Mohr Mann

2002 ABAQUS Users’ Conference 1

Automated identification and calculation of the parameters of non-linear material models Ralf Mohrmann Gregor Schmich Fraunhofer-Institut für Werkstoffmechanik D-79108 Freiburg, Wöhlerstrasse 11, Germany

1 Abstract Material models can be defined as a set of ordinary differential equations or as an analytical function. Usually, material models include model parameters. These model parameters are adjusted with respect to experimental data. A numerical tool, called FitIt [:fitit], for an automated optimisation of model parameters is described. The underlying methods are the Levenberg-Marquardt method and the simple Gradient-method. First and second order derivatives of a quality function with respect to the model parameters are used. The derivatives are calculated numerically; an empirical method for the calculation of these derivatives is established.

2 Introduction Usually, the development of complex material models is difficult and time consuming. Stability and convergence are the main aspects for choosing a typical numerical method i.e. for non-linear parameter optimisation or for numerical integration of a set of differential equations. Here, these and other numerical methods are discussed. All methods are included in a proprietary software, called FitIt, which is designed to support the development of material models. The model parameters are adjusted with respect to experimental data. FitIt can be used for models defined as a set of ordinary differential equations or as analytic functions, respectively.

3 Theory and numerical methods The theoretical background and the numerical methods are describes here in order understand the details of the implementation of FitIt. The FitIt user interface hides most of these details from the FitIt user. Therefore, readers who are interested in the application of FitIt can skip the present and the following section.

3.1 Non-linear parameter optimisation [1] 2

χ is a non-linear scalar function of a parameter vector p�

with the dimension n . The aim is to

find the minimum of 2χ . An iterative method has to be used for this purpose.

A 2nd order Taylor expansion of 2χ at the current parameter vector

curp�

gives:

2 2002 ABAQUS Users’ Conference

( ) ( ) ( ) ( ) ( ) ( ) ( )curcurcurcurcurcur

pppppppppp��

−⋅∇∇⋅−+∇⋅−+=2222

2

1χχχχ

Near the minimum the following equation can be employed

( )curcur

ppp A�� 2

min

1χ∇⋅

−−= with ( )

curpA�2

χ∇∇= .

Otherwise, a gradient step can be taken

( )curcurnext

pconstpp�� 2

. χ∇−= .

The Levenberg-Marquardt method combines the two former equations: The diagonal elements of

A are multiplied by a factor ( )λ+1 . The modified A is called A′ . The parameter increment p�

δ

can then be calculated by resolving the set of linear equations:

( )cur

ppA�� 2χδ ∇=⋅′ .

The model results ( )pxy

i

�

, and ( )pyxyi

��

,,′ are either given in closed analytical form, or they are

calculated by a numerical integration of a set of ordinary differential equations. For the integration an implicit method is used.

3.2 Calculation of the 2

χ function

The 2χ function can be calculated with different definitions. The standard definition is

( )( )

2

1

1

,

2 ,

∑=

−=

M

i iy

iiy

S

s

pxyyp

��

�

�

χ

with the number of data points, M , the value of a data point,

iy , the error in this data point,

iys ,

, and the model result for this data point, y . The second definition also takes the derivatives,

y ′ , of the variable (with respect to x) into account:

( )( ) ( )

∑= ′

′

′−′+

−=

M

i iy

ii

iy

iiyy

S

s

pxyy

s

pxyyp

1

2

2

,

2

,

2 ,,

��

��

�

χ

with ii

ii

ixx

yyy

−

−=′

+

+

1

1 and ii

iyiy

iyxx

ss

s

−

+

=

+

+

′

1

2,

21,

, for Mi <≤1

and with 1−

′=′MM

yy and 1,, −′′

= MyMy ss .

The third definition uses only derivative information for the calculation of 2χ


( )( )

2

1

3

,

2 ,

∑= ′

′

′−′=

M

i iy

iiy

S

s

pxyyp

��

�

�

χ .

