similar materials under similar conditions. Such high variability implies that the model is quite likely to be pushed into a regime where the original assumptions become invalid. Thus in rolling sheet it is the thickness of an oxide scale which determines whether the heat transfer coefficient between roll and sheet even enters the model at all. For thick oxide scales the heat transfer coefficient is irrelevant but for thin scales it must be included in the model.

1.4 Software and programming

Models now are much more computationally intensive than was the case only a few years ago, so there is a real step-change in improving the representation of real processes in the models. (Being closer to reality, "multi-physics" modelling, needs more number crunching). Doing calculations on many different physical phenomena will also change the current balance between computation spent on remeshing and computation spent on calculation: it will become cost effective to put more effort into dynamically changing the mesh size (and/or polynomial order of approximation within elements, the "adaptive p-method" [Giles, 1993]) during the solution of the problem so as to minimize the number of physics calculations. (Remeshing multi-physics problems, where each type of physics has its own criterion for triggering remeshing, is currently a research issue [Altan and Miller, 1992].) This implies that the current situation where "solvers" and "meshers" are produced largely by different companies will have to change.

Most industrial modelling uses standard packages, many are customized by the suppliers to a specific need. Some projects require extra code writing in C (or Fortran), sometimes because the data handling or representation of constitutive laws (particularly history-dependent laws) in the standard package is inappropriate. In the future, the ability of code to be run on parallel processors will become critical. This will push programmers writing this extra code towards using languages such as High Performance Fortran (a subset of FortranD) which implicitly support scalable parallelization without making the programming very much more complicated [Blankenhorn, 1993]. A recent, critical facility is the ability to do real-time monitoring of a big simulation so that a modeller can abort it if something goes wrong. Once tuned, it is run overnight [Szekely, 1988].

Comprehensive finite element packages are perceived as being clumsy to use and there is a high initiation cost in training someone to use one at all. Specialist packages tuned to a particular process or solver are easy to get started (easy to get the input data in), but may well need modification to solve the correct problem. They usually just use one specialist type of solver, e.g. a radiative heat transfer solver, or a transmission-line matrix viscosity solver. A large company’s modelling team today will typically use many of these, rather than try to tie all the solvers into one comprehensive environment comprising pre- and post-processors and data visualization tools - though that is the direction towards which large companies are attempting to move. The fragmentation of specialized skills caused by using a plethora of solvers does not encourage wider dissemination of basic modelling skills.

Integration and data management is a very real current problem and is intimately connected with the difficulty of ensuring either that the solver appropriate for a modelling problem is available for a company’s standard environment, or that data passed to several distinct packages is kept consistent.

Clearly the software situation is unsatisfactory. There are more important ramifications however: the need to become a specialist in one particular type of package or class of algorithms means that there are many fewer people with expertise in any particular range of areas. This means that collaboration and teamwork – which is always necessary in a modelling project – is done badly for two reasons: (a) because it is less likely that the appropriate expertise is in the team, and (b) the team members individually are less used to working outside their specialization. Thus attention to producing better, more modular, modelling software development environments [Wood, 1992] (preferably with some automation of the implementation of the specified mathematics [Kant, 1993]) will have benefits in both manpower availability and interdisciplinary abilities.

1.4.1 Infrastructure entry costs

Although packaged software costs alone are much greater than hardware costs, and costs of software written in-house are greater still, the transition from personal computer (PC) hardware to workstation hardware is still perceived as a significant entry cost. This is despite the fact that a fully-configured PC with adequate disk-space, memory, screen and processing power is almost invariably more expensive than a workstation which is actually more powerful.

There are two reasons for this entry barrier: first, PC hardware can be bought incrementally over a period of months as required and can be justified on an item by item basis by modellers already producing useful results. Second, workstations use the Unix™ operation system which currently has very significant training costs for the system manager. The difference between mass-market and niche-market hardware is likely to persist but will be far less significant as prices will be lower and standardization better.

The system management costs of UNIX workstations are now much larger than the hardware costs, and the more capable numerical models are currently only available for workstations. However this will be a very temporary difficulty: easy-management Unix (such as NeXt, Sun’s Solaris™ and Apple’s A/UX 3.0) and Posix-complient OS/2 and Windows™ NT are either already available or will be within months. The personal computer and workstation markets, both hardware and software, can confidently be expected to merge during the mid 1990s as projects come to fruition to make Windows™ and Macintosh software run on the full range of workstations and personal computers.

Packages tend to be licensed per "seat" or by processor. Systems to manage multiple use over a network are rudimentary but improving rapidly. However, multiple versions for different computers often require different releases of system software; which means that a modelling team’s systems are permanently out of step in any "open system" comprising computers (and system software especially) from different manufacturers. This situation will improve as standardized operating system support, especially graphics and windowing support, stabilizes, but the inter-package dependencies will remain.

1.5 People: education and training

In Britain, and also apparently in most of Western Europe and America, there is a shortage of modellers trained in both a breadth of computer science, commercial sense and materials science and engineering, and a depth of analytic and numerical analysis. There is apparently no shortage of expertise in narrow numerical and mathematical skills appropriate to devising new or improved formulations of the finite element method; but these are rarely coupled with adequate background in understanding the physical processes which actually cause the behaviour being modelled.

These problems are exacerbated by the strong tendency for undergraduate students on general engineering degree courses to choose materials engineering options if they are weak in numerical analytic ability or do not like continuum mechanics. In addition, the numbers of bright school students choosing to study materials science/engineering at UK universities has been dropping over much of the past decade and shows no signs of recovering, though this is not the case in the USA. Students who choose these courses are also traditionally weak in mathematical ability although some materials departments are now producing more mathematically challenging courses. Even where numbers remain adequate this is sometimes at a cost of reduced entry standards. Government policies requiring universities to increase cost effectiveness can have unintended side-effects, especially on interdisciplinary subjects where a smooth combination of undergraduate teaching and postgraduate research is required. Lower, or non-traditional, qualifications on entry, however, do not necessarily imply less able or less appropriately trained graduates leaving the universities.

In America, the lower level of education (compared with Europe or Japan) in both basic science and mathematics on entry to undergraduate courses means that nearly all the training relevant for materials process modelling must take place at the post-graduate level. This is a strength because such courses can be more highly focused, but also a weakness because the breadth of background is lost. In the UK, post graduate education in this area would take the form of distinct one-year MSc. courses, or two-week modular training courses; but there are very few specializing in materials process modelling and industry is not sufficiently well organized in modelling yet to provide a "customer pull" to the graduates of such courses. It is not even clear whether such courses should be based in materials engineering departments or, if they are, whether an intake of graduates from physics or chemical engineering would be more appropriate for industries’ needs than materials scientists.

The most serious manpower shortage is for people able to evaluate critically a range of possible modelling options and pick the most appropriate for the problem in hand. This requires, as described above, the ability to perform the analytic modelling intelligently and to set up boundary conditions and materials models properly. Awareness of the industrial and business problem context is almost completely absent from process modelling education (where it exists) which concentrates on numerical methods.

The non-numeric education of modellers coming from a materials science background should in future include at least a qualitative appreciation of "boundary conditions": the formal statement of specific types of assumptions that any model, even a purely qualitative model, must make. These boundary conditions, and formal statements of the degree of coupling between fields and physical mechanisms, form the bridge between what have historically been two distinct modelling communities, a situation which cannot be allowed to continue. The current problem is that education systems are not producing either materials engineers or numerical analysis graduates capable of working effectively in multi-disciplinary modelling teams. There is an irreducible minimum of shared concepts which are necessary before such a team can begin to work. These concepts are fundamental and have to be introduced early, not late, in bachelors degree courses.

The finite element approach and other meshing methods are usually the default techniques not because they are the best, but because the modeller is not capable of posing or solving the appropriate analytic model based on the materials science and physics of the problem. The software system leads the modeller through the modelling task by imposing a procedure (which may be inappropriate to the problem) which untrained modellers do not otherwise know how to begin. The essence of modelling skill is knowing how to manipulate assumptions about constitutive laws and boundary conditions appropriate to the physics of the situation.

Finite element models also have the great advantage of producing some kind of visual "result" within a defined period of time, whereas if a modeller attempts to derive an analytic model and cannot find a reasonable description, then there will be no result at all. The ease of producing complex coloured graphical results from a numeric simulation, compared with a handful of numbers from analytic equations, should not be underestimated as a determining factor. Modellers work in organizations, and reward for a job performed well is dependent on impression and communication as well as accuracy and validity [Handy, 1993].

1.5.1 Manpower costs

University research teams are not always appropriate as the primary point of contact for a commercial company wishing to model some practical processing problem. University researchers are in a situation which rewards specialization on a particular technique or class of techniques, which may be only partially appropriate for the company’s process. If the company has little modelling expertise (which is the case for the vast majority of companies), then this fact is not appreciated by any of the parties concerned. This is a direct result of the lack of broadly trained materials process modelling engineers and the small size of most academic research teams in this type of work. Larger departments expose academics to a wider range of modelling techniques, especially if they pool their programming teams.

Some research academics have the resources and ability to develop specialized techniques in the context of a broader appreciation of what is feasible with unrelated methods. However, many academics are not in a position to be able to take broader issues into account and so develop modelling techniques which may be inappropriate in an industrial context. The practical application of modelling technology would appear to be best done through shared industry research and development laboratories, trade association laboratories, large consultancy companies or an equivalent of the 60 or so German Fraunhofer Institutes or the 170 Japanese Kohsetsushi (public testing laboratories).

As far as the cost of modelling a processing operation is concerned, there seems wide agreement that it is dominated by manpower of analysis and programming (80%), excluding the cost of acquiring materials parameter values (which can be very high if measured from a fully functioning plant which would otherwise be producing product). Szekely estimated a total cost of $120 000 for a reasonably complex model taking one man-year to develop [Szekely, 1988]. The range was from $20-30 000 for "a simple exercise" to many millions for a "really complex model" (there are complementary costs in physical modelling, pilot plant work and plant-scale experimentation). Szekely’s estimate was made in 1987 but is reckoned to be much the same now.

Staff in the UK are paid less but computers are relatively more expensive compared with the USA. So an annual per-person cost in the UK of £25-35,000 (for a member of a company’s internal process modelling team) would be approximately equally split between salary, computer systems and general local support but not including any company-wide overheads. Costs would be approximately doubled for contract or consultant modellers (more than £350 per day). Hardware for process modelling is apparently not an issue. Costs are dominated by salaries, software and support.

SECTION 2

Models and Modelling

2.1 The process of modelling

2.1.1 The modelling cycle

A spectrum of skills is required to complete any manufacturing process modelling task. The outer loop of Figure 2.1 is managed by someone close to the process who understands the business context of the problem and can concentrate on specifying the objectives and providing raw data. The innermost loop requires mostly computational skills and the intermediate loop consists of activities mediating between these disparate worldviews.

Figure 2.1: The modelling cycle [Bathe, 1986; Wood, 1993b]

These loops are present in any modelling exercise. The communication problems and lack of common skills between the inner and outer loop personnel are acute for materials processing because the physics, and the mathematical and numerical techniques are all more complex than is the case in, say, stress analysis in design. (Some metallurgists may know some software engineering, but they are rarely the same metallurgists who are competent in continuum mechanics.) The extra complexity requires correspondingly greater specialization of skills. Historical factors mean that, in Britain at least, there is also a significant cultural gap between modellers and machinists or foundrymen.

These characteristics of industrial process modelling highlight the needs for initiatives in educational programmes, training and appropriate software tool development.

Modelling teams

Within a commercial organization, a modelling "team" manages the interpretation of modelling needs and creates a model. It may run the model itself and also interpret the results, or the customized model may be handed out to other engineers in the company for them to use themselves. A modelling team manages the disparity in worldviews between the innermost and outermost loops of Figure 2.1 and its place in managing models, engineers and software is shown in Figure 2.2.

Our program of industrial visits has shown that modelling "teams" which fit the roles described here, invariably exist in practice (though many consist only of one person). This is unsurprising since the different viewpoints have to be managed somewhere. One refinement in certain industries is the "contract team" where the modelling expertise is contracted out to a specialized company. Another is the modelling software supplier where a highly-specialized software package is produced that models only one particular process (e.g. casting automotive wheels in a specific range of aluminium alloys), which is then sold back to be used by engineers in the larger client company.

Figure 2.2: A partial illustration of how modelling teams work

A useful objective, referred to in the Overview section, is to consider what educational and software system changes would be required to increase the number of people using process modelling by a factor of 10. Clearly this can not be achieved by increasing the size of modelling teams by this amount: it can only come by making such teams more effective so that, say, 5 times as many engineers (clients of the teams) can use the models and the teams become twice as efficient. The former certainly requires an engineer better educated in modelling principles and the latter implies better software tools for constructing robust, safe, easy-to-use models more quickly.

One last issue concerns the management of modelling teams. Software development is always an important activity. This is frequently the configuration and customization of general purpose codes, with original coding much less common. However, in either case the particular difficulties of managing software construction are present with the added complication that the teams are often part of a conventional engineering organization, which is usually unaware that the systematic methods of controlling software development and techniques for client management can be applied to considerable benefit in the development of process models [Brookes, 1982; DeMarco, 1982].

Figure 2.3: Skills in joint academic/industrial modelling teams

Sometimes in an industrial group the right mix of skills does exist, or almost exists. However policy set down by managers above the modelling team manager may not allow these skills to be exercised effectively, or at all. This raises the need for education that is somewhat different from that needed by future modellers. One example of poor use of personnel is in giving the creators of innovative modelling methods the task of running specific modelling problems using pre-existing methods. Education is required for senior management to give them an overview of the important themes in process modelling: starting with business benefits, then introducing the spectrum of techniques and technology, then identifying important skills, and finally describing where those skills should be deployed for best returns.

Where methods come from

Modellers and modelling teams do not come from nowhere. A long-term industrial commitment to supporting modelling means that a few companies, and a few countries, have institutionalized education and integration methods to form teams. The "raw material" from which teams are formed is described in Figure 2.3.

Thus many of the current arrangements for funding research world-wide can be deeply flawed because they do not ensure that projects include all three essential components of a process modelling project: mathematical methods, materials science and computer/software science (see especially the recommendations by Goh and Sinclair for managing SERC grants which involve software production [Goh and Sinclair, 1991]). In the UK, the ACME division of SERC does better than most by monitoring projects and requiring researchers to consult materials and/or mathematical specialists if the project appears weak in either area. Software production and computer science skills have been less well managed in the past.

2.1.2. Models are better than what ?

The usefulness of a model can only been seen in comparison with other ways of estimating the same information. Human expertise in predicting, say, defect severity and location is not based on a physically accurate model inside an engineer’s head: it is based on experience and pattern-matching the results of an immense range of component shapes and process variables. Human experts cannot reliably predict the behaviour of complex shapes they have never seen before, and their expertise can only be used to find workable methods, not optimized methods.

Thus modelling techniques which deal with shape, such as finite element and finite difference techniques (FE/FD), can be expected to lead to significant competitive advantage if shape variants are an important part of the commercial use of a materials processing technology. Unlike analytical methods, FE/FD methods make it easy to change boundary conditions without otherwise affecting model construction or interpretation, so if physical results are strongly sensitive to boundary conditions then FE/FD has a strong advantage.

Models are not necessarily cheaper or faster than performing a real experiment. The decision of where to make the trade-off depends on the generality of the experimental results and relative costs of setting up a model. For "die and mould" processes the machining cost of dies is high so there is a strong incentive to reduce the number of trials of new dies required to achieve a particular final component shape.

Even where the process involves no dies (e.g. most laser processes), sensitivity to material variation can often mean that trials have to be performed on nearly finished components. For example, tests can be extremely expensive in the case of laser-drilled holes in nickel alloy aeroengine combustion chambers and turbine blades, so there is a real role for modelling here.

The asymmetry in information between simulation and experiment is also very significant. For example, a shaped casting thoroughly instrumented with thermocouples is more expensive than a simple trial, but a model can display results with "virtual" thermocouples everywhere. A trial casting will show the location of porosity but its microstructure gives insufficient other information about the process conditions which led to the porosity (unlike a trial weld in a steel plate where the process can often be tracked throughout the part from post facto metallographic examination).

Therefore, despite the fact that FE meshing complex shapes is very expensive in man-months, despite the fact that CV-FD calculations (Control-Volume Finite-Difference) can take 24 hours per trial, and despite the fact that approximate analytical predictors may exist, meshed models of novel complex shapes are commercially appropriate if the company infrastructure can support a modelling team.

2.2 Classes of modelling problem

The knowledge gained from this study, apart from clarifying needs in training and collaboration, is significant in producing the basis for a scheme for classifying materials processing in relation to modelling techniques. The goal is to show where each technique is applicable in terms of the physics and mathematics of the process being modelled, and how the data and parameters required are very strongly conditioned by the boundary conditions and materials behaviour whose form is also determined by the physics.

A common set of principles for classifying modelling problems in materials processing serves a number of purposes: (a) as a guide to identifying the appropriate modelling methods for the problem in hand; (b) as a means for improving the rational allocation of resources between and within modelling projects, both in academia and in industry; (c) as a mechanism for integrating the different approaches to modelling. Once such a common framework is established, different approaches can be compared rationally and precisely. This section proposes tentative classifications based on: (i) the types of physical process involved, (ii) the types of coupling between these processes, (iii) the types of differential equation which describe each process, (iv) the types of boundary conditions of the system, (v) the types of materials data required as input for the model’s constitutive laws and boundary conditions.

Figure 2.4: Summary map of modelling problems and techniques

This generic view (originally expressed by Wood [1992]) places all the problems of computational power, parameter measurement, software modularity technology, model useability etc. in a single context.

In attempting to classify process modelling problems, it is useful to consider the following (in sequence):

• Identifying the classes of physical phenomena that influence operation of the target process (the process to be modelled).
• Identifying the mathematical equations which describe the relevant physics.
• Using the parameters within the descriptive equations as a guide, identifying data gathering needs: experiments for (a) material properties and constitutive laws, (b) boundary conditions, and (c) eventual validation of the model.
• Identifying the best numerical analysis algorithms to solve the descriptive equations given the processes’ boundary conditions and constitutive behaviour.
• Finding where the numerical algorithms can be obtained from: (a) commercial packages such as ABAQUS, MARC etc., (b) academic research groups, (c) published literature, leading to the development of new software, (d) publicly down-loadable from Internet.

With respect to the last point, one example of complex process modelling software freely available from Internet is Brakke’s "Surface Evolver" which models the dynamic behaviour of droplets wetting planar surfaces [Brakke, 1991]. It has been used extensively in modelling solder joint formation. The first 4 steps of the above list are discussed in more detail below.

