MULITSCALE AND META MODELLING FROM HIGH ACCURACY …metal2014.tanger.cz/files/proceedings/02/reports/111.pdf · MULITSCALE AND META MODELLING – FROM HIGH ACCURACY TO HIGH EFFICIENCY

23. - 25. 5. 2012, Brno, Czech Republic, EU

MULITSCALE AND META MODELLING – FROM HIGH ACCURACY TO HIGH EFFICIENCY IN

SIMULATIONS OF METAL FORMING PROCESSES

Maciej PIETRZYK, Danuta SZELIGA

AGH Akademia Gorniczo-Hutnicza, 30-059 Krakow, Poland

Abstract

Models used in metal forming are classified in the paper with respect to their predictive capabilities and

computing times. Classification covers various models from simple closed form equations through various

advanced numerical methods (finite elements, boundary elements, finite volume etc.) to advanced multiscale

solutions. The procedure for development of the reliable and efficient model is proposed in the paper. This

procedure comprises deriving of mathematical formulation, sensitivity analysis and identification of

coefficients for a selected material. Capabilities of the commonly used thermomechanical-microstructural

finite element model are demonstrated using forging of crank shafts as an example. Metamodels, which are

formulated on the basis of advanced numerical methods, are proposed as a substitution for the simple,

approximation equations. The advantage of metamodelling in applications to design optimal technological

parameters is exposed. Accuracy of metamodels is much higher comparing to the simplified models, while

very short computing times are maintained. Results of research performed at AGH on modelling

manufacturing cycles and on multi scale modelling of metal forming processes are presented in the second

part of the paper.

Keywords: Meta modelling, Multiscale modelling, Metal Forming

1. INTRODUCTION

Diversity of models proposed in the scientific literature for metal forming processes was the motivation for the

Authors to undertake these research. Different models of various complexity and various predictive

capabilities are now available. Two aspects decide about accuracy and effectiveness of the modelling:

Selection of a relevant model for a particular application,

Correct identification of models (boundary conditions and material properties).

Problem of identification of models was investigated in several Authors papers, see for example [1,2], and it

is not discussed here. Selection of the relevant model for a particular application is the general objective of

this paper. This selection has to be made by searching for a balance between model predictive capabilities

and computing costs. Primary classification of the models with respect to the two mentioned criteria was

made in [3]. Similar classification focused on rolling processes is presented in [4]. The particular objectives of

the present paper are twofold. The first is extension of classification of the existing models to cover the whole

manufacturing cycle, including heat treatment. Presentation of the recent research performed at AGH on

metamodelling and on multiscale modelling in metal forming manufacturing cycles is the second objective.

2. CLASSIFICATION OF MODELS AND MODELLING PROCEDURE IN METAL FORMING

2.1 Classification of models

Historically, slab method [5] and upper bound method [6] were commonly used for simulations of metal

forming processes up to late 60-ies of the last century. Since early 1970-ies, finite element (FE) method has

become the most popular simulation technique [7]. In 1990-ies the FE codes were connected with the

microstructure evolution models and thermomechanical-microstructural simulations became possible [8].

New challenges in modelling metal forming occurred at the beginning of the XXI century. Possibility of


prediction of microstructural phenomena accounting explicitly for the structure of polycrystals is the first

challenge, which led to development of multiscale models. In these models usually FE codes are connected

with such discrete methods as cellular automata (CA), molecular dynamics (MD) or Monte Carlo (MC).

Review of multiscale modelling methods can be found in [9]. Problem of computing time is the second

challenge, which is particularly important when optimization of the process is performed.

Extension of the classification of metal forming models with respect to predictive capabilities and computing

costs presented in [3,4] to whole manufacturing cycles is shown in Fig. 1. Models in the left bottom corner

are mainly used for on-line control and are generally limited to rolling processes. They are usually based on

slab [1] or upper bound [2] methods. The metamodels have recently appeared in this group. The

metamodels are often used in optimization of metal forming processes characterized by complex flow of

metal and long FE computing times. Models in the centre of Fig. 1a are based on FE or alternative methods

and they are commonly used in technology design and optimization of processes. Microstructure evolution

equations and information concerning grain size are often implemented in these models. Beyond strains,

stresses and temperatures, grain size and phase structure are additionally available for the design.

a)

b)

c)

Fig. 1. Classification of models in metal forming with respect to their predictive capabilities and computing

costs (a), production chain for manufacturing of fasteners (b) and production chain for manufacturing of

crash box made of DP steel (c).

