Advanced Numerical methods for F. E. Simulation of Metal

Forming Processes

Jean-Loup Chenota, Marc Bernacki

a, Lionel Fourment

a and Richard Ducloux

b

aMINES ParisTech, CEMEF - Centre de Mise en Forme des Matériaux, CNRS UMR 7635, BP 207 1 rue Claude

Daunesse 06904 Sophia Antipolis cedex, France

bTransvalor S. A., 694, av. du Dr. Maurice Donat, 06255 Mougins Cedex, France

Abstract. The classical scientific basis for finite element modeling of metal forming processes is first recalled. Several

developments in advanced topics are summarized: adaptive and anisotropic remeshing, parallel solving, multi material

deformation. More recent researches in numerical analysis are outlined, including multi grid and multi mesh methods,

mainly devoted to decrease computation time, automatic optimization method for faster and more effective design of

forming processes. The link of forming simulation and structural computations is considered with emphasis on the

necessity to predict the final mechanical properties. Finally a brief account of computation at the micro scale level is

given.

Keywords: Plasticity, Numerical Analysis, Finite Element, Metallurgy, Metal Forming, Micro modeling.

PACS: 61.72.Bb, 62.20.fq

1 – INTRODUCTION

The first attempts to utilize the finite element method for metal forming simulation were achieved by several

scientists with their pioneer works in the seventies, treating the feasibility of simple geometries in two dimensions

[1-4]. More than ten years after, three-dimensional computations were attempted on very simple geometries [5].

Under the pressure of industrial needs and taking advantage of the rapid increase of affordable computer

performance, many developments were tested in laboratories and introduced in commercial codes. To-day there is

still a strong demand from many companies:

- to improve the reliability,

- to decrease the execution time, notably by utilizing the power of parallel computers,

- to treat more complex problems,

- to be able to optimize automatically industrial processes,

- to perform several thermal and mechanical treatments successively,

- and to introduce more realistic physical behavior.

In this presentation we review the main recent developments implemented in Forge3, a commercial and research

multi purpose computer software.

Moreover we analyze a very promising field for improving the physical description of materials at the

microscopic scale. This approach, which is also based on a Finite Element discretization, is now considered as an

alternative, or a complement, to the expensive experimental identification of incremental metallurgical laws, which

are coupled with the macro computation.

2 - BASIC FORMULATION OF THE FORMING PROBLEM

The classical finite element approach was described in details in [6], to which the interested reader is referred, as

only a very brief summary will be recalled here.

2.1 – Mechanical and Thermal Description

For cold or hot forming processes, an elastic plastic or viscoplastic constitutive equation is utilized. A

widely used approximation, which reveals accurate enough in most applications, is based on an additive

decomposition of the strain rate tensor into an elastic part e and a plastic (or viscoplastic) one p :

e p (1)

The Jauman objective derivative of the stress tensor is introduced and the hypo elastic law is written:

e e e eJdtrace I 2

dt( )

(2)

where e ande

are the Lamé coefficients. The plastic or viscoplastic component of the strain rate tensor obeys a

Perzyna rule:

1

eq eq3 2σ (σ R)/K/

ε / σ'mp (3)

where eq is the equivalent strain, ’ is the deviatoric stress tensor, is the equivalent strain rate and is the

equivalent strain, K, R and m are material parameters.

Isotropic work hardening can be modelled if K depends on the equivalent strain . Kinematic anisotropic hardening

is defined by introducing the back stress tensor X and replacing by - X in the expression of eq.

At the interface between part and tool the friction shear stress can be modeled by a “viscoplastic Coulomb” law:

1 p

f n v v/

(4)

Where αf and p are friction coefficients, n is the normal stress and Δv is the tangential velocity.

For an incompressible or quasi incompressible flow, it is desirable to utilize a mixed formulation. In the domain

of the part, this formulation can be written, for any virtual velocity and pressure fields pv*, * :

c

dV pdiv v dV v dS 0: * ( *) *

(5)

And the mass conservation equation:

div v p p*dV 0( ( ) )

(6)

Where is the compressibility coefficient of the material.