2.2.1 Classification by type of physics or process

If we take it as understood that manufacturing engineers cannot and should not be expected to be experts in metallurgy, numerical analysis and continuum mechanics, then classification of problems by mathematical type is not a reasonable starting point for classifying processes. (Neither will it help progress towards the eventual aim of hiding these types of expertise within routinely usable software.) This leaves classification by type of underlying physics (e.g. plastic yield, convection etc.) or type of process (e.g. forging, welding etc.) as possible methods. Conventional classifications by type of process [Edwards and Endean, 1990; Lenau and Alting, 1988]) are limited in the guidance they give to selecting modelling method. For example, from the modelling point of view, friction-welding has as much to do with forging as with gas-metal-arc welding.

Modelling engineers could reasonably be expected to identify whether any of the following occur in the process to be modelled: convection, radiation, conduction, elastic deformation, small plastic deformations, large plastic deformations, water diffusion, chemical reactions, magnetic fields, acoustic energy transmission etc. Future software systems could conceivably exist where such a classification would be adequate to construct an appropriate model of the whole process from software components determined by each type of physics (this has already been achieved for some types of multiple-fluid diffusion problems [Kant, 1993]).

Classification by coupling

Identifying the physical processes, heat conduction and plastic deformation, for example, is one type of classification. Identifying the direction and strength of coupling between these processes is another classification which it is reasonable to expect a manufacturing engineer to estimate. Such estimates are useful guides for discovering the important boundary conditions and for constructing algorithms [Wood and Sargent, 1993].

Figure 2.5 shows a deformation process where stress/strain and temperature are strongly coupled and where the microstructure is determined very largely by the temperature alone. There is a second order influence of the microstructure on the stress (the strength of the material). There is almost no effect of the stress on the microstructure or the microstructure on the temperature. These very weak dependencies can become observable for some materials, for example the stress-triggered martensitic transformation of partially-stabilized zirconia, or the significant evolution of latent heat in the solid-state transformations of rotator crystals such as succinonitrile.

Figure 2.5: Several types of coupling for three physical processes

The significance of this line of thought is the potential for making large savings in computer effort (making software more widely affordable) by only making couplings which are strictly necessary. For example, in hot rolling it is often possible to calculate the temperature field (by FD) using an average deformation field estimated from the process geometry. This field may be computed separately and then superimposed on a subsequent FE computation of the detailed strain field ([Beynon and Sellars, 1992]). This is also a good (and rare) example of applying sensitivity analysis to a complex numerical problem to correctly identify the first-order effects from the rest. Note that the description of a parameter as "first-order" is to some extent a description of the level of coupling between that parameter and others in the problem.

Physical understanding and supercomputers

Although it is obvious to those developing models for research purposes, it is worth reiterating to those planning to use models in industrial applications that no amount of computing power compensates for poorly understood physics or metallurgy: "the use of sophisticated software or supercomputers is unlikely to help" [Szekely, 1990a]. Thus initial problem classification is vital and unavoidable.

The most obvious way of making modelling expertise available to engineers is for them to use highly specific models, e.g. Magma’s cast-iron casting modelling system, or specialized codes for super-plastic forming of a particular titanium alloy or for deep-drawing anisotropic, rolled aluminium alloy sheet. Such specialization always makes the model much easier to use since irrelevant data is not requested and unnecessary configuration controls are not present. Unfortunately developing specialized programs has historically been infeasibly expensive.

Commercial developments over the past decade or so have concentrated on the production of general purpose solvers of various kinds (finite element, computational fluid dynamics etc.) and significant progress was then made in the late 1980s in making these formidably complex packages easier for casual modellers to use [Szekely, 1990a]. The development of process models based on these packages implies a specialized way of using the software. This requires a complex mix of skills in order to decide which facilities (in the general purpose code) should not be used.

However in order to get an order of magnitude increase in the number of process modellers (as has been very strongly suggested as an important industrial need by our correspondents in the course of this study), more specific packages will be required.

2.2.2 Differential equation classification

The development of materials processing models almost invariably involves describing the underlying process physics quantitatively. Materials processing models almost invariably involve description in terms of sets of partial differential equations (PDEs). Different types of physical problem are representable by elliptic (e.g. Laplace’s equation), hyperbolic (e.g. the wave equation) or parabolic (e.g. heat conduction) equations; dependent on time, spatial dimensions, field variables and internal states. (There are also many more specific and general classifications: thus both Laplace’s equation and the wave equation are specializations of Euler’s equation [Stephenson, 1970])

If the modelling problem can be simplified so that shape is not relevant, then the problem can reduce to being of the lumped-parameter type, representable by a set of ordinary differential equations (ODE). For lumped parameter models, close attention is paid to constitutive laws and physical insight guided by dimensional analysis (as used successfully by Ashby and others for modelling materials processes [Ashby, 1992; Shercliff and Ashby, 1991]) If PDEs are necessary, then the complex apparatus of numerical analysis is required for their solution (e.g. [Davis, et al., 1992; Hartley, et al., 1992]) whereas lumped-parameter models can be solved by simpler means. Solution techniques are discussed below, after the classification of problems in terms of their data requirements.

All differential equations contain parameters whose values must be determined prior to solving the equations. In process modelling, these parameters divide into those relating to boundary conditions and those relating to material behaviour. Further, these parameters are the source of any non-linearity in the model. So, what we choose to do about these parameters has a profound influence on subsequent activities in model development and use.

Boundary conditions

The boundary conditions (BCs) form another classification system (e.g. Dirichlet, Cauchy, or Neumann type [Stephenson, 1970]) and the most appropriate solution techniques depend rather critically on the exact types of PDEs, BCs and, for multiple physics models, the types of coupling between the different PDEs (e.g. the mutually dependent stress/strain and temperature fields in deformation processes such as forging). For many materials processes the boundary conditions and the constitutive law of the material so constrain the situation that the type of PDE is almost irrelevant. An extreme case is the strain field in finishing sheet rolling which is fixed by kinematics (the BCs determined by the shape of the rolls) and depends only very weakly on the metal being rolled (though the temperature and stress fields, and thus the microstructure, are strongly dependent on the material).

In practice, the need to specify quantitative values for the boundary conditions and the material parameters within the equations creates a need to acquire physical data concerning the manufacturing process and the processed material. This is discussed in the next section.

2.2.3 Classification by input data

Processes can be further classified by their needs for different types of materials property data and for the measurement difficulties of some types of boundary conditions.

Gathering of physical data

This activity has 3 purposes:

• it further specializes the mathematical classification,
• it identifies the need for experiments to validate the model, and
• it provides input to the selection of algorithms.

The relatively general classification based on the type of mathematical equations and boundary conditions is now specialized by obtaining numerical values for material property and boundary condition parameters. Where such data does not exist or is of suspect quality, the need for experiments is indicated. The actual decision of whether to institute an experimental programme depends on its costs with respect to the value of parts produced by the process and also costs in terms of model development timescale and accuracy.

Experiments needed to validate a model should be planned before the model is constructed. Typically they are a mixture of full-scale instrumented tests on production equipment and laboratory trials using analogue materials: isostatic pressing using plasticene spheres, for example.

Selecting algorithms usually requires simplifying assumptions to be made, such as whether certain material properties are constant. A varying material property is a common source of non-linearity which then requires an iterative and computationally expensive algorithm. Whether such an approximation is justified is a business decision since it depends on type, accuracy and value of the information required from the model by the end-user. However, not all modellers test the influence of making the assumption of constant properties. A common situation is to use, for example, temperature-dependent properties because the software can handle it, rather than because the greater accuracy is meaningful to the end-user.

Consideration of data needs and how it is to be gathered is an important step in the construction of a model. Its perceived importance depends on the background of the modellers. Model developers with a strong bias towards numerical analysis techniques typically play down this topic, classing it as an "industrial implementation detail". In contrast, industrial practitioners place a greater emphasis on data gathering. This probably stems from: their appreciation of the cost and timescales associated with experiments on production-scale equipment and their stronger reliance on measured cause and effect. Regardless of subjective views, the reality is that this step is often very costly, time consuming and increasingly sophisticated.

More accurate models require reassessment of the assumptions made about what were previously second-order effects: for example, a detailed understanding of gross microstructure formation in casting requires a better model of the generation of latent heat. This in turn requires an improved quantitative understanding of atomic-scale solidification physics and more accurate values of thermal parameters for solids and liquids. The experiments needed to obtain such data are not easy to design, so if an accurate model is known to be required, such experiments should be planned early in the model development project.

Another example might be strain-path sensitivity measurements in deformation modelling: whether an alloy’s strength depends only on the total equivalent strain, or also depends on the detail of the strain-path, needs to be known before a mathematical model can be constructed. Such assumptions require numerous and difficult experiments.

The above discussion has concentrated on material properties, but boundary conditions are important in the same way. Two common examples are (a) measuring values of effective heat transfer coefficient (e.g. between cast metal and mould, or between dies or rolls in metal deformation processes), and (b) measuring friction behaviour in forging, rolling, extrusion and machining. Two general methods are now being applied to these types of contact problem:

1. treating them as aggregate phenomena to be modelled in terms of more localized processes on a finer scale (with concomitant extra experimental programmes), and
2. attempting to measure them directly from full-scale experiments using inverse modelling techniques.

Aggregate physics and inverse modelling are both treated in more detail below.

Models limited by data availability

There are two major complicating conditions which mean that a valid research model of, say, a casting process may exist but might be useless for industrial transfer because of a lack of data for any particular alloy. These conditions are that

1. there is no industrially-sensible method for obtaining the data (e.g. parameters that have to be measured from extensive transmission electron microscopy) and that
2. any parameters so measured imply nothing about the values for even very similar alloys: i.e. they are "capricious".

If the parameters were not capricious then eventually the expensive measurements would be made and useful data accumulated because it would be profitable for a large enough group of companies to set up a joint measurement project. If they are highly variable, then because of the continuous changes to alloy compositions, no useful data will ever accrue.

There are two reasons why properties may be highly variable between similar materials [Sargent, 1991]

1. Several mechanisms involving state variables competing in the same small volume, yielding capricious behaviour from small parameter changes.
2. The "aggregate" properties, where the overall behaviour of some volume is variable because different things are going on in different sub-volumes and the aggregate physics is not known (see section on constitutive laws below).

Many surface properties are of this "aggregate" nature so boundary condition determination can also be capricious (see section below on materials phenomenology). It is the cost of measuring these difficult parameters that forces the current commercially based classification of process modelling, e.g. as sand-casting, pressure die-casting, lost-foam casting, investment casting etc., where the particular alloy class must also be specified for the modeller to have an idea of where to begin. Many constitutive laws for aggregates are not only highly non-linear, but also history dependent. This is especially true of slurries and mushes [Sekhar and Dantzig, 1992]. The lack of data and constitutive relationships is a serious practical problem [Jain, 1991; Szekely, 1990a].

2.2.4 Solving the equations

Techniques for solving equations, both analytical and numerical, introduce a great many "modelling artefacts": concepts not part of the original problem statement but which are required in order to set up the problem to be solved. One of the clearest examples of an artefact is a finite element mesh. It is only used in the solving process in the innermost loop of the modelling cycle (Figure 2.1) and should not be seen in either the problem input or results output (the outermost and middle loops of the modelling cycle). Another artefact is the type of solving formulation: explicit and implicit methods can easily produce different results because they make different assumptions about the system behaviour [Brown, M.R. and Spittle, 1993]. These artefacts should be invisible to the practising manufacturing engineer using a completely-packaged modelling system; but rarely are.

If there was some unambiguous method for classifying modelling problems into specific mathematical classes based on the physics involved then it would be feasible to ask the modelling engineer to do only the classification and then software systems could handle everything else automatically. This is contentious for two reasons:

1. some types of physics can be mathematically modelled in several ways,
2. numerical analysts are typically unfamiliar with transformational functional programming.

These will now be discussed in a little more detail. First, some types of physics can be described in several ways, e.g. diffusion problems can be reformulated in terms of the lossy-wave equation and solved using transmission-line matrix methods, in addition to the usual finite difference methods [Pulko, et al., 1992] (finite element and boundary element methods are also used in practice).

Second, the concept of building software that generates a numerical analysis solver from a structured description of the problem is, perhaps, a little hard to swallow, especially by those long-involved with the nitty-gritty of applying a particular analysis technique to specific classes of problem. However, the credibility of this type of approach has a long history in computer science (general problems are often actually easier to solve than specific problems [Bentley, 1989]) and has already been demonstrated for the automatic generation of FD solvers [Kant, 1993] (this is discussed in the section on "Software factories" below). Similar achievements in the FE area are just as practical, but they simply have not yet been done. The reasons are simple: the mix of skills required is unusual and only exceptional individuals have specifically set out to acquire them, and funding agencies often do not have the appropriate committee structure to evaluate such proposals adequately. It is noteworthy that Kant’s work was performed at an industrial laboratory and not a university.

Meshed solution methods

There are a variety of numerical techniques for solving sets of equations over a shape or volume. The major types are finite differences (FD), finite elements (FE) and boundary methods (BM) and each is most appropriate for specific types of equation structure and boundary conditions (i.e. different types of physics and/or analysis objectives [De Biase, et al., 1993]). Generally an FE method is faster for complex geometry but can be inelegant for multi-physics problems, whereas it can be easier to represent a simple view of multiple mechanisms in an FD method. A given materials process may require all types to solve different aspects of a single problem.

It is sometimes said that there is no role for boundary element methods in modelling materials processing since these do not find the values of field variables over the whole of the material volume but only at surfaces, so they are inappropriate for modelling variables to be used in estimating microstructural evolution (for example [Meltsner, 1991]). However there are many processes for which some variables are only required at the surfaces, especially where it is the overall flow, or pressure drop, or drag that is required [Phan-Thien and Tanner, 1992]. Boundary elements can efficiently solve for such variables leaving the more computationally intensive FE and FD methods for variables that do have to be solved throughout the volume.

Structured meshes

1. Structured meshes are created using rectilinear, brick-like elements where the boundaries between them run completely across the part. A very large number of elements is required (over a million) to represent any usual part, but despite this, the faithfulness to the original shape is not good because many uninteresting, uniform volumes are forced to be covered in fine detail.
2. They can be generated entirely automatically in a few minutes from a 3D solid model or well-formed shell model.
3. Finite difference (FD) models used to require this type of mesh and current commercial control-volume CFD packages still do. FE models never use these types of mesh because the computation would take forever.

Unstructured meshes

1. Unstructured meshes can be any shape: tetrahedra, bricks, hexahedral prisms etc. and good faithfulness to original shape can be achieved with a relatively small number of elements.
2. Automated mesh generators are currently poor. Working from an existing 3D CAD model, generating a usable mesh takes typically 6 man-months for a truck differential casing.
3. Finite element (FE) models have always used these types of meshes but until recently FD models could not [Cross, et al., 1992].

Because their mesh generation is so much quicker, FD models would appear to have a very significant time advantage in producing results over FE models, but this is not always the case in practice. Many companies will already have produced an FE model (based on an unstructured mesh) in the course of the design in order to model strength, stiffness, deflection in service etc.

The chief barrier to more widespread use of all commercial meshed models in the future remains the difficulty of automatically generating good initial unstructured meshes. This is an entirely geometrical and mathematical problem.

This barrier can, however, be circumvented if automatic and robust remeshing and mesh refinement algorithms can be developed which can be accessed from the solving programs during simulation. This is a challenging research area since many different types of remeshing criterion are appropriate depending on the type of physics being represented by the model at that location at that time. It requires research collaboration of a particularly close nature simultaneously in numerical methods, geometry and physical metallurgy. Very little current remeshing research is apparently performed in such interdisciplinary teams.

Primary and secondary manufacturing

Complex shapes are nearly always found in secondary processing whereas primary processing nearly always produces simple shapes. Thus meshed methods are expected to dominate in the modelling of secondary processing and non-meshed, lumped-parameter models in the primary processes.

There are many exceptions however. The start-blocks for direct-chill continuous casting have complex shapes and the simple shape of primary extruded billet is misleading because the thermal and strain fields of that process have very complex shapes. Many secondary processes, such as welding and laser surface treatment, can have very simple thermal fields in some materials. The thermal diffusivity strongly affects the shape of the thermal field and so different metals can require quite different modelling techniques for the same manufacturing process.

Lumped parameter models

The situation of engineering modelling PDEs is immature compared with that of ODE lumped parameter models. Chemical engineers now routinely use numerical models to the practical exclusion of analytic models. The clarity of insight afforded by analytic models is more than compensated for by the fact that slight changes in problem formulation solved by a numerical system do not require entirely different actions by the user (the software systems take care of selecting the appropriate methods). A single procedure (as seen by the user) always works and always gives an understandable answer. The fact that an analytical approach may give a better answer in some fraction of cases is insufficient incentive to make attempting it worth the trouble.

An example of a "slight change" in problem formulation might be a change from modelling the finishing stage of a rolling mill to modelling the first pass. In the finishing stage the problem is effectively 2D plane-strain, since the width of the strip is very large compared to its thickness and the roll contact length. For the first pass the width and thickness would be comparable, and very significant sideways spread occurs during rolling. Models which are accurate for finishing stages can be completely indeterminate for early stages.

A lumped-parameter formulation does not model spatial variation directly and the parameters may or may not be physically meaningful in themselves. If the parameters do describe some identifiable physical reality, such as temperature, solute concentration or strain-rate, then this type of model is termed a "state variable" model. Note that these are not necessarily the same as the more common use of the term "state variable" used to represent an intensive property [Moore, 1972] since state variables in these types of model can be history-dependent: accumulated strain, for example (e.g. [Lowe, 1988]).

Both intensive and history-dependent variables may or may not be measurable in practice. Thus an internal stress variable, which may be calculated by measuring the curvature of dislocations in a transmission electron microscope, would be "unmeasurable" in an industrial context. Clearly for simple direct modelling the most useful types of lumped parameter model are those expressed in measurable state variables. However there are circumstances in which the fact that the quantity is unmeasurable is a prime motivation for modelling. An example is the dissolution of secondary particles during extrusion of aluminium alloys which is vital from the point of view of subsequent ageing, but could never be measured on-line and would be difficult experimentally.

Graphical displays of results give an excellent intuitive feel for the behaviour of the systems, often much better than an analytical solution. The argument between analytic and numerical modellers, still heard with respect to materials processing, is now effectively dead for chemical engineering and has not been proposed seriously for 25 years.

2.3 Materials and modelling

Much of this report has centred on the problems of modelling industrial processes in general: problems with commercial focus, software, manpower etc., and with abstract problems to do with equation solving, shape representation, data gathering, parameter formulation and temperature distributions.