In many cases different types of experiments with various numbers of data points are used within FitIt simultaneously. Therefore, each experiment should be assigned to a class. A useful

definition for 2χ is then

( ) ∑ ∑ ∑= = =

=

J

j

K

k

M

ijj

j kj

SUMK

p

1 1 1

2

,

1ωχ

�

with the number of classes J , the weight jω for each class, the number of Experiments jK of

the class j and the number of data points kjM,

of Experiment k of class j . SUM is chosen

according to one of the three definitions for i

S given before.

3.3. Integration of differential equations [1] The integration of differential equations is done numerically with an implicit method. Implicit integration enables the solution of stiff and non-stiff problems, respectively. The non-linear set of equations

( )11 ++

⋅+=nnn

yfhyy�

�

��

with ( )pxyfy��

�

�

,,=′

is solved iteratively. A first order method is the so called semi-implicit Euler method

( )nnnyf

y

fhhyy

�

�

�

�

��

1

11

−

+

∂

∂−+= ,

which is not guarantied to be stable but is very robust. Higher order implicit methods can result in advantages in computing time and/or accuracy. A Rosenbrock method, which is a generalised Runge-Kutta scheme, worked for any practical problem. This method seeks a solution of the form

∑=

++=

s

i

iinnkcyy

1

1

�

��

,

where the corrections ik

�

are found by solving s linear equations

siky

fhkyfhk

y

fh

i

j

jij

i

j

jijni,...,1,1

1

1

1

1

=∂

∂+

+=

∂

∂− ∑∑

−

=

−

=

�

�

�

�

�

��

�

�

γαγ .


The coefficients ijic αγ ,, and ijγ are fixed constants, independent of the problem. For both

methods, the calculation of the Jacobian yf�

�

∂∂ is done numerically.

3.4 Calculation of 1−

A : Singular value decomposition [1]

The inverse of the matrix A has to be calculated within the Levenberg-Marquardt method. If the

current value of a parameter component i

p does not (significantly) influence the 2

χ result,

A becomes singular and cannot be inverted directly. Therefore, the singular value decomposition

(SVD) method is used. First, A is decomposed with the QR-method

T

VWUA ⋅⋅= with 1=⋅UUT

and with 1=⋅VVT

.

All matrices are square. W is a diagonal matrix; the components of its inverse denote to singular

components. Second, the singular components have to be identified. The singular components are zero; these components near to zero are set to zero in order to enable the calculation of the

inverse of A

T

UWVA ⋅⋅=

−− 11

.

3.5 Numerical calculation of derivatives

Derivatives cannot be calculated numerically with an equation like

( )( ) ( )

h

hxfhxfxf

2

−−+=′

without any precautions. Truncation and roundoff errors should be taken into account. In our

case, f is the numerical result of a 2

χ function and therefore the finite accuracy of f is an

additional limitation for the choice of the step size h .

An empirical recipe was found to work well in practical applications:

The step size h is adjusted such, that ( )hxf + and ( )hxf − differ in one half of the

significant digits. If no h is found to fulfil this criteria within the interval xh ⋅<< 1,00 , the

derivative is set to zero. This recipe is used for first and second order derivatives.


4 Verification of numerical method

4.1 Exponential function with finite resolution This examples should illustrate how the recipes for the calculation of derivatives works. The

exponential function x

e was used as a model function and evaluated with finite accuracy at

1=x . Figure 1 gives the results of the derivatives for three resolutions calculated with

( ) ( )h

xfhxfy

−+=′ .

Clearly, the truncation errors can be seen for step sizes larger than 10% and roundoff errors are present for small step sizes. In case of intermediate step sizes, the numerically calculated derivative well reproduces the analytical result plotted as the horizontal line in figure 1. The

vertical lines in figure 1 indicate the number of equal digits for ( )xf and ( )hxf + ; these

numbers are derived from figure 2. Comparing the results for a resolution of 6 digits (4 digits) with the vertical line for three (two) equal digits, it is obvious, that the above given recipe works well in

this case. Figure 2 gives the difference and the relative difference of ( )hxf + and ( )xf as a

function of the applied step size h , respectively.