The most widely used time integration method remains the simple one step Euler scheme. In this simple

approach, the total time is decomposed into small increments t and the displacement field is proportional to the

velocity field at the beginning of the increment:

u t v (7)

In the same way the stress increments are introduced, so that eqs. (5) and (6) are rewritten:

c

dV p p div v dV v dS 0( ) : * ( ) ( *) *

(8)

p t div u p dV 0( / ( )) : *

(9)

For hot forming process the heat equation is introduced:

cdT dt div kgrad T r 0/ ( ( )) : (10)

Where is the material density, c the heat capacity, k the thermal conductivity and r the fraction of plastic work

transformed into heat. The thermal and mechanical coupling comes from heat generation by plastic work, thermal

dilatation which modifies eq. (9), and the dependency of the material parameters on temperature, e.g.:

n

0 0 0 1K K ε ε exp( , m m m TT / ) (11)

2.2 – Finite Element Discretization

Many different finite element formulations were proposed, and developed at the laboratory level, but it is now

realized that the discretization scheme must be compatible with other numerical and computational constraints. To-

day a satisfactory compromise is based on a mixed displacement and pressure formulation. The pressure field is

discretized using tetrahedral elements, and a bubble function is added to the velocity or the displacement field, in

order to stabilize the solution for incompressible or quasi incompressible materials. The increment of displacement

field is discretized with shape functions Nn, in term of nodal increment of displacement vectors Un: n n

n

u U N ( ) (12)

Using isoparametric elements the mapping between the physical space with coordinates x and the reference space,

with coordinate is:

n nn

x X N ( ) (13)

The strain increment tensor can be computed with the help of the conventional B discretized linear operator:

n nn

U B ( ) (14)

After time discretization, the mixed integral formulation for the mechanical problem is:

c

Un n n f n n

vR ( ) dV (p p) trace(B )dV N dS 0

v

(15)

R ( ( ) ) 0

P

m mdiv u p M dVt

(16)

To which the discretized heat equation is added:

t t 0C. T H .T F (17)

Where C is the heat capacity matrix, H’ is the conduction matrix and F is a vector gathering the boundary

conditions and the heat source terms.

At this stage, several methods can be used, depending on the thermal and mechanical coupling. When coupling is

weak enough, the mechanical and thermal problems are solved separately and the coupling is postponed to the next

increment. For intermediate coupling a separate resolution is still used, but fixed point iterations are used. For strong

coupling and localization the global system is solved by a Newton-Raphson method on V, P, and T (or U, P and

T) as mentioned in [7].

3 - NUMERICAL PROBLEMS

3.1 – Meshing and Remeshing

For complex geometries, it is now well recognized that tetrahedral elements are more convenient for initial

meshing of the preform and for remeshing. The remeshing step is compulsory when deformation of the work-piece

results in too distorted elements, which may introduce inaccuracy. In the mixed formulation the pressure and

velocity shape functions must be compatible. The pressure field is discretized with linear elements, while a bubble

function must be added to the linear velocity approximation, in our case the tetrahedron is subdivided into four

tetrahedra and a velocity is added in the center, as is shown in FIGURE 1. In fact the contribution of the bubble

function to the system of equations can be eliminated at the element level by simple matrix algebra, resulting in a

better conditioned system.

FIGURE 1: Interpolation in a mini tetrahedral element

For a more reliable control of accuracy, an estimation of the discretization error is performed and the

elements must be refined locally in the zones where the strain is higher. This is achieved by prescribing a local size

of the elements and the mesh is rebuilt accordingly [8]. But this approach may lead to generate a very large number

of elements. This last drawback can be partly overcome, especially for localized metal flow with shear band, by

introducing anisotropic meshes having narrow elements in the direction of high strain gradient and elongated in the

orthogonal direction [9].

3.2 – Equations Solving

This step is particularly important as for each time increment, several linear systems are generated by the

Newton-Raphson procedure, their resolutions representing the more expensive contribution to the total CPU time.

Iterative methods are effective on the reasonably well conditioned systems we get due to the stabilization induced by

the introduction of the bubble function (which is eliminated before resolution). Moreover these methods are easier to

parallelize, provided a domain partitioning is defined, each sub domain being treated on a separate processor. After

this first step it is now necessary to analyze the parallelization of the other steps of computation: remeshing,

partitioning, contact, etc. [10].

3.3 – Multi Material Coupling

The problem of multi material coupling appears when thermal conduction is taken into account between several

domains, when the tools are considered as elastically deformable, or when a part is formed with several materials.