Solid and liquid materials have characteristics which add significantly to the complexity of these general problems and which introduce entirely new problems: multiple scales, multiple physics and multiple models.

2.3.1 Size scales and multi-models

One classification of processes is based on the size of the phenomena involved. For a shaped casting, the shape itself is a characteristic only at the largest size scale, say above 1mm, whereas the growth of nuclei involve phenomena at atomic length scales.

The effects due to shape are modelled by a meshing method: finite element or finite difference, used to predict shape changes, thermal and stress fields. All "smaller" processes are represented by micro-models and the optimal coupling between these micro-models and the meshed macro-models is a matter of current research [Marsh and Glicksman, 1993; Tjotta and Langsrud, 1992]. The recent cellular automata models of nucleation and dendrite colony growth (e.g. [Rappaz and Gandin, 1993]) are particularly good examples as they link models of different types of physics at different size scales using different representations and different algorithms.

Micro-models, for solidification, for example (see Appendix A), which arise from academic research are rarely immediately appropriate for a manufacturing modelling project. The underlying physics may be the same in some sense, but if integration with a macro-model is required, the basic physical description of the micro-model usually needs to be reformulated completely.

For example, a micro-model which describes behaviour under conditions of constant temperature may be an accurate and valid description of the physics involved, but if the macro model imposes the condition of a strong temperature gradient over the relevant size-scale, this perfectly valid micro-model may be perfectly useless. In this example, the isothermal model may have led to sufficient insight that it becomes much easier to reformulate the model appropriately, but more commonly, such reformulations require that some previously important effects become irrelevant and new effects need to be discovered and analysed.

It is important to realise that the micro-models, even when re-formulated using appropriate boundary conditions, are never actually plugged-in to a macro-model used for commercial work. An appropriate micro-model is used to generate a higher-level approximate description of the phenomena in the regime of interest, it is this approximation which is plugged-in to the model at the next higher level. That model in turn is exercised, and its behaviour explored, so that it too may be used in the derivation of different phenomena at a still higher level of abstraction and (typically) at a larger size scale.

An extreme view, sometimes forcefully expressed, is that models which lead only to insight are in fact useless; and that the insight that they provide is an illusion of progress whereas in fact no progress has been made; and that this particularly applies to models which contain unmeasurable or uncontrollable parameters. Academics tend to reject this viewpoint outright, but our survey over the last year indicates that there is at least a grain of truth in it, at least so far as modelling manufacturing processes is concerned. However, part of the problem is simply the long lead-time (often over a decade) for the qualitative insight from an early academic model to work through to being applied as a useful modelling tool.

2.3.2 Materials specialization

Some specific types of alloy and some types of shape, for any particular process, are not handled well by current software either because the basic science of some important phenomena are not understood sufficiently to be modelled, or because the data for such phenomena has not been measured, or because efficient numerical methods have not yet been devised for these phenomena.

Thus for some types of alloy and shape, the engineer has to use models knowing how they are inappropriate and where their results will be misleading. This will continue to be the case for many years to come, even after some currently inexplicable effects (e.g. air gap behaviour in shaped casting) are fully understood, so software and educational aids to enhance the intelligent use of slightly inappropriate models is a long-term requirement.

The degree to which a science-led micro-model is inappropriate is defined largely in the degree to which the boundary conditions (assumptions) of the model match the boundary conditions of the real situation. (Differences between the model’s constitutive formula and behaviour in reality is more readily understood and appreciated.) Thus explicit description of boundary conditions, in the same terms as they would be defined for a large-scale meshed model, can greatly aid common understanding between the members of a modelling team (see Section 2.1), and also significantly aid the integration of new micro-models within established engineering modelling frameworks.

Extending a model which has been validated for one material to another often requires this type of reasoning. The effect of thermal diffusivity on thermal fields has already been mentioned and it implies that a model for one material, say steel, can be used for another, say aluminium, but only within a certain "processing window" (process variable settings), which may be entirely different from the window used for the first material. The thermal diffusivity of aluminium is 4 times that of steel, which in turn is twice that of cast iron. Thus a welding torch travelling at the same speed over a sheet of steel or aluminium of the same thickness produces an entirely different thermal field and heat treatment. (This was also described in the Overview section.)

Differences between materials often produce qualitative differences in processes. A complication of modelling the rolling of steel is the growth of a thick oxide scale which then deforms plastically when rolled but which dominates the heat-transfer because it is a poor conductor of heat. Modelling the rolling of aluminium might be expected to be simpler because no scale is formed, but in fact it is much harder. Heat transfer is now dominated not by conduction in a solid but by the heat transfer coefficient between the aluminium and the roll, which is a chaotic stick/slip phenomenon with steam and lubricant pockets requiring an aggregate, average model on its own.

Sheet pressing ("stamping") is yet another type of processing where the appropriate model depends on the type of alloy. This too is illustrated by a contrast between steel and aluminium. The behaviour of a steel sheet, including many types of stainless steel sheet, is well-described by a "forming limit diagram" (FLD) [Chan, 1990; Tseng, 1988] which predicts where stretch failure will occur given just the major and minor applied strains. This approach is simply invalid for most aluminium alloys. In fact steels are unusual materials in that forming limit simplifications are valid. For many aluminium alloys even the Von-Mises failure criterion is inadequate and finite-element models are hard to use because solutions are "pathologically mesh dependent" [Owen, 1992].

There are large programs of experimental work currently underway in some automotive companies which are aimed at producing the data from which FLDs can be derived. If these experiments record only the major and minor strains then the entire programs will be wasted effort. Note that this class of modelling error is only made by end-users and rarely by the original material producers.

General trend

Aluminium alloys are hardly new materials but these examples illustrate a general trend: the great majority of materials modelling, both for secondary and for primary processes, has been performed for steel. In many schools of numerical analysis, the mathematicians are unaware that many of the assumptions built in to generations of numerical methods and software packages are actually material-specific and relate to steel (e.g. a failure criterion based only on major and minor strains, [Demeri and Tang, 1992]).

Such is the bulk of numerical model development, particularly of deformation processes (e.g. [Hartley, Pillinger and Sturgess, 1992]), that it is entirely possible that steel-specific idioms will lurk within software for decades to come, unrecognized and undetected, until investigation of some severe accident brings them to light.

It is not simply that numerical analysts are ignorant of materials issues, most materials engineers are also narrow specialists in their chosen classes of material. Thus it can be the metallurgists responsible for transferring a process from steel to aluminium who make unwarranted assumptions. Strain-path sensitive yield phenomena are not taught in any detail in materials’ bachelor’s degrees in most universities (e.g. [Edwards and Endean, 1990]), even though the crystallographic issues in texture development may well be included.

2.3.3 Phenomenological modelling

Bulk constitutive behaviour

The constitutive behaviour of solid materials underlies nearly all complex material processing. Often the same mechanisms are important in quite different processes: thus solidification kinetics are relevant in both primary continuous casting and secondary shaped casting, and high strain-rate deformation occurs in forging and extrusion.

A lack of basic quantitative understanding of these fundamental mechanisms, their high variability from material to material, and the different modelling needs of people working with these different processes all conspire to reduce the usefulness of general models and to encourage specialized models.

Aggregate physics

Many problems will involve some physical phenomena which are not completely determined by basic atomic physical mechanisms but also depend on some collective mechanisms which cannot easily be averaged out in a statistical manner. This is most obvious in such substances as sand-piles or compacting powders where the rearrangement of particles dominates the dilatancy and the ease with which the aggregate can be deformed [Mehta, 1992]. Examples of aggregate mechanisms are texture (grain orientation in metals), dendrite nucleation, recrystallization, and stick/slip friction. In all cases some average behaviour is required but this is really produced by many distinct and different individual events. Sometimes a simple constitutive law can be applied, as has been found for many saturated soils, but for others, such as dry sand-piles or stage II hot isostatic pressing (HIP), state variable models based on unobservable internal parameters are often the best that can currently be achieved. Simple models can sometimes describe the evolution of such complex systems without involving either atomistic (small scale) or continuum (large scale) effects, but by describing the behaviour in terms of aggregate features [Choy, et al., 1990].

Meshed models dealing with macroscopic shape have to treat each mesh element as a continuum, so an average constitutive law is necessary. Currently a closed-form analytical lumped-parameter constitutive law is required for ease of integration with commercial packages. Plugging in an averaging sub-model would be catastrophically computationally expensive and may introduce iterative coupling problems as the gross behaviour affects the individual events of the sub-model [Sasikumar and Exner, 1992]. The solution is to produce microscopic models with the intention that they will be reduced and simplified for inclusion in a model constructed using a commercial package [Lowe, 1988].

An aggregate sub-model which produces as a result a constitutive "law" requires enough individual entities (grains, for example) to properly represent a continuum in their average behaviour. This means enough grains so that the atypical behaviour of those on the surface of the clump does not distort the gross result. For a 3D model, a clump of 50 grains is less than 5 grains in diameter. Since the coordination number of grain to grain contacts is typically between 6 and 12, models thus require about 5000 grains to be realistic [Asaro and Needleman, 1985]. Most metals have 12 slip systems in each grain and a step in the continuum behaviour must involve at least a few events at the grain scale, so the computational load is obvious (and it does not map well to parallel computers either).

Such models are often termed "network models" or are implemented using a programming technique known as "cellular automata" or by Monte Carlo methods. These models simulate multi-body interactions where the driving force is typically energy minimization [Ashby, 1992]. Three types of phenomena, all in metals, are the concern of active research in aggregate modelling (though the researchers may not see it in this way). These are recrystallization [Furu, et al., 1990], hot-deformation [Lalli, 1992; Wilkinson, 1986] and solidification [Stefanescu, 1993].

Network models start from a constructed microstructure [Parse and Wert, 1993; Wilkinson, 1986] and then simulate competitive growth mechanisms with impingement. Using these methods, modern results for solid-state recrystallization and for nucleation and growth of solids in a liquid medium are very similar to analytically-derived Kolmogorov-Johnson-Mehl-Avrami kinetics [Humphreys, 1992; Rappaz and Gandin, 1993; Sasikumar and Exner, 1992]. Modelling nucleation and growth is more complex in the solid state than in solidifying melts, especially in processes such as rolling where some material recrystallizes over and over again, so that a great deal of "book-keeping" is involved. With thousands of grains, this requires some innovative computer science to program efficiently.

Small size scale models of aggregates which reproduce the same type of behaviour as approximate aggregate-scale models are useful in that they illustrate the applicable ranges of behaviour where the approximate models are valid [Stefanescu, 1993].

However these small scale models nearly always require kinetic parameters which are almost unmeasurable in practice, so they will not commonly be used as components of large scale (macro-transport) models for some time to come (apart from the computational expense). Examples are the terminal velocities of falling dendrites or the permeability of packed dendrite beds. In principle, however, the same physical field (temperature, composition etc.) is viewed as being integrated over several different length scales in sequence, from the smallest to the largest, where the field at each scale is not identical to the field at the next higher scale but has some well-defined relationship (e.g. the compositions of interdendritic and extradendritic liquid) [Beckermann, 1993].

Network models are not the only route to deriving constitutive behaviour from deeper physics. There have been attempts to model a small number of deforming metal grains using finite element models where the elements are small enough such that the change in shape and orientation of the grains can be tracked [Lalli, 1992].

Other aggregate properties are those determined by interacting particles [Mehta, 1992; Nohara, et al., 1988]. These types of aggregate physics will eventually lead to improved accuracy and generality in the reduced constitutive laws so that they become more suitable for inclusion in the modelling engineer’s toolkit.

Surface boundary conditions

In terms of materials research, the view of parameters and assumptions as boundary conditions to sets of PDEs will make research into necessary surface property measurement more focused. Almost all boundary conditions involve surfaces, those that do not, such as symmetry conditions, require no physical modelling and therefore no property measurement. "Slight" changes in problem definition (from a manufacturing engineer’s perspective) can strongly affect surface boundary conditions and thus change the type of solution method and hence the accuracy and validity of the result. Attention therefore focuses on the most variable material properties, over a range of materials and over the duration of the process.

Surface properties such as heat transfer and friction coefficients are highly variable and they directly affect the character and type, as well as the magnitude, of the boundary conditions. These properties are also aggregate properties: friction occurs by the formation and breaking of microscopic asperity contacts, heat transfer in rolling is an average of lubricated and unlubricated regions, and air gaps. Numerical and computational developments in inverse modelling will be necessary to fully capitalize on research on aggregate physics, and will also be necessary to provide the measurement techniques to validate such new theories.

Although surface properties are common to many processes and always difficult to deal with, there is less scope for common research work than might be expected. The reason is that they generally have to be measured by inverse modelling, i.e. a properly constructed model must already exist so that the property measurements can be derived from an instrumented process. There is always the doubt that the "property" in question may not be in reality a single, indivisible entity (it rarely is, it is almost invariably a process in its own right [Edwards and Endean, 1990], requiring its own micro-model). So if there is an alternative direct method of measurement, there is always serious doubt whether the direct measurement is measuring the same "thing" as is appropriate for the real process.

2.4 Software modularity

Computer science techniques have and will continue to evolve considerably. This, coupled with explicit representation of knowledge of numerical analysis methods, will revolutionize the mathematical aspects of developing new finite element and finite difference techniques. This is because it will enable new ideas to be integrated more quickly into industrial-strength code, rather than being laboriously hand-coded into software which can then only be used for testing the algorithm. Some of the important computer science developments relevant to materials process modelling are discussed in this section. Inverse modelling methods are considered separately in section 2.5.

Modelling software systems which are more modular are required if significantly more engineers are to be given adequate modelling tools. There is scope for improved institutional arrangements to enable future modellers to use more of each others work than has hitherto been possible. This may involve no more than support to write a text book or handbook of "vanilla" modelling procedure libraries [Wood, 1993a; Wood, 1993b].

Historically, the rate of speedup due to new algorithms (0.25x106 over 30 years, for 3D elliptic PDEs [Bentley, 1988]) is more than half that due to new computer hardware (0.5x106 over 40 years), so increasing the rate of algorithm technology transfer is well worth while for speed alone. These advantages will only accrue to researchers willing to embrace new techniques, such as learning new computer languages and specification techniques; and leaving the Fortran to be generated automatically.

2.4.1 Existing software systems

Almost without exception, research process models and all commercial software are written directly in a third-generation language (Fortran, Lisp, Pascal, C, C++) [Cocks, 1993; Ion, Shercliff and Ashby, 1992; Lalli, 1992; Szekely, 1990b; Towers, 1989] with only user interface code derived from libraries or code generators [Bailey, et al., 1993].

Modelling systems produced for other branches of engineering (e.g. i-Think/Stella, VisSim, Mathematica [Wolfram, 1991], Excel etc.) have not been used even for the initial stages of materials process modelling because they were perceived to "run out of steam" (both conceptually and in terms of run-time) too quickly. This is now much less of a problem as hardware becomes more powerful and these packages become more robust (becoming comparable to the chemical engineering oriented systems such as GAMS). It is here, at the early stages of a model’s development, that computer hardware progress will have its greatest impact because it now enables shrink-wrapped easy-to-get-into software to be a viable method for experimenting with process models.

2.4.2 Modelling skills

One essence of modelling skill is the transformation of the physics and mechanics of the problem into a formulation of appropriate equations, parameters and boundary conditions. This is a precursor to the numerical analyst’s skill in constructing bespoke or specializing general purpose programs which efficiently and robustly solve these formulations (see section 2.2 above).

While both sets of skills should be promulgated and enhanced by better educational arrangements, there is also the proven capability of supplying some of these abilities through sophisticated software which (a) aids the construction of model formulations from a statement of the physics and thermodynamics [Vázquez-Román, 1992] (for lumped parameter models) and (b) automatically constructs efficient solving programs from model formulations, exemplified by the creation of finite-difference based solvers [Kant, 1993].

2.4.3 Software factories

The need for highly-specific packaged models has been demonstrated in the case studies: they are clearly required if the number of industrial modellers is to be increased by an order of magnitude. Thus the companies and organizations producing packaged software have to become an order of magnitude more efficient. The only realistic way this can be done in the medium term (4 to 8 years) is through the development of automatic aids for program production.

The idea that finite element or finite difference software could and should be generated for specific purposes, both for research and for end-user packages, dates from the mid-1980s when it was shown that the necessary technology and computer science already existed [Rehak, 1986]. This changes the view of software development from the crafting of individual solvers to the engineering of an infrastructure to produce many different solvers, with consequent benefits for "code-reuse" and software quality. (Code does not get re-used, it gets thrown away. Only the specifications are re-used, which is the appropriate level of granularity and level of abstraction for re-use.)

There are several ways in which such factories can be constructed. In the short term a "template filling" method would be quick to produce, but a purely "generative" method would give better long term robustness, efficiency and coverage of a wider variety of problem [Wood and Sargent, 1993].

Figure 2.6: The template scheme for software factories

Kant’s factory "Sinapse" uses a generative approach for relatively narrow areas of physics (e.g. sound wave propagation and heat diffusion) taking about 50 lines of input specification and producing up to 4000 lines of either parallelized Fortran (for a Connection Machine) or Fortran-77. Sinapse uses a step by step approach to transforming the input specification into output code using mathematical manipulations (it is written in 25000 lines of Mathematica [Wolfram, 1991]). A modelling consultancy could use such a system to generate polished, efficient, user-friendly but highly-specific packages to order. Such a consultancy would enjoy competitive advantage and also expand the number of people using models; perhaps by an order of magnitude.

Figure 2.7: The generative scheme for software factories

The software factory concept can also be applied to the generation of very specialized pre- and post-processors. In many ways this has already occurred, although in a fragmented way. The recent availability of programmable pre- and post-processing tools, such as Patran 3, AVS and Explorer, indicates the mainstream future direction.

2.4.4 Knowledge-based systems

Software systems containing an explicit representation of some procedure or technique, as opposed to the procedure being implicitly coded into the program, are termed knowledge-based systems. Many have been developed to help construct finite element models [Altan and Miller, 1992; Rehak, 1986] but only a few to attempt the interpretation of finite element results.

In the context of only the innermost loop of the modelling cycle (Figure 2.1), it is straightforward to produce expert systems to aid model construction and results interpretation. But this is just recognition, not interpretation. What are really required are automated builders and interpreters for the middle loop which assesses the validity of descriptions and proposes new types of model. This is very much harder to do because it requires "knowing" the industrial context of the problem. In sufficiently specialized modelling systems, however, e.g. for large-batch high pressure die-casting of commercially-pure aluminium for light-fitting housings, the industrial context is "given" so useful knowledge-based systems are possible with current technology. Building such systems cheaply enough to be viable for (inevitably) restricted markets again indicates the need for appropriate software factories.