Right top corner in Fig. 1a contains multiscale models, which combine FE or alternative methods for macro

scale simulations with discrete methods in micro and/or nano scales. These models are characterised by

long computing times and their applications are rather limited to scientific research on materials processing.

Problem of computing costs becomes particularly serious when optimization has to be performed or

simulation of the whole manufacturing chain is needed. Two examples of manufacturing cycles are shown in


Figs 1bc. The cycle for fasteners is composed of hot rolling of rods, controlled cooling, pickling,

phosphatizing, cold drawing, annealing, multi step forging, thread rolling and heat treatment [10]. In the case

of manufacturing of the crash box made of the DP steel this cycle is composed of hot rolling, laminar cooling,

pickling, cold rolling, continuous annealing in the intercritical region, stamping and welding [11]. Examples of

application of various models to simulation and optimization of selected processes or whole manufacturing

chain are shown below.

2.2 Modelling procedure

As it has been mentioned in the introduction, selection of a relevant model for a particular application is

crucial for the efficiency and accuracy of modelling. The procedure of the development of the best model

should be composed of:

1. Deriving of the mathematical formulation,

2. Sensitivity analysis,

3. Identification of the coefficients in the model.

The first point is obvious and it has been from the very beginning the main part of the development of the

model. The last point is essential for the accuracy of the model. Inverse analysis is now commonly used for

determination of the coefficients in the model, which give the best description of the investigated material.

Inverse method is composed of the experiment, numerical model of this experiment (usually FE model) and

optimization techniques. The coefficients, which give the most realistic description of the behaviour of the

considered material, are determined by searching for the minimum of the goal function, which is defined as a

square root error between measured and calculated output parameters. This procedure is well researched

and described in numerous publications, see Authors publications [1,2], and it is not discussed in the present

paper. The focus is put on the sensitivity analysis, which allows to select the most important coefficients of

the model and to design the model as efficient as possible.

2.3 Sensitivity analysis

Sensitivity analysis (SA) allows to assess the accuracy of an analyzed system or process model. It

determines the parameters, which contribute the most to the output variability, indicate the parameters which

are insignificant and may be eliminated from the model. Beyond this, the sensitivity analysis evaluates these

parameters, which interact with each other and determines the input parameters region for subsequent

calibration space. The steps of the sensitivity analysis are [12,13]:

Sensitivity measure. The measure expresses the model solution (model output) changes caused by the

model parameter variation.

Selection of the parameter domain points. Design of experiment techniques are commonly used to select

the lower number of points, which guarantee searching through the whole domain.

Method of sensitivities calculation. The sensitivities are estimated by global indices or by local ones.

Application of the information from the sensitivity analysis to the inverse method has the following meaning:

Verification if the objective function is well defined – it means whether it is possible to estimate the

parameters, which are looked for, based on the information included in the objective function. In the case

of no sensitivity or low sensitivity of the objective function to the parameter changes, identification of this

parameter cannot be performed and the objective function hast to be transformed to another form,

including verification of the model output space norm.

As the preliminary step – to select the starting point/the first region of interest or the first population for

optimization algorithm.

In the optimization process – to construct the hybrid algorithms (e.g. the combination of a genetic

algorithm to select local minima and a gradient method to explore those minima) or modified algorithms

(e.g. the particle swarm procedure enriched with the local sensitivities information) to increase the

procedure efficiency.


In this work the sensitivity analysis was applied to evaluate the influence of the model parameters on the

model outputs. In consequence, the model is better understood and its calibration and modification was

become possible. Two algorithms of the SA were applied. The first one, Morris Design [14] that is screening

technique, gives qualitatively information of the model response sensitivity to the parameters. The method is

used to determine the group of the parameters of the highest impact on the model. The second algorithm,

based on variance analysis, called Sobol method, is applied to estimate sensitivity quantitatively. Since

Sobol method is of the higher computation cost comparing screening techniques, the algorithm is run for

selected by MD parameters.

Sensitivity analysis will be applied in section 3.2 below to investigate the phase transformation model.

