Fundamentals - Preventive Biomechanics - page 109

3.4 Parameter Identification (Material Identification)

3.4.1 Introduction

The significance of finite element simulation depends on the quality of the material
parameters and the material models employed in modelling the involved struc-
tures. Besides modelling techniques (mesh quality, element size etc.), emphasis
must be put on material parameter identification of the employed continuum
mechanical material equations. In the majority of applications here, material
parameters are used which are non-linearly related to the stress, cf. e.g. equation
( 3.272 )or( 3.274 ), such that non-linear optimization algorithms must be used for
parameter estimation. In this context, the term determination of a material equa-
tion is often used. A crucial point in optimally determining and identifying
material parameters is good agreement and representation with experimental
findings such as force-displacement data of the material. This process is presented
in a general way and is referred to in the relevant Sects. 4.3.1 , 4.3.2 and
Sects. 5.3.3 , 5.3.5 .

Quality Functional. To optimally reproduce experimental findings, a measure
must be introduced to quantify the deviations between measured experimental data
and simulated data. The measure is based on a material equation to indicate the
quality of the reproduction or mapping of the experimental data with a theoretical
model equation. Commonly, this is accomplished using a quality functional U of
the form

m s

X

n

U ð p Þ : ¼ 1
n

m ¼ ! f i h i ; ð Þ f i ð h i Þ

½ min : ð 3 : 364 Þ

i ¼ 1

In ( 3.364 ), U ð p Þ denotes the quality, f i M (h i , p) is the model function, i.e. the
function based on the material equation or more precise, f i M are the model function
values, p is the parameter vector, i.e. the set of material parameters and it holds
n-independent components (variables) x i which can be adjusted to minimize (or
maximize) the quantity U, the h i are the independent variables and the f i E are
experimental data (e.g., the h i may represent the displacements and the f i the force,
stress or strain data), n is the number of data points and m is the ''norm parameter''
(m, n [ R + ).
If no restrictions exist, the variables x i can be freely chosen from the parameter
domain E n , also commonly denoted R n , denoting n-dimensional Euclidian space.
The parameter vector p can thus be represented by any point in E n .
Although p represents a column ð n 1 Þ matrix of rank n and thus does not
follow tensor transformation, the term parameter vector will still be utilized, since
it is commonly used.