Many modern complex software systems contain an element of knowledge-based processing and such techniques are particularly useful in two areas:

• in constructing software factories (as well as in their results) [Rehak, 1986], and
• in bypassing modelling completely in the solution of industrial problems which more usually would be thought to require it.

Some problems involving shape, particularly those where the process is basically the same but where there is iterative development, are potential candidates for non-modelling problem solving. Examples include anything requiring dies, jigs, moulds, shaped-rolls etc. where the physics of the process does not change but where a new shape of tooling usually requires several iterations to get right.

The obvious technique is a meshed modelling method: used to reduce the number of trials by giving a good first estimate for a die shape [Dean, et al., 1990]. In such situations it is often the case that the entire complex geometry is not relevant, only certain qualitative features together with a few simple numbers determine the success of a trial, and a pattern-recognising knowledge-based system can provide a good estimate most of the time, using a computational run-time only a tiny fraction of that of a finite element model.

2.4.5 Parallel processing

Numerical modelling, either using meshes with tens of thousands of elements or using simple meshes with very high order polynomial solution methods, will certainly require parallel computing hardware for cost-effective solution in the future. This matter can not be simply left to hardware developers: the physical and mathematical characteristics of modelling problems, and the software development issues described above, all directly affect the best route to process modelling on parallel computers.

In current modelling practice, however, most of the time and effort is spent in getting the appropriate model constructed in the first place; the computational cost of running the model in a production sense is small by comparison and only important in a few, very mature processes. This situation can be expected to change dramatically as it becomes cheaper and quicker to generate process models through a combination of better-educated manpower, better communications with other modelling efforts (e.g. via Internet), and automated aids for software construction. Therefore costs of running production models can be expected to be relatively more important in the future even discounting the inevitable extra complexity of more sophisticated models.

Software for "parallelization" can be classified under four headings [Furtney and Taylor, 1993].

• Scale of parallelism
• Uniformity of parallelism
• Granularity of synchronization
• Global or local communication

The "scale" of parallelism is the number of processors that could be kept busy if an unlimited number were available: some types of computing problem are inherently serial, one thing has to be done after another, in which case the "scale" of parallelism is 1. A totally uncoupled problem, say of a 10,000 grain microstructural development which depended entirely on a pre-specified temperature cycle and where adjacent grains had no effect on each other, would have a "scale" of 10,000.

The "uniformity" of parallelism is simply the uniform average of the "scale" as a function of time: so a program which spent some of its time at a scale of 10,000 and some of its time at a scale of 1, would have very low uniformity because the scale changes rapidly.

The granularity of synchronization depends on how much processing can be done independently (on different processors) between communication (and therefore synchronization). Thus in a coupled stress/temperature model, if the equations are explicit and there is significant heating from deformation, then the stress model and the thermal model will both be fast and will each need to update the other every calculation cycle. Conversely, if the equations are implicit (requiring iteration to solve for each element) and if there is very little thermal input from low-strain deformation, then the stress model will be relatively slow and will also only need to be synchronized with the thermal model perhaps every 10 cycles. The former has fine-scale granularity and the latter has coarse-scale granularity.

Communications are local if calculations only affect their "neighbours", e.g. in a finite difference model. They are global if the entire system of equations over all elements has to converge together, e.g. most finite element models. (This is one of the major distinctions between rival FD and FE casting simulation packages, see appendix on shaped casting.)

Ideally-good models

Ideal models which are well-suited to run on almost any type of parallel computer are those with high parallelism scale and uniformity, coarse-grain synchronization and local communications. These will run on everything from massively-parallel processors (tens of thousands of very simple computers) to workstation clusters (a dozen or so ordinary networked workstations).

Ideally-bad models

The hardest models to run on parallel computers, or those which will only run at all well on very specific types of parallel computers, are those with low parallelism scale and uniformity, fine-grain sychronization and global communications. Unfortunately this category includes, among others, Nastran, Ansys and Fluent: i.e. nearly all of the most-used commercial modelling packages [Furtney and Taylor, 1993].

Since the ability to take advantage of parallelization for common process modelling problems depends so much on the detailed characteristics of the particular problem, including the precise nature of the boundary conditions, automated aids to construct software from physical specification becomes imperative. This adds significantly to the many other reasons for encouraging mathematically-aided software construction kits, as described above.

One encouraging feature is that genetic algorithm approaches for solving inverse problems (see below) are inherently uniformly parallel at a large scale with very localized communication needs, so their high computational requirements may not be so much of a problem as might be expected.

2.5 Inverse modelling

2.5.1 Introduction

There are many situations where a modeller wants to "run the model backwards". This is termed "inverse modelling" [Szekely, 1990a] and may be required to:

• calculate the shape of dies, given the shape of the component to be made from them,
• calculate the load on a rolling mill, given the required thickness of the sheet to be produced, or
• calculate material properties and boundary condition coefficients, given an instrumented experimental process.

This is rarely a simple matter, and may be impossible even when the usual way of running the model gives no problems. An example is the case of rolling described earlier: the deformation is almost unrelated to the material strength and microstructure, so predicting the stress in the material from the deformation involves extremely "stiff" equations and loading answers which can vary by a factor of 2 over trivial deformation perturbations.

The "field" problems solved in current generation process models can be termed "direct problems": where the distribution of a field (stress, temperature, displacement etc.) is calculated from a complete set of input parameters. Whilst the input to such software may be complete in terms of the number of parameters being adequate to the number of degrees of freedom of the problem, it is often the case that there is significant doubt associated with the accuracy of some input parameters. Surface and interface properties, such as effective thermal resistance, may be difficult or impossible (even in principle) to measure. This uncertainty can be reduced by measuring the field quantities that are influenced by the poorly measured parameters and then attempting to solve the "inverse problem" to calculate those parameters.

Inverse problems can be loosely defined as those in which part of the answer is known, but part of the input information required to calculate a complete answer is unknown. For example, given thermocouple measurements at various locations and times, it is possible to estimate variations in effective thermal contact that give rise to those measured temperatures. The inverse problems of particular interest in process modelling can be termed "inverse field problems". The variety of such problems is wide: for example, the estimation of thermal and mechanical contact characteristics, estimation of time or temperature-varying material properties, even continuum mechanical properties derived from 1-dimensional tests.

Regardless of the nature of inverse problems, they all possess at least two characteristics that make them difficult to solve in comparison with the associated direct problem:

• In field problems the response at any internal measurement location lags behind the imposition (or other variation) of the boundary conditions that caused it. Also, smaller magnitude responses are obtained as the distance increases between the boundary condition and the measurement location. Both of these phenomena result from the finite propagation characteristics of real materials.
• All inverse problems based on the use of experimental measurements are ill-posed, having multiple solutions due to errors or noise in the measurements. This problem cannot be solved by increasing the measurement rate since this simply causes successive measured values to be increasingly correlated within the bounds of the measurement technique.

2.5.2 Solution techniques overview

Beck’s book [1985] provides a comprehensive discussion of techniques for the solution of inverse field problems which yields considerable insight (given patience and tenacity on the reader’s part). Several approaches to the solution of inverse field problems have emerged in recent years, all of which rely on repeated solution of the associated direct problem, combined with various mechanisms for estimating the unknown quantities based on a comparison of measured and calculated values.

In any field problem, different locations within the medium in which the field exists demonstrate different sensitivity to the imposed boundary conditions. This alternative expression of the first characteristic (described above) leads to the concept of sensitivity coefficients, which are defined as the rate of change of known (measured) quantities with respect to the quantities to be estimated. Sensitivity coefficients are important for three reasons:

• to obtain maximum information from experiments,
• to determine whether the inverse problem is linear or non-linear,
• solving by minimizing least-squares of differences.

Most direct models do not produce sensitivities directly, instead they must be calculated by repeated runs of the direct model. An exception is if the direct model can be reformulated to calculate sensitivities at each iteration. This can be done fairly easily if access can be obtained to the source code of the model; an analytic differentiation (which may be done using modern compiler technology) of the relevant equation gives the required answer [Dantzig, et al., 1993; Pantelides, 1988].

Since the computational effort for any inverse problem is directly proportional to the effort for the direct problem, there is a great deal to be gained by making the direct model as simple as possible. Using a boundary element method, where applicable, rather than a finite element or finite difference method, can lead to great savings [Das and Paul, 1993].

Experiment design

To obtain maximum information from experiments, sensors should be placed at those locations where the medium is most sensitive to variations in the quantities to be estimated. Hence a knowledge of sensitivity coefficient distribution (in both space and time) provides valuable insight into the design of experiments on which inverse field solutions are based.

Linearity

Sensitivity coefficients are important in determining whether the inverse problem is linear or non-linear. If, when the governing equations are differentiated with respect to the unknown parameters, the resulting expressions are functions of unknown quantities, then the inverse problem is non-linear. It is possible to have a non-linear direct problem and one or more associated linear inverse problems, depending on the sources of non-linearity and the objectives of the inverse analyses. It is also possible to formulate non-linear inverse problems associated with a linear direct problem.

For linear inverse problems, or for non-linear problems where linear piece-wise linearity is a good approximation, variation in sensitivity coefficients throughout the analysis domain can be obtained by solving a modified direct problem: where the governing equations and boundary conditions are re-cast in terms of sensitivity coefficients. In many cases this can provide a practical means of designing the experiments required for the solution of inverse field problems.

Solving

The basic method for solving inverse field problems is to minimize a least-squares difference between measured and calculated field values. This difference implies a differentiation which therefore implies a need for sensitivity coefficients in the solution algorithm. A major problem with this approach is that sensitivity coefficients are derived from experimental differences which amplify measurement noise, making accurate estimation of the unknown variable parameters very difficult. Various modifications to the procedure which aim to mitigate the effect have been proposed (e.g. in Beck’s book [1985]), however each approach tends to exhibit its own limitations when applied to realistic industrial problems.

Limitations can include the need to make a comparison between calculated and measured field variable at all locations simultaneously, for each analysis time step (as in the dynamic programming and other "whole domain" approaches), or the need to assume constant material properties in non-linear boundary condition estimation problems. An additional feature of the current methods is that they require an assumption about the general behaviour of the unknown parameters: for example, that they are piece-wise constant or piece-wise linear.

Implications and alternatives

In contrast to the above techniques, the use of genetic algorithms (GAs) offers an alternative approach. Recent work [Wood, 1993c] has indicated (though in a very simple inverse field problem) that the approach may offer certain advantages: since sensitivity coefficients are not explicitly used, the GA approach is relatively immune to measurement noise. Also, in the work reported, the GA approach was less sensitive to sensor location, measurement rate and the amount of available measured data. These improvements are at the cost of significantly greater computational effort than is required in any of the more well-established techniques. In particular, GAs cannot easily use sensitivities which may be directly calculated by a model (e.g. [Dantzig, Tiller and Tortorelli, 1993]). However, GA computations are inherently uniformly large-scale parallel and so may be efficiently solvable in the near future, they also inevitably involve several thousand solutions to the associated direct problem and so will always be slower.

It is generally the case that all of the above methods have demonstrated their capabilities in 1-dimensional problems. However their application to industrially relevant 2-dimensional inverse field problems is very difficult, and their application to 3-dimensional problems has not yet been attempted with any success. The difficulties encountered in applying the more well-established techniques to such problems usually stem from the need to obtain accurate values of sensitivity coefficients, and, for many problems, the arbitrary assumptions (in an engineer’s mind) that must be made regarding the distribution of multiple unknown boundary conditions. In the GA approach, currently limiting factors are the definition of suitable objective and fitness functions [Goldberg, 1989] and the need for much faster solution of many direct problems. However, in principle the GA approach does offer an ability to solve inverse problems involving many unknowns.

2.5.3 Future developments

Up to this point the discussion has focused on the occurrence and solution of inverse field problems in the development of process models. There is much scope for further work here, with attention likely to be aimed at two main areas:

• Characterization of the detailed interaction between processed parts and the processing machine: for example, the estimation of thermal resistances due to shrinkage in castings and the detailed phenomena that underpin contact and friction in many other processes.
• The development of models of microscopic phenomena in processed materials, such as nucleation and texture evolution and their influence on macroscopic properties.

Not only does the need for inverse analysis arise within each of these distinct areas, but also in the interactions between these areas and the activities required to develop macroscopic process models. This interaction can occur on two levels. First, the exchange of numerical input data that can be immediately used within a process model (which are typically estimates of parameters obtained from research at a macroscopic level). Second, the transfer of increased understanding about microscopic phenomena, which cannot be directly integrated into macroscopic process models. An additional role for inverse analysis occurs in the latter, where "higher level" or "global" approximations within a model must be matched to localized microscopic behaviour. This is a model-matching exercise, whose principles are discussed in [Guillochet, 1993].

In addition to their use in the development of process models, it is likely that inverse analysis techniques described above will be further developed and applied to the optimization of the whole of the manufacturing process. It is expected that highly stylized solutions to specific inverse problems within process optimization will emerge over the next 3-5 years. One basis for this expectation is the growing interest reported at conferences [Chenot, et al., 1992; Lewis, 1993] and the imminent launch of a specialized journal ("Inverse problems in engineering", Gordan and Breach Science Publishers, Reading RG1 8JL, UK).

SECTION 3

Conclusions and Recommendations

Conclusions and recommendations

The most obvious conclusions from our study concern education and training. This may not be apparent from this report because it covers the whole range of different specialized skills, but from our industrial and academic visits, and particularly from our study of published literature, it is plain that very few modelling teams are capable in all the skills and sciences described here. In Britain neither the undergraduate nor postgraduate courses appear to be producing the people able to use and develop the new modelling tools produced over the past decade. Although this study is not in any way exhaustive, the same difficulty also appears to be relevant in North America and non-Scandinavian Europe.

The cheapness and ease of use (when used by numerical analysts) of current finite element modelling packages is extremely worrying because the lack of metallurgical awareness implies that a great deal of superficially convincing, but fundamentally flawed, materials process models will be produced during the next decade.

Modelling materials processing is seen as a strategic commercial issue in America, Europe and Japan; but most public money appears to be directed towards research and technology development demonstrator projects or towards developing more robust or efficient solver algorithms for existing or new processes. Our review has shown that for large companies in high technology industries these are indeed important for competitiveness (though perhaps only slightly more important than the data management and multiple software version problems).

However, for significant impact on manufacturing industry as a whole, several other issues are much more important: manpower training, well-integrated software environments and a great deal more help in setting up problems. This is particularly so for the small and medium-sized enterprises which make up the bulk of manufacturing industry. Modelling efforts should perhaps be supported on a much more specifically industry by industry or process by process basis than has been the case hitherto. (This would be at a very fine level: certainly different software for pressure die casting compared with aluminium alloy sand casting, and probably different packages for aluminium as opposed to zinc die casting). A significant step is represented by the initial phase of the UK "CAST" initiative which is intended to introduce simple simulation models into the many small companies of the largely uncomputerized UK aluminium alloy shape casting industry [Adams, et al., 1992].

One idea that could help progress the fundamentals of process modelling skills would be the establishment of a UK centre or "club" which could form a focus for the interdisciplinary research specifically in modelling materials processing and model deployment. Either of these could coordinate the production of a syllabi for modular degree courses and be a clearing house for dissemination and guidance on "current best practice". A centre could provide a repository for materials data and constitutive laws (as has been suggested for the USA [Jain, 1991]).

We have observed that many modelling development projects have duplicated effort on the same problems and that many more projects now face some of the same problems (such as surface property measurement and contact problem mathematics). These difficulties could be alleviated and joint opportunities exploited by national and continental (e.g. European) centres.

A centre could take a form along the lines of the Laboratoire de Méchanique et Technologie at the École Nationale Superior de Cachan (Paris 6) or the Centre for Process Simulation and Control at Imperial College (both of which could claim to be doing some of this type of research already); or the Institut for Energiteknikk (Norway) which works directly with the primary casting industry supplying a "contract modelling" service for projects which it is not sensible for any one company to support on a continuous basis. All these centres have semi-permanent research teams, but a process modelling centre could alternatively take the form of the Isaac Newton centre for mathematics at Cambridge where there is a very small permanent staff and research is carried out in 6-month projects by selected groups of visitors. This last option is particularly attractive since the permanent staff could offer expertise in a broad range of computational and materials skills and the constant throughput of visitors would widely disseminate the expertise generated.

SECTION 4

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Wood, R.L. An inverse thermal field problem based on noisy measurements: comparison of genetic algorithm and sequential function specification method. Processing of Advanced Materials Int. J. Computer-Aided Engineering (submitted) (1993c) .

Wood, R.L., and P.M. Sargent. Possible designs of software factories for producing finite element, finite difference and boundary element packaged software, Technical report, (to appear) (November), Loughborough University of Technology, 1993.

This report is issued by the Department of Engineering, Cambridge University, as CUED/C-MATS/TR.206 and copies are available from The Librarian, Trumpington St., Cambridge CB2 1PZ for a nominal charge.

APPENDIX A

Shaped castings

A.1 Commercial status

The modelling of shaped castings is dominated by one over-riding consideration: parts are made by casting because they are complex three-dimensional shapes, difficult to fabricate by other processes. Thus complexity of shape is fundamental.

The shaped casting industry is based largely on aluminium-silicon alloys and cast iron. Moulds are either sand or "permanent" (steel) and cycle times are much longer, minutes to hours, than the few seconds necessary for high or low pressure die-casting. Large castings in titanium, steel, bronze etc. are a more specialized business, as are directionally solidified turbine blades, medical prostheses and investment cast jewellery.

Software modelling of shaped castings to improve "methoding" (the placement and sizing of risers, feeders, chills etc. and the design of moulds to yield defect-free parts) was insignificant in the mid 1980s except for specialized, very high value industries, and even there it was new. Now it is widespread [Dudley, et al., 1992]; though only in large foundries. In Britain it has been estimated that there are over 500 small foundries, but only a handful employ more than 500 people: most employ between 20 and 50.

This section on the modelling of shaped castings reviews the technology from the point of view of industrial technology development and deployment, not from the point of view of a research scientist interested in casting phenomena.

A.1.1. Software

In 1993 there are three leading-edge commercial packages employing computational fluid dynamics (CFD).