3. FE MODELS AND METAMODELLING

As it has already been mentioned above, the FE method is commonly used as a model of metal forming

processes. FE simulations usually need long computing times. This is even more critical when FE method is

applied in optimization tasks. Since long computing times are needed to determine the objective function and

usual optimization algorithms require large number of evaluations of this function, optimization based on the

FE method becomes inefficient. Even if a simple FE model with coarse mesh is used in simulations of metal

flow, the time necessary to calculate one simple process is about 10-30 min on typical PCs. Thus, searching

for alternative models, which can accelerate optimization, is the key to efficient optimization. Application of

the metamodel is such an alternative. Process of forging of crank shafts using TR technology (TR - from the

name of the inventor Tadeusz Rut) was selected below to demonstrate capabilities of those two approaches.

3.1 FE simulations of the TR crank shaft forging

Details of FE simulations of the crank shaft forging according to the TR technology are given in [15].

Schematic illustration of this process is shown in Fig. 2. FE simulation allows to predict metal flow, as well as

distributions of strains, stresses and temperatures during the whole process. Loads acting on the tools are

also calculated. These information is used in the design of the best forging technology, as it is shown in [15].

FE model was coupled with the microstructure evolution equations for the investigated steel and calculations

of recrystallization and grain growth were performed at each Gauss integration point. Results of simulations

at two selected points are shown below to demonstrate capabilities of this method. Locations of these points

in the stock material are shown in Fig. 2a. Point 1 lies in the area, where shaping of the shoulder involves

intensive deformation, while contact with the tool causes drop of the temperature. Contrary, smaller strains

and no temperature drop are observed in location 2. Calculated changes of process parameters and grain

size in both locations are shown in Fig. 3, where: - strain, T – temperature, X – recrystallized volume

fraction, D – grain size. It is seen that grain refinement due to recrystallization occurs in location 1. Contrary,

smaller refinement and significant effect of the grain growth due to high temperature is observed at point 2.

Recapitulating, FE or alternative methods allow advanced simulations of metal forming processes and

prediction of various parameters of metal flow, as well as microstructural parameters. Computing costs are,

however, an important factor limiting applications of this approach. It becomes not efficient when optimization

of the process has to be performed and numerous calls of the objective function based on the FE simulations

are needed. It is not efficient in applications to simulations of manufacturing cycles either. An alternative

approach, based on metamodelling, is presented in the next section.


a) b)

c) d) Fig. 2. Schematic illustration of the TR method for forging of crank shafts: a) beginning of the unsymmetrical

pre-upsetting, b) end of the unsymmetrical pre-upsetting, c) beginning of the forging of the crank throw, d)

end of the forging of the crank throw.

-50 0 50 100 150 200

time, s

0

40

80

120

160

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1

700

800

900

1000

1100

1200

D, m X T, oC

point 1

-50 0 50 100 150 200

time, s

40

80

120

160

D

X

T0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1

700

800

900

1000

1100

1200

D, m X T, oCpoint 2

Fig. 3. Changes of the temperature, strain accounting for recrystallization, recrystallized volume fraction and

austenite grain size for locations 1 and 2 in Fig. 2a. Meaning of symbols is the same in both plots.

3.2 Optimization of forging of crank shafts using metamodelling of the modified TR method

Metamodels of processes are recently developed to enable efficient application of new generation

optimization techniques, which require a large number of calculations of the objective function. Artificial

neural networks (ANN) are often used as metamodels. Once trained, such metamodel is extremely fast and

can be effectively used to perform optimization. Any optimization technique can be used, even that which are

inspired by the nature and require a large number of evaluation of objective function. In the case of forging of

crank shafts generation of data for training of the ANN would be difficult, therefore, the surface response

method [16] is used as metamodel of the TR forging process. An example of such optimization is shown in

[17]. The aim was to obtain the required shape of the crankshaft. Two parameters were chosen as decision

variables: bending tool displacement during forging of the crank throw (d1) and initial spacing of face die

inserts (d2), see [17] for details. Nine simulations were performed for each scheme. Absolute value of

difference between obtained and ideal shape of the crankshaft was assumed as the objective function. The

plot of this function approximated by the second order response surface is shown in Fig. 4. Response

surface is the convex that allowed determination of the minimum for d1 = 273 mm for bending tool


displacement during forging of the crank throw and d2 = 1369 mm for the initial spacing of the face die

inserts.

Fig. 4. The second order response surface constructed on the basis of simulations.