The thermal problem will not be summarized here as it is the less critical. From a purely theoretical point of view, in

the mechanical problem, there are few modifications with respect to the contact with a rigid body: at the interface

between different materials, we must impose a unilateral contact condition with friction (and possibly with a force of

cohesion). However challenging numerical problems appear to take into account this situation with non coincident

meshes at the interface between materials.

A convenient way to treat this problem is to utilize the “master and slave approach”. In this first method a

Lagrange multiplier (or penalty) contribution of the non linear equations to solve is introduced to avoid penetration

of the slave surface in contact contact

B , into the master surface contact

A . But this approach is effective when the

surface mesh of the slave is more refined than the surface mesh of the master in contact. If this is not the case, it

appears that the action of the slave on the master is not well defined and no effective contact condition may be

imposed on some nodes of the master.

In the fully symmetric Lagrange multiplier formulation, the additional term can be written:

1

( ) ( )2

contact contactB A

SYM B A

A B B B A Ah u ds h u ds (18)

pressure p

velocity v

Where A and B are Lagrange multipliers corresponding respectively to the master and slave. A

h is the distance

between a node of the slave and its projection on the master surface, while B

h is the distance between a node of the

master and its projection on the slave surface. It is obvious that, in general, this formulation is not acceptable as it

introduces too many constraints on the contact surfaces and results in a kind of locking.

Fourment et al in [11] proposed a quasi symmetric formulation which is designed to avoid locking. The

basic idea is to decrease the number of constraint in eq. (18) by replacing the Lagrange multiplier A by B , which

is the projection of B on the surface A

. With a nodal formulation, the quasi symmetric approach imposes a

number of constraints equal to the number of nodes of the slave mesh in contact. This method was applied

successfully to forging with deformable tools.

3.4 – Multi grid and multi mesh

A constant concern in F. E. simulation is to reduce computational time in order to be able to solve more complex

problems, involving always more refined meshes. However the CPU time is not a linear function of the number of

unknowns, even for classical iterative solvers. The multi grid method is potentially a way to achieve a quasi linear

dependence of resolution time and consequently to reduce dramatically the computational cost. In ref. [12] a

promising node-nested Galerkin multigrid method is described for solving huge systems originating from mixed

formulations and linarization of 3D metal forming problems. The smoothing and coarsening operators are built using

node-nested meshes made of unstructured tetrahedra. The coarse mesh is only used to accelerate the resolution of

linear systems; they are built by an automatic coarsening algorithm based on node removal and local topological

remeshing techniques. A research version of the FORGE3 finite element software is utilized to test the effectiveness

of the multigrid solver, for several large scale industrial forging problems. In particular, the linear rate of

convergence of the methods is verified on different scales simulations and the decrease of CPU time can reach a

factor higher than 7.

Another method for saving computational cost is to use different meshes as developed in ref [13]. In hot

incremental forming, such as cogging or ring rolling, a unique mesh for mechanical and thermal simulation is not the

optimal choice. In these processes, on one hand the mechanical problem is very expensive due to non linearity and

the necessity of four unknowns per node, but the deformation is localized. On the other hand, the thermal problem is

linearized and involves only one unknown per node, while the whole mesh of the part must be refined. A Bimesh

method will take advantage of this particularity by using different finite element meshes for the resolution of the

different physical problems:

- a main fine mesh to store the results and to carry out the thermal computations,

- and an intermediate mesh for the mechanical calculations, which is refined in the zone of localization and much

coarser where deformation is negligible.

The numerical development of the Bimesh method consists mainly in building the embedded meshes and

managing the data transfer between the meshes. The Bimesh method leads to a CPU reduction of about 4 on

industrial examples and is compatible with parallel calculations.

4 – OPTIMIZATION, LINK OF SEVERAL PROCESSES

4.1 – Optimization of a Single Process

The optimization of forming processes has been achieved mostly by trial and error, firstly from experiments on

actual equipments and materials and more and more utilizing a simulation code. But it was realized that these

optimizations can also performed by coupling an optimization module with a F. E. computer code. Several methods

were attempted using complex derivatives of the cost function which represents the targeted goal of optimization

(see e. g. [14, 15]).

To-day the necessity to treat a wide variety of problems, and the availability of relatively cheap parallel

computers, allow us to utilize evolutionary algorithms which need only the computation of the cost function. In

addition, in order to reduce the number of evaluations of the cost function, it can be combined with metamodelling

with a Meshless Finite Difference Method. This method was developed by Fourment et al in [16], fully coupled with

the Forge3 computer code and it is shown on several industrial examples that less than one hundred simulations are

necessary to reach a satisfactory optimum.