• Magmasoft is a control-volume finite difference code (CV-FD) using structured meshes from Magma (which grew out of the Foundry Institute, RWTH Aachen) which claims that 60% of the world’s automotive pistons are cast in moulds designed using Magmasoft [Dudley, et al., 1992; Lacaze, et al., 1992; Pickin and Travers, 1992].
• Procast is a finite-element code (FE) using unstructured meshes from UES Inc. (Dayton, Ohio) and is commonly used in the USA.
• Simulor, a third package, was developed by Pechiney Research Centre (Voreppe) with CISI Ingenierie (Grenoble). It is very similar to Magmasoft and is in use in Pechiney subsidiaries world-wide [Lacaze, et al., 1992].

Magmasoft and Procast have many more users than Simulor. There are at least another 5 commercial codes available world-wide and 5 more in Japan, not counting proprietary software used only internally and many research projects [Cross, et al., 1992; Lacaze, et al., 1992; Piwonka, et al., 1993]. In the USA it is estimated that 200 foundries are using simulation software.

Many empirical software models exist, such as those based on Chvorinov’s geometrical "law". These run satisfactorily on personal computers, are based on 2D drawings data as input and require very little materials property data. They are correspondingly limited in their predictive capabilities [Pickin and Travers, 1992].

The reason that FD and FE methoding modelling software is now widespread is primarily that it does not now require an expert numerical analyst to use it: it is packaged software targeted directly at foundrymen. Secondarily, the computer infrastructure costs (hardware and the overhead from system mangement) have become cheap enough for larger foundries to afford. These developments both occurred in the late 1980s. One key development in the packaged software is that it provides an easy framework for setting up the geometry and casting conditions. Now that this framework exists, improved scientific results in understanding casting phenomena can be plugged-in as a series of continuous upgrades without requiring foundrymen to be retrained [Adams, et al., 1992].

A.1.2. Computer infrastructure

The Japanese foundry industry is built of many small companies, as in Britain, thus the packages developed there were strategically aimed at personal computer hardware so that deployment into industry would be straightforward (however it did mean that the packages were largely empirically-based rather than simulation-based). Only larger foundries have 3D CAD facilities which are largely a prerequisite to performing methoding modelling based on CFD [Pickin and Travers, 1992]. In Britain, one leading British casting simulation package apparently runs on a workstation in precisely one company (Kaye Presteigne: [Lewis, et al., 1992]).

The availability of easy-management workstation operating systems and the merging of the personal computer and workstation computer markets (as described in the main body of the report) will have a particular impact in this industrial sector.

A.2 Science used in models

From the industrial perspective, the science used in the models is entirely secondary to the provision of a useful tool; and some empirical tools contain no science at all. However, casting technology has stimulated research in physical processes of casting to such a degree that in addition to industrial models without science, there are also novel scientific models which have been developed but not yet applied to industrial problems (e.g. [Rappaz and Gandin, 1993]).

A new physically-based model of a casting process may not be implemented in an industrial tool for the same 6 reasons as described in the main report: too new, too specialized, no better than previous methods, or unreasonable demands of speed, data or expertise. These types of barrier to the use of some types of models are considered in more detail below. Many such models are still of great use to the casting research and development community, they would just not find their way into end-user packages.

A.2.1 The levels of science

Casting is an unusual manufacturing process in that modelling activities are well-developed at nearly all the size-scales for all the multitude of physical processes involved: nucleation, dendrite tip growth, liquid metal flow through pasty, semi-solid dendrite colonies, thermodynamics of latent heat evolution etc. (though not all scales for the same alloy) [Overfelt, 1992]. A recent conference reviewed progress in modelling these processes and mechanisms (as well as continuous casting and welding modelling [Piwonka, Voller and Katgerman, 1993]) and this is definitely a growth area as assessed by activity in conferences, publications and support by funding agencies.

Shaped casting thus provides an excellent illustration of a manufacturing process where modelling has produced a very wide variety of scientific and industrial tools and where comparison of these models in terms of areas of application, relevance and use can provide general lessons.

As described in the main report section "Models and modelling", one classification of processes is based on the size of the phenomena involved. Casting modelling has to couple macroscopic modelling of geometric shape with micro-models of materials processes. For a shaped casting, the shape itself is a characteristic only at the largest size scale, say above 1mm, whereas the growth of nuclei involve phenomena at atomic length scales [Marsh and Glicksman, 1993; Rappaz and Gandin, 1993; Stefanescu, 1993; Tjotta and Langsrud, 1992; Towers, 1989].

The rapid growth of activity in casting modelling over recent years is partly due to the fact that there has been long-standing academic research in phenomena at many of the size scales involved. The principles discovered have rarely been amenable to being plugged-in directly to macro-models of shaped casting. However, experience with the relevant phenomena and the relevant experimental measurement techniques, has meant that when a macro-model framework appeared, appropriate micro-models could be developed fairly quickly from the basic science (e.g. [Hunt, 1991; Stefanescu, 1993]).

Figure A.1: Some levels of modelling in shaped castings

Simplistically, casting modelling can be considered on two size scales: a "methoding" scale where the shape of the component is relevant at which gross fluid velocities, temperatures and stresses are calculated by meshed methods coupling temperature, fluid velocities and stresses in the solid e.g. [Chan, et al., 1991; Chow and Cross, 1992]), and a "solidification" scale where several levels of microstructural process are modelled with a variety of meshed and lumped-parameter models e.g. [Campbell, 1991; Funkenbusch, Lambropoulos and Li, 1986; Rappaz and Gandin, 1993; Stefanescu, 1993].

These two type of model are sometimes termed MT, "macro transport", and TK, "transformation kinetics" models respectively. Commercial casting software packages place most emphasis on the "methoding" scale, with a programme of continuous improvement for the solidification models which contribute to more accurate latent heat modelling and which produce more sophisticated defect and microstructure predictions. The kinetic parameters which go into the TK models are very hard to obtain.

It is important to realise that the micro-models, even when re-formulated using appropriate boundary conditions, are never actually plugged-in to a macro-model used for commercial work. An appropriate micro-model is used to generate a higher-level approximate description of the phenomena in the regime of interest, it is this approximation which is plugged-in to the model at the next higher level. That model in turn is exercised, and its behaviour explored, so that it too may be used in the derivation of different phenomena at a still higher level of abstraction and (typically) at a larger size scale.

A.2.2 Solidification modelling

Solidification begins with nucleation, a process that is inherently stochastic if it occurs homegeneously, and continues with the growth of dendrite colonies from both the walls of the mould and free-floating in the liquid.

Nucleation

The usefulness of modelling nucleation depends on the alloy and the casting procedure, if the nucleation potential of the liquid cannot be controlled to within ±5% then no model will be of much help. If nucleating agents are added, as is commonly the case, then the availability of nucleating sites will depend only on the movement of these entrained particles near the solidification front and not on any kind of stochastic process.

Only recently have modelling techniques been devised to predict individual nucleation colonies probabilistically so that overall behaviour can be obtained by modelling many such events [Rappaz and Gandin, 1993; Spittle and Brown, 1989b]. Such models use a nucleation "mesh" much finer than the FE or FD mesh used for the associated macro-transport modelling, but it is still at the size of the colony, not individual atoms.

Figure A.2: Some types of process coupling in shaped castings

Colony growth

Colony growth involves multiple models of several size scales: a grain (colony), is an order of magnitude larger than the primary dendrite spacing, which is an order of magnitude larger than the secondary arm spacing. One useful simplification is that the ratio of thermal to atomic diffusivity implies that at these scales the temperature can be considered to be nearly uniform.

Time-dependent modelling of the shape of isolated dendrites at a size scale of less than 10 microns has been performed and reproduces stable and unstable behaviour including shape-changes and the transition between dendritic and cellular growth patterns [Hunt, 1990; Hunt, 1991]. These have been incorporated into physics-based dendrite colony growth models of great realism [Rappaz and Gandin, 1993].

Colony growth models will be necessary for more sophisticated macro-transport solidification models which currently depend on experimentally-determined or simple thermodynamic solid-fraction versus temperature latent heat models. Starting with the Schiel equation in 1942 there have been about 7 generations of attempts at these latent heat evolution models, 5 of them since 1981 (e.g. [Stefanescu, 1993; Wills and McCartney, 1992]). Better cooling-curve models will require data for metastable peritectics, divorced eutectics and non-equilibrium atomic diffusion in both solid and liquid phases.

2D versus 3D growth models are not simply related. Because growth involves impingement, a 2D model of a casting produces quite different results from a 2D section through a 3D model of the same casting. 2D models are, however, probably adequate to predict realistic recalescance and cooling curves to be used in macro-transport models.

A.2.3 Defect prediction

Many shaped castings contain fluids in use, so the most important defects are those which can lead to leaks, either through loss of strength or directly through pits in notionally flat machined surfaces.

Isolated porosity defects in shaped castings can be predicted using monotonic local-criterion-based approaches such as Niyama’s which depends on local field variables [Viswanathan, Sikka and Brody, 1992] and can be applied after the model has been completed and run to full solidification. This is because these defects have no coupling with the controlling independent variables. Substantial volumes of porosity, however, will have a coupling effect because of a change in local density and reduced thermal conductivity: so while a local criterion may provide an adequate description for such porosity, it cannot be applied after the model has finished but must be fully coupled during the model run. The porosity prediction must also be coupled to the tensile hydrostatic stress in the casting caused by shrinkage [Fryer, et al., 1993] .

A.3 Barriers to use

The major barriers to increasing use of models in foundries are to do with how existing modelling technology is implemented, packaged and delivered. This has been discussed earlier in a general context but it is particularly noticeable for foundries.

Because casting involves so many processes and is performed using such a variety of alloys, some types of alloy and some types of shape are not handled well because of lack of data or understanding of the specific phenomena which are peculiar to those specific alloys. Thus for some types of alloy and shape, the foundry engineer has to use models knowing how they are inappropriate and where their results will be misleading.

There are also many fundamental physical phenomena in shaped casting which are not understood at all to a level that permits quantitative models to be developed. A classic example is liquid flow through solidifying, mushy alloy; or the deformation of pasty solids in general [Bridgewater and Benbow, 1993; Davis, et al., 1992; Sekhar and Dantzig, 1992].

A.3.1 Implementation technology

Modelling shaped castings is justified commercially by the need to predict the effects of different shapes for a particular, well-characterized alloy. Modelling new materials for established shapes occurs much less often. This implies that meshed methods are the major technique and that generating the mesh for complex 3D shapes is unavoidable. This is where the difference between structured and unstructured meshes, and the trade-offs between finite-element and finite-difference techniques become important (as described in the main report).

Magma are now developing an FE version of their software because they see a commercial need there. Competition in the marketplace between the two main methods, as exemplified by Procast (FE) and Magmasoft (FD), is strongly affected by secondary considerations such as the availability of databases of materials properties, algorithms to model solid-state transformations in cast-irons, solidification and defect prediction models (see section on data gathering in Section 2 of the main body of the report).

In research terms, the leading proponents of the two schools have added so many extra numerical and computational techniques that the "core" algorithm (FE or FD) is so buried and reinterpreted that to an outsider the distinction is lost. Even the one crucial difference in the type of mesh necessary has now been overcome [Chow and Cross, 1992; Cross, et al., 1992; Fryer, et al., 1993].

Although the computer science and mathematical problems with mesh generation are still severe, it is to be expected that improved (probably patented) algorithms will be available within a few years which will automate the production of unstructured meshes for nearly all practical shapes.

A.3.2 Infrastructure problems

The small size of foundries in Britain impairs the ability of the industry to adopt modelling methods compared with larger conglomerates in other countries or the larger primary processing companies responsible for direct chill casting. However this is likely to be a temporary situation.

The injection moulding industry is similarly fragmented but has a software lead of several years (Moldflow was first sold in 1978, Magmasoft in 1988) aided by less computationally intensive models but similarly limited by the need for workstation computers (typically Unix). The polymer component companies apparently computerized their design offices using CAD long before the foundries, so the initial hurdle of installing sophisticated computer infrastructure was already surmounted. Adding Moldflow was then a relatively simple step. The foundry industry, more traditionally minded and possibly with less capital to invest in design support, has lagged but can be expected to catch up.

A.3.3 Software development

Most engineers using and developing casting modelling software say they cannot see a time when the expertise to perform the modelling can be embedded in the software itself so that shop-floor foundrymen can use it. This viewpoint is coloured by the fact that current commercial metal casting simulators are general tools for many different types of casting, requiring a deep knowledge of which software options are suitable for which type of casting. Highly specialized casting simulators, e.g. specifically for automotive wheels, could clearly be run by less broadly educated users.

The problem of increasing the number of people engaged in process modelling is thus transposed into one of generating, configuring selling and supporting hundreds of different specialized software packages for the many variants of shape casting markets. Thus software engineering technologies and the software factory concept which enable modules of different simulation options to be written by one company but reconfigured and adopted by specialist re-sellers will have a big impact on the use of modelling by small and medium sized enterprises.

Note that this is the exact opposite of the early 1980s viewpoint that led to the development of general purpose CFD codes. These were seen as an advance because research teams could then build on a common foundation rather than constructing their large Fortran programs entirely from scratch [Bui, 1992] Now that software engineering techniques enable consistency and modularity to be managed more easily, [DeMarco, 1982] it is possible to build individual simulators not on top of a monolithic general purpose code, but out of a library of carefully designed modular components [Wood, 1992]. At least, it would be possible: no research funds have been allocated in any country covered by this review to address this modularity and reconfigurability issue but the way forward is demonstrated by Sinapse [Kant, 1993].

A.4 General lessons from shape casting

A.4.1 Heat flow modelling

New shapes of component require new process settings and new shapes of feeders and risers ("methoding"). Simple heat flow modelling is sufficient to predict defect dangers for new shapes which are too complex for human expertise. The complexity of the basic science controlling the process is low (conduction) but complexity of shape and boundary conditions is high.

The assumption of pure conduction from a solid casting exactly at the solidus temperature into a mould at uniform preheat temperature is invalid, but for many castings, rectifying these assumptions makes only a quantitative difference of levels of defects predicted. Since the task is to predict the effect of new shapes based on the performance of old shapes, the practical benefit of more sophisticated models can be low for some sections of the industry.

A.4.2 Mould filling

For industries where radically different shapes occur frequently, and for large components where the fill-time is a sensible fraction of the solidification time, the thermal response of the mould (die) is important to the thermal field of the liquid metal and hence on its solidification and production of defects.

Two cases can be identified: very slow filling where flow is near-laminar and thermal conduction is dominant (such as for castings over a few tonnes in weight), and the more usual case where flow of liquid alloy must also be modelled using a fluid dynamics method, including some approximate model of the contribution of turbulence.

The result of such "filling models" (including liquid flow) is a better starting point for the simple heat flow modelling after solidification and hence gives better predictability of the locations of defects [Campbell, 1991]. A further refinement is the use of turbulence compensated conduction and convection in the liquid as solidification occurs [Salcudean and Abdullah, 1988] .

The most important aspect of turbulent mould filling in practice is the possibility that surface debris and oxides will be incorporated into the casting, producing massive defects. This is also rather difficult to model since only the gross flow patterns are repeatable from casting to casting. The position of the free surface can be tracked by a variety of methods [Waite and Samonds, 1993], all with significant problems (such as "losing" some metal from the system if splashes are too small to be tracked). Viscosities can also be deceptive, two liquids with the same viscosity can have radically different flow patterns which then affect free surface movement [Ohnaka, 1993] if one is a single phase and the other a mixture of fluid with suspended particles (as is the case with a solidifying melt).

A.4.3 Heat evolution

Solidification can be modelled as an instantaneous latent heat evolution, or over time and temperature change as the metal cools between liquidus and solidus (using the Schiel heat evolution equation for example [Campbell, 1991]). Heat evolution models based on more sophisticated solidification models [Rappaz and Gandin, 1993] also improve the starting point for simple heat flow modelling in the same way that filling models do. Concurrent heat evolution and cooling models show the movement of "thermal centres" as cooling continues. These should remain in the risers if the part is to be fed properly with liquid metal as it cools and shrinks so their location indicates porosity defects which can be highlighted on model output [Dudley, et al., 1992].

More complex solidification models also represent liquid flow through mushy solid and hence predict defects directly rather than via some defect criterion. This is much more important for some alloys than others (e.g. wide versus narrow freezing range alloys). So solidification models have two roles: better thermal field representation for defects predicted by existing criteria, and better criteria.

A.4.4 Residual stress

Residual stress can only appear in the solid through thermal or transformational contraction being restrained by the mould or by other solid parts of the casting, so simple types of residual stress model which ignore the dependence of yield stress on temperature can calculate approximate stresses directly from the mould shape. More complex models add the time evolution of the thermal field, and strain/time/temperature dependent yield. Additional sophistication can come from solidification models of the flow of liquid within mushy zones and hence local density and strength differences.

Residual stress models predict kinds of defect (e.g. hot tearing) that criterion approaches (i.e. those that set a critical level in a monotonic function of local field variables) cannot predict. Criteria are always local, but hot tearing depends on events some distance from the tear. This local/global distinction is another reason why some solidification and transformation defect predictions based on global models are more accurate than criteria-based predictions of the same types of defect.

The thermal contraction itself can be fed back as an input to thermal field models because air gaps form between mould and component which strongly affect heat transfer coefficients, i.e. the strain field and temperature field evolutions are closely coupled.

In current modelling practice the formation of these air gaps is estimated by an approximate empirical model where the heat transfer coefficient is simply set to be a function of time and temperature. This is surprisingly accurate, and it does not require a stress calculation so greatly speeding computational run time. Over the past dozen years, attempts to produce real physical models of heat transfer as a function of air gap (actually largely full of hydrogen, water vapour and organics; not air) have not been successful and are no more accurate than the empirical correlation. This situation will change with time, but the open question is whether the parameters which will go into improved air gap models are themselves any easier to measure or more stable across various alloy types and casting methods [Sargent, 1991].

A.4.5 Models are better than what ?

The usefulness of a model can only been seen in comparison with other ways of estimating the same information. A foundryman’s expertise is based on pattern matching results based on an experience of a range of component shapes and methoding procedures. Thus casting models can be expected to give particular benefit whenever a component shape/alloy combination is required which is outside the foundryman’s direct, personal experience.

"Heat centre" and "modulus" calculations based on Chvorinov’s law are adequate for predicting the likelihood of specific types of defect in families of similar, relatively simple shapes [Piwonka, 1992; Viswanathan, Sikka and Brody, 1992] and they have the added prediction of the ease of extracting the part from the mould. However they fail for complex shapes and even many common shapes such as those including fins or cores.

It must be remembered that there are many non-modelling functions which are vital to support a foundryman’s use of a modelling package: these include cycle-time calculations, optimization rules for feeder sizing, and a database of past casting projects. Also vital are the controls available to the user to drive the package in a particular way, such as the frequency of, and criteria for, snapshots of an evolving mould fill simulation. These can make or break the introduction of software into foundry practice.