3.2 Modelling of manufacturing cycles

Problem of modelling of manufacturing cycles is in the field of interest of several research laboratories in the

world. Selection of the best models for subsequent stages of the cycle cannot be unified. It has to depend on

the complexity of individual processes of the cycle and the general rules presented in Fig. 1a remain valid.

Authors have performed simulations of manufacturing chain for forging of Cu-Cr alloys [18] and forging of

fasteners [19]. In both these example FE method was efficient enough to perform simulations of the cycle

and prediction not only flow and microstructural parameters, but also tool wear. Problem of simulation of

manufacturing chain for the DP steel crash box (Fig. 1c) is discussed below. Since dual phase microstructure

is obtained during continuous annealing after cold rolling, typical ferritic-pearlitic microstructure is obtained

after hot rolling and laminar cooling. Simple hot rolling model, usually based on 1D FE solution combined

with microstructure evolution equation, is used [11]. Cold rolling is the process where the demanding

challenge for modelling begins. Microstructure phenomena occurring during continuous annealing are crucial

for the properties of product. On the other hand, these phenomena depend on the cold deformation of the

ferritic-pearlitic microstructure. Any strain localization in this microstructure has an influence on ferrite

recrystallization and on the following phase transformations during annealing. Such factors as carbon

segregation and kinetics of transformation from the intercritical region have to be accounted for by the

models. It is expected that realistic modelling of all these phenomena becomes possible when such discrete

methods as Cellular Automata (CA) or Monte Carlo (MC) are used. Simulation of the carbon diffusion during

transformation using FE solution with the moving boundary (Stefan problem) is an alternative [20]. Predictive

capabilities of these methods are discussed in the next section. In the meantime, modelling of the cold rolling

assuming continuum of the material and modelling of the continuous annealing based on the conventional

phase transformation model is shown below.

The phase transformations model is based on Avrami equation:

1 exp nX kt (2)

where: t – time, X – transformed volume fraction.

Equation (2) is combined with the Scheil additivity rule, which accounts for the temperature variations.

Theoretical considerations show that a constant value of n in equation (2) can be used. This coefficient is

introduced in the vector of the decision variables a as a4, a15 and a24 for ferritic, pearlitic and bainitic


transformations, respectively. On contrary, value of the coefficient k must vary with temperature. The

formalism of the function k = f(T) must be carefully chosen to describe properly the temperature dependence

of transformation kinetics. Functions selected for the ferritic, pearlitic and bainitic transformations are:

4

6 3

5

7

400

exp

a

a T AeDa

kD a

ferrite (3)

15

1413 12exp

a

ak a a T

D

pearlite (4)

23 22 21k k a a T bainite (5)

where: D – austenite grain size.

Incubation time is introduced for bainitic b and pearliticp transformations. The following equations are used:

11

19

9 10

1

17 17 18

expˆAe

expˆ

P a

b a

b

a a

RTT

a k a

RTT T

(6)

where: T – temperature in oC, T̂ – absolute temperature in K.

Start temperatures for the bainitic Tb and martensitic Tm transformations are functions of steel composition:

o

20

o

26 27

[ C] 425[C] 42.5[Mn] 31.5[Ni]

[ C]

b

m

T a

T a a C

(7)

Koistinen and Marburger equation is used for modelling martensitic transformation. Full model for all

transformations contains 23 coefficients [11]. Sensitivity analysis described in section 2.3 was performed to

evaluate importance of the coefficients. This analysis has shown that, since carbon content in the DP steels

is low, the output of the model is not sensitive to the parameters a12-a15 and model for the pearlitic

transformation can be significantly improved. Constant coefficient k = a12 was assumed for the pearlitic

transformation and, finally, 20 coefficients remained in the whole model. These coefficients were determined

using the inverse analysis for the dilatometric tests. Values of the coefficients obtained for the DP steel used

for manufacturing of crash boxes are given in Table 1.