4.3 – Structural Computations and Optimization

At first sight it may appear that linking the computations of several processes and transferring the data from one

step of thermal and mechanical treatment to the next one, and finally to the structural computation software, is only a

very basic problem. In fact, due to large deformation and possibly high temperature, the material undergoes

important metallurgical transformations which affect the mechanical behavior. A complete treatment is described in

[17] to evaluate quantitatively the improvement of fatigue resistance which is given by forging to highly loaded

work pieces. Increase of mechanical strength is due to the local evolution of material micro structure which is

induced by large plastic deformation involved in the forging process. At the end of the process, the local mechanical

behaviour of the part is anisotropic and also the fatigue resistance. The skill of the engineer is to take advantage of

this phenomenon to optimize design of forging steps in order that the part will have maximum resistance to in

service load. A micro approach is necessary to evaluate the final orientation of the micro structure after forming and

the corresponding mechanical behaviour. The evaluation of fatigue resistance is performed by structural

computation, using data transfer from the forging steps. The final goal is to be able to perform “integrated

simulation”, from thermal and mechanical treatments to structural computation, utilizing the same software with

automatic data transfer between each step. From the point of view of the final user, this is probably the most

convenient way for complete optimization of the process.

5 – COMPUTATION AT THE MICRO SCALE

5.1 – Feasibility of Multi Scale Coupling

It is well known that the micro (or nano) structure of metals is a key factor for determining the constitutive

law during forming and for predicting the final properties of the work-piece. To treat in an average way, the

evolution of the material micro structure during thermal and mechanical treatments, the classical method consists in

fact in a macro description, selecting representative material parameters (grain size, phase percentage, precipitates,

etc.) and to identify physical laws which govern the evolution of these parameters, and their influence on the

mechanical behavior [18]. The macro approach is quite convenient for coupling thermal, mechanical and physical

computation, but it suffers severe limitations and needs a large amount of experiments to identify the physical laws

describing micro structure evolution.

On the other hand, computation at the micro scale is now possible and is developed for a potentially more

realistic description of materials. Micro modeling is potentially much more accurate but, due to heavier computer

cost at the local micro level, direct coupling with macro thermal and mechanical simulations seems limited to 2D

problems and simple parts, even with large clusters of computers.

One way to view the short term applications is to use micro modeling of material in post processing, to

predict micro structure evolution for a limited number of locations in the work piece, neglecting coupling effects.

Another method is to utilize the micro approach to help identification of macro laws.

The basic ingredients of the general micro model developed at CEMEF are summarized in [19].

5.2 – Generation of Representative Element Volume suitable for F. E. computation

Success in the integration of multiscaling modelling activities requires an effective means to organize the data that

defines structure, and its variability over many scales, and to facilitate its access by a heterogeneous collection of

modelling tools. This requirement will be addressed through the use of a modelling system referred to here as the

Digital Material. A digital representation of the material:

(a) is based on essential geometric features of the microstructure,

(b) and includes statistical distributions that quantify the critical attributes of those features.

An example of an important geometric feature is that of a grain. Grains in a polycrystalline material exist in a variety

of sizes, shapes, and crystal lattice orientations that depend on the processing history. These attributes of the grain

may be described by probability distributions. Collectively, the features and their attributes provide the means to

represent a material’s microstructural state and its natural variability. This leads directly to the ability to determine

statistical response distributions (properties) from the input state distributions (microstructure). The construction of

the Digital Material is based on experimental measurements of these geometric features, and is done either by

mapping exactly the real microstructure, or by forming a statistical equivalent. A convenient way of constructing a

large number of grains is given by the Voronoï tesselation. FIGURE 2 illustrates the current development of the

DigiMicro software developed in Cemef - Mines Paristech. A recursive Voronoï tesselation algorithm is available

and allows us to define regions (grains) and subregions (subgrains). The obtained geometrical features can be

combined according to desired criteria of sizes and shapes, a detailed discussion of this issue is given in ref. [20]. A

sampling algorithm allocates all relevant properties to the different geometries created. Probing algorithms can

measure particular instantiations of a microstructure, e.g. grain size, grain shape distribution, or crystallographic

texture. Each individual grain shape is described by a fitting ellipsoid. A digital microstructure can also be cut along

a prescribed plane, which then allows us to perform digital measurements along the chosen plane, e.g. Electron Back

Scattering Diffraction (EBSD) virtual experiments. A second format based on voxels is available. The Voronoï

format can readily be converted into the voxels format. The voxel format is very useful when data must be

transferred from 3D microscopy experiments, or from other model results using regular grids to discretize the

microstructure. Moreover, numerical algorithms were also developed to take account morphological characteristics

and statistical description of inclusions or porosity in the polycrystal as illustrated in FIGURE 2f , see also ref. [21].