A.4.6 Modelling versus experiment

The decision of where to make the trade-off between modelling a casting and performing an experiment depends on the generality of the experimental results (shape and alloy), the relative costs of setting up a model and the time available to get an answer. For shaped castings the cost of trials is very significant because moulds have to be designed and machined (or the patterns machined).

The asymmetry in information between simulation and experiment is also very important because it means that a good simulation gives much more information than a good experiment; though an experiment’s data is in principle more reliable. A casting thoroughly instrumented with thermocouples is even more expensive than a simple trial, but a model can display results with "virtual" thermocouples everywhere. A trial casting will show the location of porosity but its microstructure may give insufficient other information about the process conditions

A.4.7 Summary

Shaped castings clearly show a series of models of improving sophistication where some improving features are relatively independent (e.g. the several types of turbulence model) and others (e.g. convection) can only be sensibly applied if a more fundamental feature has already been modelled (e.g. conduction). This industrial sector in the UK is dominated by small companies, yet modelling efforts (like most research) are concentrated at the needs of the largest companies. Greater benefit to more of the industrial base of the economy would be generated by bringing the smaller companies up to speed rather than further extending the leading-edge activities of the larger companies. However such projects would be harder to manage and to cost-justify on an individual basis.

• Castings have complex shapes by definition, otherwise they would not be made by casting. Thus this type of manufacturing process determines directly some of the geometric modelling problems.
• Software engineering technology which enables modules of different simulation options to be reconfigured by specialist re-sellers would have a big impact on the use of modelling by small and medium sized enterprises.
• Remeshing research requires collaboration of a particularly close nature simultaneously in numerical methods, geometry and physical metallurgy.
• New shapes of component require new process settings. Very simple physics coupled with complex geometry can have big payoffs because FD and FE can cope with arbitrary shapes where human personal experience cannot.
• Improving simple models by adding new physics involves double the amount of science and numerical analysis for a less-than double benefit in predictions, and even then only for some industries.
• Simple physics defect criteria which are "local" are easy to manage in constructing and using a model but while accurate enough are eventually dead-ends. So progress to better models means going through a stage where the current, more sophisticated models actually give worse predictions than the simpler models.
• Shaped casting is an industry like injection moulding, pressure die casting, and rolling or extruding complex sections, where the cost of machining new shapes of dies and moulds is very significant, unlike the simple jigs for welding and gluing, or the simple shapes of sheet rolling or direct chill casting.
• The tradeoff between modelling and experiment for shaped castings is currently biased towards modelling only for complex shapes in companies big enough to manage the technology.

APPENDIX B

Laser Processing

B.1 Current status

B.1.1 Industrial status

Processing of materials by laser offers a wide range of fabrication and finishing techniques, most of which are now reasonably mature. CO₂ lasers of 5kW and above have been available for well over twenty years, and have been used for almost all laser operations. Recent developments in lasers are primarily aimed at broadening the potential of laser applications, rather than breaking completely new ground. Examples are the moves to develop very high power CO₂ lasers, the work on shorter wavelength Excimer lasers and portable solid state lasers, integration with robotics, and growth in pulsed rather than continuous wave processes.

In Europe, the country with the greatest investment in laser technology is undoubtedly Germany (especially the Institute for Laser Technology, a Fraunhofer Institute in Aachen). The Nordic countries have a number of laser centres, based in national industrial research laboratories or universities. A major UK effort is concentrated in the joint Laser Centre of the Welding Institute (TWI) and Culham Laboratory. A number of university groups have a long tradition of laser processing work, and the UK has several laser manufacturers and industrial users. It is fair to say however that lasers have not lived up to their early expectation, and have not been adopted as mass production tools on the predicted scale either in Europe or elsewhere.

The overriding reason for this is the continued high cost of the machines themselves, which has been a particular hindrance to growth in the number of laser users during the recent years of recession. The expense would not be so important were it not that engineering managers are often not aware of the practical applications of laser processing; partly because information (from experiment and models) is presented in terms which are not easily quantified, i.e. "quality" rather than in more familiar terms such as productivity gain.

B.1.2 Modelling activity

There are two main purposes behind modelling of laser processing (at all levels there is a desire to reduce the extent of experimental work required, and thereby to minimize cost) :

1. to provide predictions of operating conditions and final microstructure properties for use on-line with a laser in a production or prototyping environment;
2. to provide physical insight into the complex mechanisms involved in laser processing, leading to improved process control.

The first of these objectives is to some extent motivated by the desire to make laser systems more turnkey in their operation, with a reduction in the level of skill needed in the operator. A greater degree of on-line control, and routine selection of operating conditions, are therefore current goals. Process models clearly have a role to play here, but must be highly simplified - either by directly building an approximate model, or by "reduced modelling" of a more sophisticated, but inherently slow and complex, model. Intelligently deriving a simplified model from a complex model is perhaps more difficult, and certainly requires quite different modelling skills, from devising a complex model in the first place.

A database of parameters and procedures covering a wide variety of geometries, materials, beam types and process conditions would be more useful industrially before even simplified models could be used practically; and any such model would have to be a demonstrable improvement on such a database.

A considerable proportion of the laser processing modelling work in the academic literature has been directed at understanding the physics of the interactions. Laser processing introduces some new complex phenomena (laser-material interactions, laser melt pool convection, keyhole stability etc.). This "deep physics" modelling is a necessary long-term contribution to the field, and has clearly provided stimulating challenges to a large number of scientists. Only some of this work however is taken to a point where the results can be usefully extracted by those whose aims are development and control of industrial processes. In other cases, the results of a potentially useful study are lost due to insufficient attention to clarity in publication, and poor choice of groupings of process parameters in producing graphical output. In common with many sophisticated modelling activities in other fields, provision of useful modelling tools for industry is cited as an objective far more frequently than is in fact the case in practice.

In many ways it might be expected that the laser industry would have a higher level of integration with computers than some of the traditional industries - lasers are a relatively recent development, and have a degree computer control built in from the beginning. Computer-based models are however rarely found on a laser facility; parameter selection from handbooks of data and previous experience is generally the norm. So while modelling efforts in laser processing have been very extensive, this modelling activity has not led to the development of much software for use on-line with a laser facility. The work discussed in this case study is therefore all found in the published literature, rather than in software.

Possible reasons why the models produced by academics do not reach a production environment include: (a) lack of support in writing user-interfaces - being of insufficient interest for academia, and too costly for companies themselves (perhaps intermediate research institutes have a role to play here in commercialising academic work - and not just in laser processing); (b) the results produced are not expressed in a way which the end-user finds helpful or even of interest - again indicating a need for more liaison between universities and industry; (c) the physics of complex processes are not sufficiently developed to describe a wide range of conditions - the models at present are highly specific; (d) economic viability is crucial to end-users, but is rarely given any consideration in academic process modelling efforts.

Figure B.1: Laser modelling activities

Figure B.1 illustrates the "steady state" for many process modelling activities, including forging, some types of casting, semiconductor device fabrication, and possibly some rolling and heat treatment processes. The diagram could be annotated to indicate the cost and time associated with the development and use of any current generation process model development. The important point about this diagram, in the context of laser processing, is that almost all of the physics-based model development to date has concentrated on the initial stage of trying to understand the physical phenomena that underpin laser-material interactions.

B.2 Science base of models

B.2.1 Levels of complexity

A convenient classification of laser processing is into three divisions: laser heating, laser melting, and laser keyholing. This classification is essentially based on the surface peak temperature being below melt, between melt and vaporization, or above vaporization. It also divides the physics of the processes neatly, with a substantial increase in complexity in moving from heating to melting to keyholing. A fourth area which receives considerable attention and could be regarded as an area in its own right is laser-materials interactions, but this is predominantly applied to keyholing problems.

Figure B.2: Levels of modelling in laser processing

We can also envisage a hierarchy of modelling approach, starting from (a) simple analytical methods based on judicious use of dimensional analysis, experimentation and a degree of empiricism, running through to (b) multi-physics numerical methods with as much physical detail as possible. The first approach has progressively less of use to say as process complexity increases, while the reverse is true for the second category. Figure B.2 shows a schematic of the levels of modelling found in laser processing, summarizing the characteristics of the two limiting modelling approaches.

B.2.2 Current modelling methods

Modelling of laser processing, depending on the particular process, can include prediction of temperature fields, residual stress fields, phase transformations (at high heating and cooling rates), and structure-property relationships. Few complete process models exist, even for the least complex processes such as transformation hardening. This reflects the large number of process variables, as well as incomplete physical understanding of some of the processes. Continuous wave lasers have received most attention, though pulsed lasers are increasingly being considered. The current Industrial Laser Handbook includes a review of modelling of laser processing [Ion, 1992]. The approaches to modelling laser processes are summarized below, divided into the three main categories of process introduced earlier.

Laser heating

Heating processes without melting predominantly means transformation hardening of steels, but also includes carburising and annealing treatments. Modelling in these cases essentially requires solution of the governing differential equation for heat flow, subject to various boundary conditions for different distributions of laser energy as delivered on the sample. Numerical approaches abound, either by integration of point source solutions or finite difference methods. Analytical approximate solutions have also been derived, when sufficient simplifying assumptions can be made.

The principal advantages of numerical procedures are the ability to include complex boundary conditions such as temperature-varying properties, and to describe complex geometries. It has been argued however that uncertainties in quantities such as surface absorptivity and some of the material thermal properties will render any solution somewhat approximate, so that simple analytical solutions for idealised geometries (calibrated to selected data sets) can be as accurate in comparison with the experimental data, and are sufficient for guiding the first choice of process parameters. Numerical approaches have generally been time-consuming, and have not therefore been tested across wide ranges of conditions and materials.

Computer speed has now made thermal field computations for complex geometries competitive. So far the output of these calculations has not been used as effectively as analytical solutions for predicting microstructure and property evolution or for conducting sensitivity analyses on the effect of changes in the values of single parameters. The intuitive feel which results from single parameter sensitivity can be highly misleading in this, as in any highly non-linear process. However many solutions in the literature have clearly not been "interrogated" in this way, leading to even more misleading statements such as "the model is accurate to within 10% of the data". The stated accuracy may, for example, be with respect to case depth in laser hardening, but the corresponding error in predicting velocity for a given case depth could be a factor of two.

Arguments about imprecise data and parameter measurement abound in all process modelling and in the extreme situation can lead to stagnation. In general, the best available model should be used to guide experiment, consonant with the costs of modelling versus experiment and the asymmetries of information which result (cf. the discussion on this point in the Models and Modelling main section of this report).

Laser melting

Surface melting treatments include laser glazing, remelting, alloying, particle injection and cladding. Cutting of thin sheet usually involves melting, with removal of the melt using an assist gas. Surface melting introduces the complexity of melt pool convection, and changes in laser-material interaction during the treatment. Approximate treatments modify the heat input for the latent heat of fusion, and use averaged thermal properties, in otherwise purely conductive solutions. Combined thermal and fluid flow solutions by numerical methods have been produced. These have demonstrated that the physical processes are much more material specific than heating processes, principally because of the wide variations in the surface tension - temperature behaviour of different materials, in particular, the effect of trace impurities. Convection flows can be either against or with the direction of heat flow, and the number of convective vortices which form in the melt pool depends on the operating conditions in a complex manner. Most models therefore give physical insight into melting processes, are specific to single materials, and have only been tested against limited ranges of data. Reduced modelling of a wider range of computations is needed to aid in parameter selection.

Laser keyholing

Keyholing is the formation of a deeply penetrating vapour cavity by boiling the material, which greatly increases the absorption of laser energy. Deep penetration welding is the primary keyholing process, with the addition of drilling using pulsed lasers. Cutting is the most extensive industrial application of lasers, and is a keyhole process for thick sections. Keyhole behaviour has attracted the attention of many physicists and mathematicians, and has presented a major modelling challenge for many years. Numerical solutions describing heat and mass transport of varying degrees of sophistication have been proposed (see for example [Akhter, et al., 1988; Dowden, et al., 1991; Dowden, et al., 1985; Dowden, et al., 1989; Schuocker, 1990]).

A great deal is still to be learned about keyhole stability and the operative physical mechanisms. Very little is known, for example, about the surface tension behaviour of the melt-vapour interface within a keyhole. Modelling is therefore almost entirely aimed at gaining physical understanding. Laser keyholes are multi-physics problems of the most complex kind, and in some regards demonstrate chaotic behaviour. High-speed photography of keyholes in transparent materials has revealed very turbulent behaviour on a timescale orders of magnitude faster than processing times. Complete models of keyholing are beyond reach, but the challenge remains to describe the behaviour in sufficient detail to be physically realistic while providing output information which is industrially useful. The gap between approximate methods, based only on heat balances, and deep physics numerical simulations is huge in this area of modelling.

B.2.3 Models required

To date, the majority of analytical and numerical studies aimed at developing a deeper understanding of the underlying physics have concentrated on the interaction between lasers and target materials. Little work has addressed the development of "whole process" models of laser based manufacturing processes. Hence a distinction must be made between modelling all the industrially important features and modelling of those features where little understanding currently exists. Using the previous classification of laser based processes as a guide, step changes in this lack of understanding occur between the classes of heating, melting and vaporization processes.

Laser heating

In heating, laser energy can be considered as a surface heat flux boundary condition with specific characteristics. In modelling these processes, for realistic geometries of processed components, the principle difficulties are:

• adaptive meshing
• coupled heat-diffusion and mass-diffusion
• material constitutive models

Adaptive meshing is needed to track the intense, localized heat source and to represent the large thermal gradients in its vicinity. The coupling of mass and thermal diffusion would not appear to be at all close since the diffusivities are orders of magnitude apart, but compositional changes caused by mass diffusion can produce non-linear heat capacity and transformational latent heat effects. Thus a degree of coupling must be modelled in some classes of alloy.

Material models for annealing and transformation hardening are not yet available for more than a very few materials, and prediction of microstructure and residual stress evolution requires very significant further research and experiment, as it does for welding and forging.

Laser melting

The modelling of melting processes entails a step change in the complexity of laser-material interaction and in addition it includes all the modelling issues of the simple heating processes. In models of melting processes, the laser (as a thermal boundary condition) moves away from the conduction problem within the processed material, now becoming a boundary condition of models of the melt pool. Comprehensive physics based models of such processes require that the local model of fluid flow and heat transfer in the pool is coupled to a conduction model in the surrounding solid. The interface between the two models is implicitly located by the solution of the coupled models which again requires adaptive meshing, this time to track the interface between phases. The small size of the interaction zone mean that surface tension effects may be extremely significant, depending on the particular process and the objectives of the simulation project.

Solidification phenomena in these processes are similar to those found in casting and their simulation can be guided by developments in that area, but all the problems of modelling solidification in castings are also applicable to laser melting processes. The resulting development of microstructure and residual stresses are again similar to casting and to laser heating processes.

Laser keyholing

In modelling vaporization processes, the effects of energy provided by the laser are again further removed from the consequent thermal, mechanical and microstructural phenomena that occur in the processed material. It is no longer possible to consider laser energy in the macroscopic sense of a localized boundary condition in a continuum problem.

The development of a deeper understanding of keyhole phenomena requires the modelling of coupled electromagnetic phenomena, compressible gas flow, liquid flow and heat transfer: all within an implicitly bounded deforming region with time-scales of microseconds and milliseconds. In turn, to provide a "whole process" model for manufacturing engineers, this formidable laser-material interaction system must then be coupled to a conduction model in the surrounding solid in order to predict the progression of the keyhole. Again, adaptive meshing is required in the conduction solver to track both the hole’s movement and the severe thermal gradients around it, and microstructure and residual stress models may be required.

B.2.4 Approximate "whole-process" models

Many of the capabilities which "whole-process" models of laser based processes should posses already exist in models of other processes. The unique feature of laser based processing is the increasing complexity of laser/material interaction phenomena as beam energy density increases.

Currently there is a discrepancy between the industrial need for "whole process" models and the capabilities of analytical and numerical "deep physics" models so far developed; since such models focus almost exclusively on the details of laser-material interaction. However, if one takes the view that such models could be considered (regardless of their internal complexity), as boundary conditions on simpler models of the whole process, then it is conceivable that whole-process models could be constructed using currently available mathematical techniques.

Unfortunately the coupled nature of the problems (especially in the melting and vaporization phenomena) along with the complexity of the laser/material interaction models, mean that currently available general-purpose commercial solvers can not be used. This is principally due to a lack of adaptive meshing and the need to transfer data between solvers for different classes of physical phenomena. Such inter-solver transfer is complicated by:

• different mesh sizes and shapes,
• different interpolation methods between mesh-points, and
• different time-scales of the phenomena.

Two coupled processes may only need infrequent synchronization and mutual updating of field variables, and the synchronization frequency may well depend on the value of these, or other field variables. Thus where there are strong non-linearities in, say, heat capacity as a function of temperature, the coupling frequency should be increased. It should be stressed, however, that these difficulties are implementation problems and that the required numerical analysis techniques are available, but need to be implemented in a more appropriate manner. A high level view of such an implementation is shown in Figure B.3, where it can be seen that the laser-material interaction (LMI) model is a generic feature of the "whole process" model. In detail, the LMI model is different for different processing regimes (heating, melting, vaporising). However, the context of the whole model, these differences are very localized.

Figure B.3: A generic software framework for models of
laser-based manufacturing processes

An initial characteristic of whole-process models would be that they could not be expected to be accurate due to their simplistic models of laser/material interaction that are currently available. However, when used as a research tool, this type of model would provide a very rich environment in which alternative models of laser/material interaction could be examined, along with their effects on processed materials. Also the use of these comprehensive, approximate models would be significant in steering the design of the necessary associated physical experiments and measurements - an area of deep complexity in its own right.

As a final point, it is important to ask the question: "What is the benefit of developing such complex deep-physics models, when considerable progress may be achieved through the more simplistic models outlined here ?" One answer is that the "deep physics" approach is the only way to establish fundamental relationships between phenomena that can be observed during the process and: (a) those phenomena which can only be deduced to occur within the laser/material interaction zone, and (b) the effect of these phenomena on the thermal and mechanical state of the processed material. Such relationships are fundamental to the development of real-time control systems for such processes. The important industrial relationship is the connection between measurements that are practical to make, and the internal state of the finished, processed part. The deep physics models are only useful if they aid this connection.

It is worth bearing in mind that care is particularly required when it is as-yet unverified that any fundamental physical processes are generic  across a range of materials and conditions. If the characteristics of the phenomena are highly sensitive, or chaotic in nature, then there would be little predictive usefulness in understanding them in such detail. A modular approach to constructing whole process models would allow specialized models to be tested and verified, while keeping those aspects of the problem which are generic in an available form for use in a different situation.