Table 1. Coefficients in the phase transformation model calculated using inverse analysis

a4 a5 a6 a7 a8 a9 a10 a11 a12 a16

1.69 0.858 188 39.06 1.78 64.76 1106 0.618 0.153 0.007

a17 a18 a19 a20 a21 a22 a23 a24 a26 a27

1600 64640 3.495 669 0.118 0.074 0.344 1.037 421.7 1.83

Simulations of the whole manufacturing chain in Fig. 1c were performed and selected results are presented

below. Typical results of simulations of the hot strip rolling are presented in Fig. 5. Comparison with the

temperature measurements using pyrometer (surface) and rolling force measurements are shown in this

figure and good agreement between predictions and measurements is observed. DP microstructure can be


obtained during laminar cooling after hot rolling, see modelling of this process in [21]. This method is used

for thicker DP strips, which is not the case in the present project.

a)

0 4 8 12

time, s

700

800

900

1000

1100

tem

per

ature

, oC

centre

surface

measurement

pass: 1 2 3 45

6

b)

0 1 2 3 4 5 6 7 8 9

time, s

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

roll

ing f

orc

e, M

N0

10

20

30

40

50

60

grain

size, m

forces

measurements

predictions

1

2

34

5

6grain size

Fig. 5. Temperatures calculated by the FE code and measured using the pyrometer (a); Rolling forces

measured at the hot strip mill and calculated using metamodel and grain size calculated using microstructure

evolution model (b).

Conventional laminar cooling was applied after rolling in the present work and ferritic-pearlitic microstructure

was obtained. Simulations of the 4 pass cold rolling process were performed using Authors FE code [22] and

strain distributions through the strip thickness were determined. Rolling forces and torques were calculated

for all passes, as well strain distribution through the thickness. This distribution was an input for simulation of

the ferrite recrystallization during heating phase in the process of annealing, see [3] for results. Simulations

of continuous annealing are described below. Thermal profile during annealing and resulting changes of

volume fractions of phases are shown in Fig. 6. Constant heating rate of 3oC/s was applied up to the

temperature of 780oC. Heating with the rate of 0.25

oC for 10 s followed. Cooling schedule was composed of

10oC/s for 6 s, 0.5

oC/s for 20 s, 40

oC/s for 8.75 s, 0.3

oC/s for 100 s and further cooling with 20

oC/s. In

consequence, microstructure composed of 75% of ferrite, 5% of bainite and 20% of martensite was obtained.

0 100 200 300 400 500time, s

0

200

400

600

800

tem

pe

ratu

re,

oC

0

0.2

0.4

0.6

0.8

1

volu

me fra

ctio

n

temperature

recrystallization

ferrite

pearlite

bainite

martensite

Fig. 6. Temperature changes, ferrite recrystallization kinetics and changes of the volume fractions of phases

during continuous annealing of the DP steel.


Flow stress model for room temperatures and Lankford coefficient were determined on the basis of tensile

tests for the samples with the two phase microstructure. These data were used in the FE simulations of the

stamping process, see [23] for more details.

3.3 Discussion

Finite element or alternative methods remain a powerful tool of simulation of metal forming processes with

wide predictive capabilities. Calculations of strains, stresses and temperatures, as well as average

microstructural parameters, are possible. However, when optimization of the process or simulation of the

whole manufacturing chain has to be performed, a large number of the FE calculations is needed and

computing costs become very high. The metamodelling technique is proposed in these applications.

Average microstructural parameters (grain size, recrystallized volume fraction of phases) were determined in

modelling of manufacturing chain in section 3.2. These data allow to predict the properties of the DP strip

treated as a continuum. This enables conventional simulations of the stamping process and evaluation of

strain distribution in the product. However, when there is an interest in local strains distribution (localization),

an influence of the phases morphology has to be accounted for. It can be done using multiscale modelling,

which is discussed briefly in the next section.

4. MULTISCALE MODELS

Multiscale models are located on the right hand side of the axis in Fig 1a. These models have wide

predictive capabilities and can be considered the most novel solutions in the field of metal forming. On the

other hand, these models require very long computing times. General idea of the multiscale modelling is

presented briefly below. More details can be found in [9,24]. The multiscale models developed at the AGH

for simulation of phase transformations are presented as well.

2.1 Classification of multi scale modelling methods

The main challenges of the multiscale modelling are capturing of discontinuities in materials and proper

description of the multi-physics phenomena. The methods allowing to cope with these challenges are usually

classified into two groups: upscaling methods and concurrent multi scale computing [9,24]. Fig. 7 shows the

general idea of distinction between these methods.

Fig. 7. The idea of distinction between upscaling models and concurrent multi scale computing [9].