The conversion of the Voronoi tessellation into an FE mesh is illustrated in FIGURE 3. The Voronoi tessellation

is fully described by the N seeds or Voronoi sites obtained thanks the DigiMicro software as shown in FIGURE 3a

The location of the interfaces (solid grain boundaries) is implicitly defined using a level set framework as in [19, 22,

23]. For each individual cell or solid grain, a signed distance function , defined over the domain , gives at any

point x the distance to the solid grain boundary . In turn, the interface is then given by the level 0 of the

function :

0)( ,

),,()(

xx

xxdx

(19)

(a) (b) (c)

(d) (e) (f)

FIGURE 2 : Illustrations of the DigiMicro software functionalities: (a) Voronoï tessellation representing 1800 grains in a volume

element, (b) microstructure cut, (c) distribution of grain volumes, (d) distribution of the ellipsoids length of the semi-minor axis,

(e) voxelized microstructure made of 1981 grains and one million voxels, and (f) Voronoï tessellation with inclusions .

Assuming that the domain contains GN solid grains, we have G

Ni1 , i with the sign convention 0i

inside the solid grain iG , and 0i outside. The procedure to evaluate these functions at all nodes x of the FE

mesh goes through evaluating the functions:

i j i

ij i j G

i j

s s s x1x s s 1 i,j N j i

2s s

( ) , ,

(20)

which correspond to the signed distance of x to the bisector of the segment ji ss , . )(xi is then defined as:

)(min)(

1

xx ij

ijNj

iG

(21)

One can also define a global distance function as:

glob i Gx x 1 i N( ) max ( ), (22)

This function corresponds to the solid grain boundary network. FIGURE 3b displays the function )(xglob

calculated at the nodal points of the FE mesh (in white).

(a) (b) (c)

FIGURE 3: (a) Three hundred Voronoi sites in a unit 3D cube, (b) the function )(xglob of the corresponding Voronoi

tessellation, and the initial FE mesh in white, (c) anisotropic meshing adaptation at grain boundaries

Appropriate refinement of the mesh along interfaces such as grain boundaries is useful to capture the large strain

rate/stress gradients developing across those interfaces upon deformation of the microstructure. It is the result of the

heterogeneous mechanical response of neighbouring grains induced by the crystallographic orientations but also

necessary to modelling subsequent recrystallization. FIGURE 3c illustrates an example of anisotroping meshing

adaptation for the microstructure described by FIGURE. 3b. The mesh is made of tetrahedral elements, whose size

and shape are not homogeneous. Anisotropic meshing is used at the grain boundaries, with a smaller size in the

direction perpendicular to the boundary. For this purpose, a corresponding anisotropic metric is defined and the

mesh is built using the MTC iterative mesher-remesher developed by T. Coupez [9].

5.3 – Recrystallization modelling in a level set framework

Historical approaches of recrystallization were based on the Johnson-Mehl-Avrami-Kolmogorov (JMAK)

analytical model where equation (23) is used for the description of recrystallization kinetics:

nX t 1 bt( ) exp( ) (23)

where X is the recrystallized volume fraction, b a constant which depends on nucleation and growth and n the

Avrami exponent.

More accurate methods were proposed to take into account more complex phenomena at the micro scale,

such us: heterogeneous nucleation, non-homogeneous stored energy or anisotropic mobility of grain, etc. We shall

present here only the level-set method, which appears as very promising. A new 2D and 3D finite element model

based on a level set framework, was developed in Cemef and validated first for primary recrystallization modelling.

As the interface moves, periodic remeshing is performed such as the refinement zone always coincides with the

interface position. In primary recrystallization, the kinetics of interface motion is directly related to the stored strain

energy. The formalism also allows us to trigger the nucleation of new grains, based on desired criteria (mechanical,

crystallographic, etc.). The following simulation illustrates the context of this new approach.