B.3 Laser processing conclusions

This review of laser process modelling is from the point of view of industrial technology development and deployment, not from the point of view of a research scientist interested in physical phenomena. In this area it would seem however that there need to be stronger links between physicists and industrialists than is generally the case.

Analytical models are appropriate for initial process parameter selection, particularly for the less complex processes. Surface absorptivity is inherently uncertain, and data for thermal properties as a function of temperature are patchy, rendering numerical approaches less accurate than is sometimes claimed especially for melting and keyholing processes. FD solutions for real geometries can however now be made quick enough for on-line use, but have so far not been exploited for microstructure prediction and informed sensitivity analysis.

"Deep physics" models have a role to play in the long-term in developing the more complex processes. To date the emphasis has been strongly on laser-material interactions, and now needs setting in a whole process context. The inherent complexity of these processes thus not only reveals new physical phenomena, but opens up the need for research activity in methods for numerical implementation of the models. Laser-based processing can be viewed as a wide but largely generic field. This may suggest that the many laser process users and developers could benefit by taking a strategic, collaborative view of model development.

"Reduced modelling" is however essential from the start. Industrial application cannot wait for an eventual complete model that covers everything from atomic interactions to microstructure, properties and residual stress. It is more important to deploy some simple models in practical situations so that later, improved models have a route to application. For the most complex processes, this may simply be a database of treatments, organized using simple analytical tools such as dimensional analysis and approximate heat flow.

In all areas of modelling laser processes, conversion of academic modelling expertise into usable software is very noticeable by its absence. There is a need for greater interaction between university and industry, and research institutes could be the mechanism for turning models into software. A particular failing of academic models is the general neglect of process economics’ considerations such as processing rates.

Added pressure to develop selection software for laser processing comes from the steady increase in the number of standards and certification procedures associated with laser processing. Since this situation has long been the case for conventional competing processes (e.g. for welding, or surface hardening) the software and approaches to economic questions used in parallel industries will serve as a useful guide to the laser industry.

B.3.1 Summary

• There is a hierarchy of complexity in laser processing, in moving from heating to melting to keyholing. Both approximate analytical and numerical deep physics approaches have roles to play at all levels.
• Deep physics models are at present too narrow in terms of process variables and materials: so give limited insight and confidence. Laser-material interaction problems have dominated the research effort.
• Multi-physics models require development in numerical methods to handle the solution of problems with many coupled boundary conditions.
• Reduced modelling is important from the first: full models will not be directly useful except for understanding. Thus there are long-term requirements for software and educational aids to enhance the intelligent use of slightly inappropriate but much faster reduced models.
• A great deal of current research activity does not lead to usable software, though this is often claimed as a goal of a project. Research institutes may be an appropriate location for implementation of academic models into user-friendly, appropriate software.
• The development of standards provides an added incentive to develop selection software.
• Economic considerations are conspicuously lacking in most modelling efforts in laser processing, but are of paramount importance from the industrial point of view.

Acknowledgement

The assistance of Dr. John Ion of the Laser Centre at the Welding Institute, U.K., in the preparation of this case study is gratefully acknowledged.

APPENDIX C

Pressing and forging

C.1 Deformation processing

This appendix describes some features of deformation processing in general and includes very brief sections on rolling, stamping and forging. This is not in any way intended to be comprehensive. Examples taken from deformation processes have been used throughout the main body of the report and this appendix gathers these points together in a single place for convenient reference.

C.2 Current status

The sheet pressing industry ("stamping" in North America) is dominated by die and tooling costs. Automotive body panels are the archetypal product with complex surfaces to be formed from uniform thickness sheet stock. Related industries are sheet metal forming (e.g. for enclosures) and deep drawing (e.g. for soft drinks cans) but these can use modular tooling or standard sizes, so production of new dies of variant shapes is not so common.

Open and closed die forging, and shaped-section rolling have a more obvious three dimensional aspect to the material flow during deformation, but accurate prediction of the effects of variant shape in sheet pressing also require full 3D modelling since thickening and thinning of the sheet does occur and is important.

This section on the modelling of pressing and forging reviews the technology from the point of view of industrial technology development and deployment, not from the point of view of a research scientist interested in deformation phenomena.

C.2.1 Modelling

A vast amount of modelling is performed on plastic deformation processes such as forging, pressing, drawing, extrusion and rolling. The published literature, both industrial and academic, on continuum deformation processing probably comfortably outweighs the literature on all other forms of process modelling combined. The reasons are not because of an overwhelming commercial importance of deformation processes (cutting processes are commercially dominant in metal-based industries, and casting and joining are not insignificant), it is because continuum mechanics offers ample scope for intellectually challenging mathematical research without the "messy" complications of actually having to consider the microstructure of real materials (which are rarely dominant in deformation processing).

C.2.2 Commercial example

Forming Technologies Inc., of Ontario, Canada, model stamping of parts from sheet, mostly for automotive suppliers. They use both FE and FD packages on personal computer systems to predict the force-displacement required and also strain path in material. The strain path is subsequently compared with forming limit diagrams. The motivation behind these modelling activities are:

1. improved die design (a cost of C$200k per die is not uncommon). Modelling identifies likely locations of wrinkling and cracking and also predicts springback;
2. prediction of final thickness variation (and thus initial gauge of sheet required). Big weight savings are possible by specifying a thinner gauge, and getting greater thickness needed via the constrained deformation);
3. prediction of the blank shape required, which reduces scrap and post-forming trimming.

The limitations are two-fold: first, friction boundary conditions vary through the process and are difficult to model, and second, the forming limit diagram method is adequate for mild steel, but less proven for HSLA steel, and is inapplicable to aluminium alloys.

More generally, steel stamping is a C$30 billion industry (Canadian dollars), but typically buyers are conservative. Process modellers often have to simulate forming of parts which the stamping company has actually been making for 20 years or more to convince the company that it really works. The conservativeness is possibly related to a relatively low level of materials understanding. Even in the stamping companies, typical users of FE packages assume that all materials are regarded as being completely described by a stress-strain curve. In the materials area, the stamping industry itself has no confidence in using HSLA (High-Strength Low-Alloy) steel instead of mild steel, and aluminium alloys are completely beyond their experience.

For deformation processing some of the commercially most important properties are flatness, shape and surface finish as well as mechanical properties.

C.3 Materials deformation

Deformation processing is performed both "hot" and "cold" with respect to recrystallization. Hot processing: forging, hot rolling, extrusion etc., involves very significant microstructural changes and good models nearly always require some degree of metallurgical sophistication. Models of cold forming processes are less obviously affected by microstructural variation and most of the complexity arises from the continuum mechanics of anisotropic, strain-path dependent yield.

Large strain deformation at high strain-rates, as occurs in forging and rolling, generates heat so that the thermal and mechanical models have to be closely coupled (large strains imply re-meshing). All but a few percent of the work of deformation goes into heat generation, the rest is stored as microstructural defects which then determine the material properties of the product. However the strong coupling does not mean that every "result" depends on the entire process. An extreme case is the strain field which in some forming operations is fixed by the die kinematics and so depends only very weakly on the constitutive law of the metal being deformed. In such cases, when the deformation is almost unrelated to the material strength and microstructure, predicting the stress in the material from the deformation involves extremely "stiff" equations which are very difficult to solve.

Forming

Sheet pressing ("stamping") is a type of processing where the appropriate model depends on the type of alloy. This is illustrated by a contrast between steel and aluminium. The behaviour of a steel sheet, including many types of stainless steel sheet, is well-described by a "forming limit diagram" (FLD: a failure criterion based only on major and minor strains [Chan, 1990; Demeri and Tang, 1992; Tseng, 1988]) which predicts where stretch failure will occur given just the major and minor applied strains. This approach is simply invalid for most aluminium alloys. For many aluminium alloys even the Von-Mises failure criterion is inadequate and finite-element models are hard to use because solutions are "pathologically mesh dependent" [Owen, 1992]. Similar problems pertain to titanium alloys [Meltsner, 1991].

Forging

Recrystallization [Furu, Marthinsen and Nes, 1990] and hot-deformation are the active concern of researchers in aggregate modelling using network models. Network models simulate competitive growth mechanisms with impingement. Using these methods, modern results for solid-state recrystallization and for nucleation and growth of solids in a liquid medium are very similar to analytically-derived Kolmogorov-Johnson-Mehl-Avrami kinetics [Humphreys, 1992]. There have also been attempts to model a small number of deforming metal grains using finite element models where the elements are small enough such that the change in shape and orientation of the grains can be tracked [Lalli, 1992].

Rolling

In rolling sheet it is the thickness of an oxide scale which determines whether the heat transfer coefficient between roll and sheet enters the model: for thick oxide scales (as in steels) the heat transfer coefficient is irrelevant because conduction through the poorly-conducting oxide dominates the behaviour. Heat transfer in rolling aluminium (where the scale is vanishingly small) is dominated not by conduction in a solid but by the heat transfer coefficient between the aluminium and the roll, which is a chaotic stick/slip phenomenon with steam and lubricant pockets.

A single rolling process requires a number of quite different modelling techniques. In the finishing stage, the problem is effectively 2D plane-strain, since the width of the strip is very large compared to its thickness and the roll contact length. For the first pass, however, the width and thickness would be comparable, and very significant sideways spreading occurs during rolling. Models which are accurate for finishing stages can be completely indeterminate for early stages.

The critical properties for modelling rolling are aggregate properties: friction occurs by the formation and breaking of microscopic asperity contacts, heat transfer is an average of lubricated and unlubricated regions, and air gaps. Numerical and computational developments in inverse modelling will be necessary to provide the measurement techniques for these parameters.

Stress/strain/strain-rate/temperature models: currently use total "von Mises strain" (up to recrystallization) as an internal variable to predict microstructural effects. This is adequate for many steels, much less so for aluminium alloys. A number of non-physical "state variable" approaches have been attempted but none based on measurable quantities. Many kinds of "real" state variables which metallurgists might expect to influence microstructural development, e.g. dislocation density, distribution, grain shape, subgrain size, solute levels, dislocation velocity, etc. cannot be measured easily; which of these are 1st order variables (if texture is important, possibly all of them!) is not known. Very similar stress-strain behaviour can arise from a very wide range of microstructures, so a state variable approach may never be sensitive enough for many purposes.

One problem remaining to be solved in hot-rolling is the deformation of the material before the roll gap (as the material is sucked in), as in most models at present there is an abrupt change of direction at the surface on entry. Difficulties with friction boundary conditions in the nip compound this and lead to excess forward shear strains at the surface (and to a significant depth below - it is not a localized problem [Sellars, 1990; Sellars, 1992]).

The FE models of deformation processing at the cluster of grains level all use the Taylor-Bishop-Hill approximation of imposed uniform remote strain or just consider 2D solutions, which are unrealistic. This is an area of research where significant computation will be required. The output for manufacturing engineers will have to be better reduced models, or explanations of how to derive simple models from experiment for common classes of materials.

C.4 Conclusions

• Forgings may have complex shapes thus this type of manufacturing process determines directly some of the geometric modelling problems.
• Remeshing research requires collaboration of a particularly close nature simultaneously in numerical methods, geometry and physical metallurgy.
• Forging and pressing are industries where the cost of machining new shapes of dies and moulds is very significant.

APPENDIX D

Direct-chill casting (Aluminium alloys)

D.1 Modelling status

Direct-chill (DC) casting is the secondary process where liquid metal is continuously cast (in a water-cooled lined aluminium mould, followed by direct water sprays) into a strand which is directly drawn off to be heat treated, rolled or extruded. This appendix describes only DC casting of aluminium alloys, based on the work of Hydro Aluminium in Norway, whose modelling activity in this area has been in progress for over twenty years. This section on the modelling of DC casting reviews the modelling from the point of view of industrial technology development and deployment, not from the point of view of a research scientist interested in casting phenomena.

Modelling of DC casting of aluminium alloys is carried out by the large aluminium producers whose size enables them to provide continuous support for industrial modelling teams and to maintain long-term university-industry research partnerships. The structure of academic funding and higher education in Norway promotes particularly close relationships between Norwegian industry and the universities and research institutes. This means that modelling sophistication is higher and commercial application of models is greater than for many other materials processing operations.

Some of the software has been developed with the Institut for Energiteknikk. This Institute grew from a nuclear laboratory, and is now largely industry-funded doing "contract modelling". They maintain a fundamental research program on solvers, and have assembled a library of methods, which are applied on short or medium-term contract work for industry. This seems to provide a national resource in process modelling which enables companies to conduct modelling which they could not afford to support continuously in-house. It is interesting to ask whether the UK could benefit from establishing its own Institute of this type.

D.1.1 Why models are needed

A number of operational needs motivate the modelling effort at Hydro Aluminium. The particular processing demands on process models are:

1. to help predict the optimum speed for production and cooling regime that runs a minimal risk of "break-out" where liquid metal escapes.
2. the design of the starting blocks which fill the base of the mould cavities when casting begins. The aims are to minimize the length of ingot required to reach an equilibrium thermal field, and to avoid cracking on cooling. Starting blocks are expensive to produce, and even more expensive to run as trials, so this problem has many of the same organizational characteristics as die and tooling design in stamping and forging: variant shapes to be produced in short time scales with costly experimental iteration to correct design faults.
3. to control microstructure: both segregation across the ingot, and local variations in dendrite microstructure near the surface. The proportions and distributions of second phases formed during solidification and cooling are also important in determining subsequent homogenization and extrusion treatment conditions.

In DC casting the primary concern is maximising productivity. The importance of product reliability depends on the end-use of the material; in aerospace this is of over-riding importance to the aluminium producer, for common "structural alloys" less so. It is fair to say however that end-users generally are getting more stringent in all areas of use of aluminium, so consistency of quality is increasingly important, and provides a strong motivation for many modelling activities.

An additional general reason for having a modelling effort is to assemble know-how within the company, making it possible for new personnel to learn a process quickly.

D.1.2 Software

The specification for process models of DC casting of aluminium alloy are the prediction of: the thermal history of the ingot, with particular attention to surface heat transfer conditions; deformation and thermal stresses produced on solidification and cooling in the part of the billet close to the starting block; spatial distribution of microstructure. These three aspects are dealt with by a software series ALSIM, ALSPEN and ALSTRUC respectively. At present these programs are decoupled, but in future will become more closely integrated. They are described in turn below (further details may be found in the literature [Brobak, et al., 1991; Fjaer, et al., 1992; Fjaer and Mo, 1990; Fossheim and Madsen, 1979; Henriksen and Jensen, 1993; Jensen, 1980; Vorren and Brusethaug, 1987]).

ALSIM

ALSIM calculates heat flow in the casting and starting block. Cylindrical billets are based on axisymmetric 2D FD methods; rectangular rolling ingots use 3D FE (the distinction in this geometry being the need to allow for some fluid flow). Lagrangian solutions are used throughout. The mushy zone is dealt with by the common methods of using an effective heat capacity and latent heat sources. The solid fraction as a function of temperature and cooling rate for a given alloy is an input function (determined using ALSTRUC).

The most important feature of the computation, to which a considerable experimental and modelling program has been dedicated, is heat transfer between the billet and the mould and cooling media. First there is the complication of an air gap forming between the billet and mould as the billet contracts, leading to a sharp drop in heat transfer. Control of this air gap is critical for avoiding break-out of liquid metal. Below the mould water sprays provide cooling, but determination of the region in which steam forms is important. Hence allowances are made for heating of the approaching water as it passes through the mould.

The main uses of ALSIM are as follows:

1. prediction of sump depth, and how this evolves with length of billet (i.e. when is steady state achieved). The metallurgical importance of this is that it controls the likelihood of cracking, due to contraction stresses.
2. prediction of cooling history, and hence dendrite arm spacing (using a semi-empirical dependence on time-at-temperature) and its variation across ingot. The metallurgical importance is that local reductions in cooling rate in near-surface regions (due to transients in surface heat transfer) lead to coarser dendrite structures. In rolling ingots this is a problem as it upsets subsequent anodising, so the surface of rolling ingots has to be machined off first (and this is very expensive).
3. effect of starting block shape: the temperature history is fed into ALSPEN to predict the formation of air-gap between billet and starting block due to thermal contraction. At present the fact that the gap forms is not fed back into ALSIM to give modified heat transfer. This simplifying approximation will in due course be removed by fully coupling ALSIM and ALSPEN. The primary concern is to avoid cracking due to differential thermal contraction. This is not caused by constraint imposed by the starting block, as the billet is able to peel away from the block, but the block does strongly influence the thermal history of the first part to solidify. It is for this reason that full coupling of the thermal and deformation histories is now sought. The initial starting block temperature may be freely chosen, but thermal deformation of the block itself is neglected, it is assumed to be rigid.

ALSPEN

ALSPEN is a separate Lagrangian FE package, which calculates the strains and billet distortions near the starting block due to solidification and contraction of the billet. The thermal history comes from ALSIM, at present unmodified by the distortion, but to be coupled in future. A critical strain criterion for cracking is used, but increasingly the limitations of this are apparent and microstructural criteria are also sought. This is inherently difficult since the microstructural origin of cracking is trace impurities on grain boundaries. The computations of ALSPEN have improved control of DC casting and greatly reduced the trial-and-error associated with starting block design and testing.

ALSTRUC

ALSTRUC is a PC-based package of Turbo Pascal routines to predict certain metallurgical quantities. Its main purpose is to calculate composition and density of primary particles (dispersoids) and secondary particles (e.g. Mg2Si), and remnant solid solution. It includes 5 elements (Al, Si, Mg, Mn, Fe) and thus covers a range of 1000/3000/5000 and 6000 series alloys. The models are semi-empirical based on extensive experimental data, but it is non-equilibrium, including the effect of cooling rate. The program is described in more detail by Langsrud et al. [1992].

The main uses of ALSTRUC are the prediction of:

1. the solid fraction as f(T, dT/dt) for use in ALSIM in the mushy zone.
2. the microstructural variation across billets, using the thermal history output from ALSIM.

The as-cast structure is important input for determining homogenization schedules after casting. Homogenization must allow dissolution of secondary particles, and iron-out microsegregation. Note that in this context quality control is much more important than faster throughput: homogenization is a slow, continuous process anyway, so provided the equipment can handle the throughput from the casters there is no incentive to speed it up, only to get consistent downstream processing and properties.

Recent work with on quench sensitivity of heat treatable aluminium alloys after extrusion also uses ALSTRUC output to calculate dispersoid density. This work is discussed in Appendix E on the modelling of aluminium extrusion.