In the upscaling class of methods constitutive models at higher scales are constructed from observations

and models at lower, more elementary scales. The idea of the representative volume element (RVE) is

employed here. By a sophisticated interaction between experimental observations at different scales and

numerical solutions of constitutive models at increasingly larger scales, physically based models and their

parameters can be derived at the macro scale. The methods of computational homogenisation are

considered to belong to this group of methods.

In concurrent multi scale computing one strives to solve the problem simultaneously at several scales by an

a priori decomposition of the domain. Two-scale methods, whereby the decomposition is made into coarse

scale and fine scale, have been considered so far. In the concurrent multi scale computing the method used

to describe fine scale is applied to a part of the domain of the solution. It can be either the same method,

which is used in the coarse scale, for example FE method, or it can be one of the mentioned earlier discrete

methods (CA, MC). In the former case the extended finite element (XFE) and the multi scale extended finite

element (MS-XFE) methods are distinguished.

Research on multiscale modelling at the AGH is focused on the connection of the Cellular Automata with the

Finite Element method (CAFE method) and several applications of this method were developed. An example

of application to modelling phase transformations is described below.

2.2 Cellular Automata – Finite Element method

CAFE method, which is described below, belongs to the upscaling multiscale modelling techniques. In this

approach the CA model is responsible for the evolution of microstructure. Basic details of this solution can be

found in [9]. The algorithm constructs microstructure images with various distribution of grain size and

orientation. Various kinds of neighbourhood can be defined. Definition of the transition rules is crucial for the

reliability of the CA method. These rules define the state of the cell in the next time step on the basis of the

state of the neighbours of this cell and the state of this cell itself in the previous time step. The general form

of the transition rule is:

1

,

,

if

else

t

k l t

k l

newstate

(8)

where: ,

t

k l , 1

,

t

k l

– state of the cell (k,l) in the previous and current time step, respectively, – logical

function, which controls changes of the state of the cell.

It is seen from equation (8) that the cell can either not change its state or, if the function is true, it changes

the state to a new state. Function is defined on the basis of the available knowledge about the considered

phenomenon. The schematic idea of the CAFE simulation of the microstructure evolution is shown in Fig. 8.

Fig. 8. The general idea of the CAFE model for microstructure evolution. Local values of macroscopic fields

(thermal and mechanical) are passed to micro-scale CA model. Microstructural parameters are returned.


Influence of external variables, e.g. temperature, is directly accounted for in the transition rules. Modelling

recrystallization during hot forming is the main application of the CEFE approach and it is well described in

the literature, see for example [25]. Authors model of phase transformations in steels is described below.

The FE program calculates temperature distribution in the strip during laminar cooling or continuous

annealing and typical solution is used. Thus, in the presented description the focus is on the CA part only.

The following states of cells are distinguished in the model: - ferrite, - austenite, / - austenite/ferrite

boundary, B – bainite, /B – austenite/bainite boundary, - carbide, M – martensite. The transition rules for

all transformations were defined in the form of equation (8). The following logical functions were proposed:

Ferrite nucleation 1

,Yt

k l

3 , 0,1Y t

e i j nucT A l p (9)

Ferrite growth 1

,Yt

k l

3 , , ,Y Y 1t t

e i j k l i jT A X (10)

Bainite nucleation 1

,Y Bt

k l

, 1 , 1 20,1 0,1Y Y 1t t

b i j N i j N NT T l P P l P

(11)

Bainite growth 1

,Y Bt

k l

, , , 30,1Y Y 0.5t t B

b i j k l i j NT T X l P (12)

Carbide 1

,Yt

k l

, , ,Y Yt t

s i j k l k l crT B c c (13)

Martensite nucleation 1

,Y Mt

k l

, , , 0 0,1

Y Y Y Bft t t

m i j i j i j zT T P P l

(14)

Martensite nucleation on the existing plates 1

,Y Mt

k l

MM

, , 0 0,1Y M Y maxt t

i j k l w zD P P l (15)

Martensite growth 1

,Y Mt

k l

M M

, , , z , ,Y Y M Y M true 1t t t

s i j k l k l k l i jT M G X

(16)

where: t – time, ,i jX – volume fraction of ferrite in the cell i,i, ,

B

i jX – volume fraction of bainite in the cell i,i,

k,l – numbers of neighbours to the cell i,j, PN1, PN2 – probability of nucleation of bainite at boundaries and

inside the austenite grains, respectively, PN3 – probability of bainite growth, which is assumed 0.2, l(0,1) –

random number from the interval [0,1], ccr – critical carbon concentration to create a carbide, f

zP P0 – a

product, which represents probability that a cell located at the border with the phase , ,Bf can

become a nucleus of martensite, M

wD - length of the martensite plate, M

zP - probability that a cell located

at the border with the existing martensite plate can become a martensite.