A ten grains microstructure in a unit cubic domain is considered, and mechanical testing is performed using

finite element simulations where each integration point of the mesh behaves as a single crystal subjected to finite

strain increments. Classical theory of crystal plasticity [25] is considered and for computational efficiency, lattice

rotation and rates of dislocation slip are computed in a decoupled way. The objective of the test case is to analyze the

spatial distribution of stored strain energy in a digital aggregate, subjected to large deformations. A channel die test

has been chosen. Slip is assumed to operate on the 12 {111}<110> slip systems as is typically considered in fcc

crystals at room temperature. A 20% reduction in height is applied, and the stored energy is computed from:

dtvEn :

(24)

with the fraction of the strain energy which is stored in the material, considered constant in a first approximation.

The stored energy corresponds to defects (dislocations essentially), which represent the driving force for subsequent

static recrystallization when performing a heat treatment. FIGURE 4a-b illustrates, respectively, the final stored

energy distribution and the anisotropic meshing used to model subsequent recrystallization. The calculated stored

energy field is used as an input to model recrystallization. FIGURE 4c-l illustrates the increasing recrystallized

volume fractions and the corresponding recrystallized front in blue. The simulation was performed in 6 hours on 16

processors of an Opteron 2,4GHz linux cluster and the final microstructure is made of 27 grains.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

FIGURE 4: A 3D ten grains microstructure after plastic deformation: (a) external surface view of stored energy; (b) anisotropic

meshing in white, grain boundaries in black; external surface view of the stored energy for recrystallized volume fractions of (c)

1%, (d) 15%, (e) 58%, (f) 80%, (g) 95% and corresponding recrystallized front in blue (h-l).

Comparisons with the JMAK theory were performed, and are described in FIGURE 5. A first simple comparison

was done with no consideration of the stored energy field, with a random choice of nucleation sites. A least-square

regression analysis on the numerical results provided a JMAK exponent n=3.91 (FIGURE 5a), while the theoretical

value is n = 4 in 3D. This result validates our method in 3D. The second case, illustrated by FIGURE 5b,

corresponds to the recrystallization kinetics of the numerical simulation described by FIGURE 4. Interestingly, in

this case, a single value of n does not allow fitting the numerical results with sufficient accuracy. This result must be

placed in the context of repeated discussions in the literature on the reasons of deviations from the standard JMAK

theory. Heterogeneous distribution of stored energy [26], and, spatial and time distribution of nuclei [27] can explain

these deviations. These can be studied in details with the present level-set model and these results illustrate the

capability of microstructural approaches to improve conventional macroscopic laws.

(a) (b)

FIGURE 5: JMAK approximations of the numerical recrystallization kinetics extracted from Fig. 3: (a) random choice of

nucleation sites, low and constant nucleation rate; (b) considering the non uniform stored energy field and choosing nucleation

sites at highest gradient values of the stored energy.

Another interesting possibility of the methodology described is being able to take into account the presence of

inclusions whose effect on the recrystallization front kinetic could be crucial. FIGURE 6 illustrates the breaking

effect, due to the presence of two spherical inclusions, in the growth of one grain.

(a) (b)

FIGURE 6 : Breaking effect, due to the presence of two spherical inclusions, on the growth of one grain: (a) velocity magnitude

on the grain surface (b) mesh on a cutting plane.

6 – CONCLUSIONS

After this short review of the field of metal forming simulation, it is possible to consider the important scientific

breakthroughs which were carried out in many Universities in the world, and have resulted in the development of

several commercial codes. Now many companies are convinced of the necessity of numerical simulation in order to

understand better deformation processes, design new sequences and optimize them with respect to cost, material and

energy consumption and quality of the part.

But it is clear that, due to more and more tough competition between companies, new needs of industry will

appear and result in new scientific developments. Among these hot topics we can foresee:

- very large scale computation with evaluation and control of accuracy of thermal, mechanical and physical

parameters,

- integrated computation from preform to finished part followed by structural computation, where the material

properties are calculated and input directly,

- optimization of the whole process with respect to in service load,

- prediction of the material micro structure and properties, utilizing a F. E. micro model in a representative

volume element. The link between the micro and the macro scales levels being achieved using homogenized

incremental physical laws.

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