D.2 Summary

• The mature industry-university links, particularly in Norway, mean that future changes in software and hardware will not affect technology transfer since it is already excellent. An Institute specialising in numerical methods applied to a wide range of industries provides a national facility for Norway in process modelling, and could be a model for the U.K.
• Modelling of DC casting requires research collaboration simultaneously in numerical methods, geometry and especially physical metallurgy.
• The thermal problem is dominated by very complex boundary conditions, with steep transients in heat transfer coefficient down the surface of the ingot.
• The defect criteria which lead to cracking are local fine-scale physical problems - simple criteria such as critical tensile strain can never be 100% safe, but are good enough for most purposes.
• Starting block heat flow modelling is an activity like injection moulding, pressure die casting, and rolling or extruding complex sections, where the cost of machining new shapes of dies and moulds is very significant.
• Starting blocks have complex shapes thus this type of process determines directly some of the geometric modelling problems. New shapes of starting block require new process settings: modelling can have big payoffs because FD and FE can cope with arbitrary shapes where human personal experience cannot.

Acknowledgement

The assistance of Dr. Ole Myhr of Hydro Aluminium, Sunndalsora Research and Development Laboratory, Norway, in the preparation of this case study is gratefully acknowledged.

APPENDIX E

Extrusion of aluminium alloys

E.1 Modelling status

This appendix describes modelling of extrusion of aluminium alloys, based on the work of Hydro Aluminium in Norway. Extrusion is the major shaping route used by Hydro Aluminium for structural alloys, using cylindrical billets produced by direct chill (DC) casting. As in the previous case study, this section reviews the modelling of extrusion from the point of view of industrial technology development, not from the point of view of a research scientist interested in hot working phenomena.

E.1.1 Why models are needed

The overall goal of the modelling program is to produce a 3D thermo-mechanical and microstructure model, from billet pre-heating through to product ageing. Modelling is seen as a means to achieve many goals:

• improve product reliability, quality and productivity
• improve die design
• improve process control systems
• aid development and specification of new equipment
• gain knowledge and education (visualization, process parameter studies, obtain a scientific explanation of observed phenomena)

General process optimization for reliability, quality and productivity is the primary concern. In the context of extrusion product quality includes surface finish as well as shape and properties. Modelling is particularly powerful in extrusion as production involves a huge range of extrusion shapes (several thousand per year). In addition, every one of the company's extrusion presses is different from the others, as are the cooling arrangements that go with it, hence ideally production can be tailored to the characteristics of the specific machines which will be used.

Die design is viewed in relation to the influence of the dies on the product, the influence on extrusion velocity (i.e. productivity) and on improving die life. So far the emphasis has been on the influence on the product. Deformation around extrusion dies is very complex. This is a critical area because the flow pattern there largely controls product distortion and properties. There is also the need to "tune" new dies. Bought-in dies rarely perform perfectly as received, but require some "tuning", i.e. localized machining and repolishing. This is very expensive so it is attractive to make more predictions of die performance first, particularly when the number of different shapes is so large. Most press equipment and dies are bought from external suppliers, so when investing in new equipment Hydro wish to maximize their influence on the design.

Note that Hydro view modelling as a means to develop human resources in the company, as well as technology. Well-designed graphics based on model predictions serve as powerful visualization tools, particularly as almost all of the actual deformation is completely invisible on the plant itself. Modelling also serves to establish common interpretations of observed phenomena such as distortions and defects.

The current phase of Hydro Aluminium's extrusion modelling project, in collaboration with the Norwegian Institute of Technology in Trondheim, draws together expertise in production engineering, numerical methods, solid mechanics, and metallurgy. In common with so many of the modelling problems discussed in this report, extrusion is a very inter-disciplinary activity. University courses in materials and manufacturing vary widely from place to place, but a good synthesis of disciplines is rarely achieved. It appears that progress in modelling could be greatly enhanced by more integrated inter-disciplinary training. Furthermore, mechanisms for building process-modelling research teams across many institutions are frequently very productive.

E.1.2 Software

The stages in the extrusion process are: preheating, pressing, die interaction, cooling and stretching (followed by cutting and heat treatment). The only feature of preheating from the modelling point of view is the use of a "temperature taper", a deliberate initial temperature gradient to give a more uniform product. The deformation stages get the most attention in modelling.

The primary specification for process models of extrusion of aluminium alloy are thus the prediction of thermo-mechanical behaviour and microstructures formed in the press, the die and on cooling. The thermal and deformation history of the billet begins within the press, and then continues in the much more complex geometrical situation in the die (when particular attention must be paid to surface heat transfer and friction conditions). Deformation and thermal stresses are produced on cooling and stretching the extruded product and all thermo-mechanical history affects the microstructure and properties of the product.

The different demands at different stages in the process mean that FE computations are a combination of Eulerian (fixed mesh, flowing material) and Lagrangian (moving grid, remeshing). The current methods used are described below. Some further details are available in the literature [Herberg, et al., 1992; Herberg and Skauvik, 1993; Holthe, et al., 1992; Skauvik, 1993; Venas, et al., 1992]

Preheating/pressing: this stage uses a 2D Eulerian package, ALMA-2Π, developed at the Norwegian Institute of Technology. A 2D formulation is used rather than 3D because a full thermal balance is needed, including temperature history of ram, chamber wall, and die, as well as the billet itself. A plastic material response is assumed.

Dies: the deformation is inherently 3D, and is modelled using the commercial 3D Eulerian code, FIDAP-3D. This can handle most extrusion cross-section shapes (including multiply-hollow sections with porthole dies). A plastic material response is again assumed. It is felt that the limitations to the deformation model are almost entirely in heat transfer and friction, rather than in inadequate stress-strain-temperature characterization.

Cooling/stretching: the principal goal here is to predict distortion due to differential contraction, hence a full elasto-plastic material response is required. Cooling and stretching are modelled together, as the purpose of stretching is to relieve residual stress and to align the extrusion longitudinally. The choice then is ABAQUS-3D, a Lagrangian code.

Flow through extrusion dies

These computations are very intensive in computer power, typically 24 hour runs on workstations, provided initial conditions are well-conditioned. Progressive mesh refinement is needed to get sufficient accuracy within a sensible timescale. This places considerable demands on user skill.

The greatest challenge is to develop physical boundary conditions and a material model which are able to describe the behaviour in the localized deformation zones close to those boundaries. The current model, consisting of sticking friction at the boundary and a standard Zener-Holloman material response (which compensates the stress-strain curve for temperature and strain-rate), tends to predict surface temperatures which are too high. This is probably because the Zener-Holloman description fails to give a sufficient drop in viscosity as melting is approached. A modified Zener-Holloman parameter however complicates the numerical procedures considerably. Ideally more physically realistic kinematic boundary conditions are desirable (such as stick-slip behaviour, perhaps due to local melting). Physical understanding in this area is however very limited.

Predictions are also required about the two types of welds which take place in extrusions: (a) longitudinal welds, the seams where a split flow is remade to form an enclosed section; (b) transverse welds, the joint between the former butting faces of two consecutive billets in the press. Transverse welds emerge heavily stretched out along the length of the extrusion. Their location and history are important, as they include trapped oxide and also have a different recrystallization response, which can give subsequent problems with corrosion and fatigue. They may also influence formability and toughness.

Cooling stage

The cooling calculations predict the evolution of temperature across the whole section with time, and calculate distortions. For most extrusion speeds, heat flow is almost entirely lateral. Heat transfer is complicated by the intricacy of section shapes (fins, hollow sections etc.), and by the use of a combination of air and/or water cooling. 100% air cooling is generally computationally tractable, but water quenching is much more difficult. Internal ligaments are often a problem: heat transfer to an internal air cavity is negligible, so ligaments cool slowly by conduction into the outer regions of the extrusion. This leads to the largest thermal gradients and consequently to local yielding and distortion. It also leads to irregular thermal cycles: regions near internal ligaments can locally cool more slowly if the external quench is interrupted at some point. Surface markings and local changes in microstructure can therefore result.

The most important aspects of the cooling stage are warping (in-plane, and longitudinally) and quench sensitivity (inadequate quench rate giving loss of age hardening potential). These are in competition: fast quench for maximum subsequent strength, slow quench for minimum distortion. If both distortion and quench sensitivity could be predicted, cooling schedules could be optimized for every section shape. Quench sensitivity models are becoming available but so far only predict the effect on strength. Fatigue, corrosion etc. are in general much more quench sensitive, but microstructural models for these problems are long-term challenges for the future.

Microstructural issues

Coupling of the stages in production is ultimately desirable. This incudes tracking back to the DC casting and homogenization stages. Certain types of defect may arise due to effects introduced during casting, for example surface streaking. These could however be interpreted as problems due to die wear and models could help isolate the phenomena. Other examples of the influence of as-cast structure on extrusion are the need to dissolve secondary particles during extrusion (for subsequent age hardening), and the dependence of quench sensitivity on dispersoid structure and alloy composition.

Some progress has been made to carry microstructure beyond the casting stage, using the program ALSTRUC (described in Appendix D). There are many instances in aluminium processing (and other industries) where microstructural features carry through from an earlier production stage, leading to downstream processing effects. Awareness of these effects tends to be patchy and depends principally on the experience of the research staff. Process modelling methods incorporating microstructure offer great potential for monitoring the influence of these features in a global way through the whole processing history (and in principle beyond, if further structure evolution continues during component life). It is also frequently found that attempts to model processes highlights the areas of underlying materials science and engineering in which knowledge is inadequate, thereby motivating further research efforts, which are industry-led.

Texture within the press and beyond has not generally been a major issue in extrusion, compared to rolling of thin sheet for example. Variations in texture can lead to dark surface streaks, which are unacceptable for architectural applications. Texture is becoming more important due to the interest in automotive applications, when it begins to affect forming limits as well as leading to poor surface finish. The present capability is to take the strain history of a given point and run it through a Taylor analysis to get texture components; which it is hoped will continue to prove sufficient.

Real-time optimization of extrusion presses

Hydro Aluminium have implemented a real-time optimization system called the OMEX Intelligent Press. Traditional press operation involved observing surface finish, estimating acceleration time (from the sound), estimating productivity (from the extrusion speed), and adjusting the billet temperature and ram speed manually.

The OMEX system, in its first form, took extrusion pressure, ram position and speed, and measured billet temperature and calculated (crudely) the exit surface temperature. A recommended billet temp and speed were output (based on a forming limit diagram), and the operator would then make a visual check of surface finish. The latest version has more on-line measurement of ram position, extrusion pressure, billet temperature, and real-time measurement of exit surface temperature. Changes to ram speed are still fed back manually if required. Exit temperature calculations are necessarily crude, as the turnaround on an extrusion press is fast (30 seconds to change a die, and exit speeds of the order of a metre a second).

E.2 Summary

• Modelling of extrusion requires research collaboration simultaneously in production engineering, numerical methods, solid mechanics and physical metallurgy.
• The common hot working material response based on the Zener-Holloman parameter is generally adequate for this process except at temperatures approaching melt. The application is particularly demanding as the gradients in temperature and strain-rate are very steep: the deformation is concentrated in a very small volume within the complex 3D die geometry.
• The thermal and flow problem has complex boundary conditions, possibly with stick-slip friction which at present must be approximated .
• The demands on models include the common requirements in thermomechanical processing (selection of process conditions, distortion, and product properties) but also include less common demands such as surface finish control.
• Computer power is a limitation in extrusion of complex cross-sectional shapes. Considerable user experience is needed with boundary conditions to get stable solutions.
• Some progress is being made to integrate microstructural features through from DC casting to extrusion behaviour and product properties.
• The models have played an important part in process visualization for staff training, and for supporting development of a moderate package of real-time optimization.

Acknowledgement

The assistance of Dr. Stig Tjotta of Hydro Aluminium, Karmøy Research and Development Laboratory, Norway, in the preparation of this case study is gratefully acknowledged.

APPENDIX F

List of interviewees

F.1 Some of the people consulted

This is a list of some of the people who kindly gave some of their time to answering our questions and showing us something of their work. The views represented in this report and the conclusions, however, are those of the authors alone.

Dipl.-Ing. Gerald Aengenheyster Rechnergestützte Formteilauslegung CAD/CAE,
IKV (Institute für Kunststoffverarbeitung)
RWTH (Rheinisch-Westfälische Technische Hochschule)

Prof. Mike Ashby Dept. Engineering, University of Cambridge

Mr. Jinka Ashoka Dept. Civil Eng., University College of Swansea

Dr. David Aspinwall Dept. Manuf. & Mechanical Eng., University of Birmingham

Dr. Chris Bailey Centre for Numerical Modelling and Process Analysis
University of Greenwich

Dr. Peter S. Bate School of Metallurgy and Materials,
University of Birmingham

Prof. Christof Beckermann Dept. Mech. Eng.,
University of Iowa

Dr. J. Beech Dept. Engineering Materials
Sheffield University

Dr. John Beynon Dept. Engineering
University of Leicester

Prof. Yves Brechet Institut National Polytechnique de Grenoble

Prof. John Bridgewater Dept. Chem. Eng., University of Cambridge

Dr. Steven Brown Dept. Materials Science, University College of Swansea

Prof. John Campbell IRC in Materials for High Performance Appls.
University of Birmingham

Dr. Brian Cantor Dept. Science of Materials, University of Oxford

Dr. Mick Cardew-Hall Engineering Program
Australian National University

Dr. Leo Christadoulou BDM International

Dr. Alan Cocks Dept. Engineering, University of Cambridge

Dr. James Crilly Unilever Research,
Colworh Laboratory

Prof. Mark Cross Centre for Numerical Modelling and Process Analysis
University of Greenwich

Mr. David Davies Process Modelling Group
Rolls Royce

Dr. Brian Davies Mech. & Ind. Eng. Dept.
University of Illinois

Dr. Trevor Dean School of Metallurgy and Materials,
University of Birmingham

Mr. Adrian Demaid Faculty of Technology
Open University

Dr. Brian Derby Dept. Science of Materials, University of Oxford

Prof. Peter Dew Dept.Computer Science
University of Leeds

Dr. D. Dunne Dept. Mech. Eng., UMIST

Dr. Lyndon Edwards Faculty of Technology
Open University

Prof. David Embury Dept. Engineering Materials
McMaster University

Prof. Russell Evans Dept. Materials Science, University College of Swansea

Dr. Norman Fleck Dept. Engineering, University of Cambridge

Dr. Stephen Flood Alcan Intl. Ltd.
Banbury Laboratory

Dr. Suresh Garimella Mech. Eng. Dept.
University of Wisconsin

Dr. D.T. Gethin Dept. Mech.Eng., University College of Swansea

Dr. Lawrence Grasty Ling Dynamic Systems

Prof. Øystein Grong Dept. Metallurgy
University of Trondheim

Dr. Peter Hartley Dept. Manuf. & Mechanical Eng., University of Birmingham

Dr. Fredy Hediger Gießerei Institut,
Rheinisch-Westfälische Technische Hochschule

Dr. John Herberg R & D Centre
Hydro Aluminium a.s.

Dr. Joe Herbertson BHP Research

Dr. Graham Hollox Dept. Engineering, University of Cambridge

Prof. John Humphreys Manchester Materials Science Centre,
University of Manchester & UMIST

Dr. John Hunt Dept. Science of Materials, University of Oxford

Dr. John Ion Laser Centre
Welding Institute

Dr. Kai Karhausen Institut für Bildsame Formgebung,
Rheinisch-Westfälische Technische Hochschule

Prof. Laurens Katgerman Lab. of Materials
Delft University

Prof.Dr.-Ing. Reiner Kopp Institut für Bildsame Formgebung,
Rheinisch-Westfälische Technische Hochschule

Dr. Stephen Kukureka School of Metallurgy and Materials,
University of Birmingham

Dr. Lawrence Lalli Alcoa Technical Center
(Pittsburgh)

Dr. Richard Le Sar Center for Materials Science
Los Alamos National Laboratory

Prof. Roland Lewis Institute for Numerical Methods in Engineering, University College of Swansea

Dr. David Lloyd Alcan International
Kingston R & D Centre

Dr. Knut Marthinsen SINTEF

Dr. Robert McAdie Institute for Numerical Methods in Engineering, University College of Swansea

Dr. Anita Mehta IRC in Materials for High Performance Appls.
University of Birmingham

Dr. Ole Midling R & D Centre Karmøy
Hydro Aluminium a.s.

Dr. Ole Myhr Sunndalsöra Metallurgical R & D Centre
Hydro Aluminium

Prof. Erik Nes Dept. Metallurgy
University of Trondheim

Dr. Tony Nutborn FEGS Ltd.
Oakington

Dr. Tor Oscar-Saetre R & D Centre Karmøy
Hydro Aluminium a.s.

Dr. Anand Paul Engrg. Analysis
Concurrent Technology Corp.

Dr. Koulis Pericleous Centre for Numerical Modelling and Process Analysis
University of Greenwich

Dr. Susan Pulko Dept. Electrical Eng.,
University of Hull

Prof. Michel Rappaz Lab. Métallurgie Physique
Ecole Polytechnique fédérale de Lausanne

Dr. Oddvin Reiso Sunndalsöra Metallurgical R & D Centre
Hydro Aluminium

Dr. Guy-Michel Reynaud Pechiney, Centre de Recherches de Voreppe

Dr. John Richmond T&N Technology

Dr. Owen Richmond Alcoa Technical Center
(Pittsburgh)

Dr. Steve Roberts Dept. Science of Materials, University of Oxford

Dr. Pierre Sainfort Pechiney, Centre de Recherches de Voreppe

Dr. Mark Samonds Universal Energy Systems

Dr. Vivek Sample Alcoa Technical Center
(Pittsburgh)

Prof. Mike Sellars Dept. Engineering Materials
Sheffield University

Dr. Ravi Shahani Pechiney, Centre de Recherches de Voreppe

Dr. Inge Skauvik R & D Centre Karmøy
Hydro Aluminium a.s.

Dr. P.Julian Spence Process Modelling Group
Rolls Royce

Dr. John Spittle Dept. Materials Science, University College of Swansea

Dr. Jörg Sturm MAGMA GmbH
Gießereitechnologie

Prof. Julian Szekely Massachussetts Institute of Technology

Dr. Stig Tjøtta R & D Centre Karmøy
Hydro Aluminium a.s.

Dr. Chris Turner Marmara Research Centre
Turkey

Dr. Hervé Víchery Pechiney, Centre de Recherches de Voreppe

Dr. Srinath Viswanathan Oak Ridge National Lab.

Prof. Brian Wilshire Dept. Materials Science, University College of Swansea