Figure 1: Numerical

evaluation of x

e at

1=x for different

resolutions

0

2

4

6

8

10

12

14

diffe

ren

tial q

uo

tien

t (f

( x+

h)

- f(

x))

/ h

10-6

10-5

10-4

10-3

10-2

10-1

1

step size h

4 digits

5 digits

6 digitso

ne

eq

ua

l d

igit

two

eq

ua

l d

igits

thre

e e

qu

al d

igits

resolution:


Figure 2: Differences as a function of the applied step size (same condition than figure 1)

10-6

10-5

10-4

10-3

10-2

10-1

1

10

differences

10-6

10-5

10-4

10-3

10-2

10-1

1

step size h

relative difference (f(x+h) - f(x)) / f(x) at x=1

difference f(x+h) - f(x) at x=1

4.2 The dependence of the quality function on one parameter

This example discusses the dependence of the quality function 2

χ on the parameter p

Z , which

was arbitrarily chosen from a practical application using a viscoplastic Chaboche-type model

adjusted to a large set of experimental data. The variation of the parameter p

Z , which

represents the viscosity of the material, was done with all other parameters fixed to their optimised values.

Figure 3: Dependence of

the 2

χ function on one

arbitrarily chosen parameter

1

10

102

103

104

105

106

107

108

Chi2

400 600 800 1000 1200 1400 1600 1800 2000

Parameter Zp


Figure 4: The2

χ function

near the minimum (enlargement of figure 3)

0

10

20

30

40

50

60

70

Chi2

800 850 900 950 1000

Parameter Zp

From figure 3 it can be concluded, that both is possible, the 2

χ function is steep or flat. And, the

valley for the minimal 2

χ is narrow. In such a case, it is very difficult to identify the minimum of

the 2

χ function with a stochastically working method like the Monte-Carlo method. Figure 4 is an

enlargement of figure 3, showing that parabolic approximation of 2

χ is possible near the

minimum.

5 Example As an example, results for a Chaboche type model are described here. The set of differential equations used within FitIt are:

Epowtot

σ

εε

�� +=

( )powpowpow

p ασε −= sgn�� with

n

pow

pow

pow

Zp

ασ −

=�

21ααα +=

pow

ipowiiipowiiRpDH ααφεα −−= �� with 2,1=i

( ) ( )powssss

B εφφφ1,1,11

exp1 −−+=

( ) ( )powssss

pB2,2,22

exp1 −−+= φφφ


The model parameters are: E , pow

Z , n , 2,1

H , 2,1

D , R , ss,1

φ , 1

B , ss,2

φ , 2

B . The

parameters were adjusted simultaneously to a large set of experiments including tensile tests, creep tests and cyclic tests. Part of the results are given in the following figures 5-7.

Figure 5: Tensile test at 625°C Loading rate: 10e-5 1/s Application: stress over total strain units; MPa, s symbols: measurements line: Chaboche Model

Figure 6: Creep test at 625°C applied stress: 135 MPa Application: inelastic strain over time units; MPa, s symbols: measurements line: Chaboche Model


Figure 7: LCF test at 625°C Loading rate: 10e-3 1/s Application: stress over total strain units; MPa, s symbols: measurements line: Chaboche Model

6 Conclusion

The automated optimisation of model parameters is possible for models defined as a set of ordinary differential equations or models defined as an analytical function. Stability and convergence is achieved using the Levenberg-Marquardt method, a suitable choice of the

2χ function, implicit integration in case of differential equations, the singular value decomposition

method and a recipe for the calculation of derivatives, respectively.

7 Reference

W.H. Press, et al. Numerical Recipes. Cambridge University Press, Cambridge 1986

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