Detailed description of this model can be found in [26-29]. Some additional rules are given below to enable

reproduction of the model on the basis of equations (9)-(16). It was assumed, that ferrite growth is controlled

by the diffusion and the interface mobility. Carbon diffusion in the austenite is calculated at each time step by

finite difference solution of the diffusion equation:

1, 1, , , 1 , 12 2 2 21

,

,

1 2 2 1 1if

else

t t t t t

i j i j i j i j i jtx x y yi j

t

i j

D t c c c c cD th h h hc

c

(17)

where: D – diffusion coefficient, hx, hy – size of the cell, and:

, ,Y Yt t

i j i j

The transition rule (12) is combined with the definition of the most probable directions of bainite sheaves

growth, see [27]. The transition rule (13) is based on the assumption that a carbide occurs when there are at

least 3 neighbouring cells with c > ccr. When carbide is created, the whole carbon excess above 0.0218%

remains in this cell and each neighbour of his cell changes its state to /B. Typical results of simulations of all

transformations are shown in Figs 9-11, where: Q – cooling rate, Dα – ferrite grain size. In simulations in Fig.

11 slower cooling was applied until about 50% of ferrite and bainite was obtained. Fast cooling was applied

next to transform remaining austenite into martensite.

(a) b) c) Fig. 9. Calculated austenitic-ferritic microstructures: a) input microstructure (austenitic); b) Q = 5°C/s (Dα =

9.2 µm); c) Q = 15°C/s (Dα = 8.6 µm) [26].

a) b) Fig. 10. Results of simulation of the isothermal bainitic transformation at the temperature of 511

oC, a) after

200 s; b) after 300 s (white – austenite, grey – ferrite) [27].


Fig. 11. Results of simulation of martensitic microstructure.

5. CONCLUSIONS

Classification of models of metal forming processes is presented in the paper. It is concluded that advanced

numerical methods, mainly the finite element method, are still most commonly used for simulation of

processes, design and optimization of technology, design and optimization of tools etc. These methods are

often combined with equations describing microstructure evolution and have wide predictive capabilities. On

the other hand, they usually require reasonably large computing times. The objective of the paper was to

explore possibilities of application of the methods, which are an alternative for the FE method in either

decreasing of the computing time or extending the predictive capabilities. Metamodels are proposed as the

former alternative and optimization of forging of crank shaft is presented as an example. Multiscale models,

based on the connection of the FE method and the Cellular Automata approach, were explored as possibility

of extension of predictive capabilities of FE and alternative methods. Research on application of metamodels

and multiscale models in metal processing allow the following conclusions:

Metamodel of the process is a certain abstraction defined on the basis of the lower level model developed

using mathematical techniques. Artificial Neural Networks are often used as metamodels.

Training of the metamodel is usually time consuming, it often requires large number of the FE process

simulations. But once trained, the metamodel is effectively applied to fast process simulations. Computing

times of metamodels are extremely short, while their accuracy is close to that of the FE model, which was

used to train the metamodel.

Application of metamodels in optimization problems is particularly advantageous. Very short times of

evaluation of the objective function allow selection of the optimization algorithm on the basis of its

efficiency and robustness only. No care is taken for the computing costs. Any optimization technique can

be used, including algorithms inspired by the nature requiring a large number of evaluations of the

objective function.

Metamodels are very efficient as the direct problem models in applications of the inverse analysis to the

identification of the flow stress model on the basis of plastometric tests.

Performed research allow following suggestion regarding selection of the best metal forming model:

FE method and alternative advanced numerical methods still remain the main simulation and optimization

techniques for metal forming processes.

Simple models with low accuracy should be substituted by metamodels.


Multiscale modelling supplies new, extensive predictive capabilities regarding behaviour of materials

during processing. This approach is costly and should be used in certain specific applications, when

detailed information regarding phenomena occurring in materials is needed.

ACKNOWLEDGEMENTS

Financial assistance of the MNiSzW, statute project of the AGH no. 11.11.110.080, is acknowledged.

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