Fundamentals - Preventive Biomechanics - page 124

Expressed mathematically, the inverse problem can be formulated as

i¼ 0 for given F and S ð p Þ¼ S Exp :

Find p such that G F ; p ; S h

In the experimental part of this investigation, described in more detail in
Sects. 4.3 and 5.3 , experimental force and displacement data from foam and tissue
indentation are evaluated at discrete time steps. In the particular cases of tissue and
foam parameter identification, simulation force and displacement measures,
instead of stress measures, are used as comparable quantities in the parameter
optimization process.
Inverse modelling as a form of parameter optimization is associated with
several difficulties, briefly considered in the following. An overview on the topic is
provided in Bertero (1997); Santamarina and Fratta (2005) and Tarantola (2004).
One of the problems that arise regarding inverse modelling, aside from the
problems of existence and continuous dependence, is the issue of uniqueness.
These problems represent a typical mathematic property of inverse problems, and
they are summarized under the term of ill-posedness. The opposing property is
well-posedness, a concept first introduced by Hadamard (1923). A problem is
called well-posed if the three previously introduced requirements are satisfied
(Courant 1989).
While the direct problem, in a deterministic context, is well-posed and has a
unique solution, the inverse problem does not. Especially, the issue of non-
uniqueness was observed in the parameter identification processes described in the
subsequent chapters. In this context, non-uniqueness leads to more than a single set
of parameters yielding minimum values for the objective function, and thus good
correlation with the experimental data. Depending on the model function structure
this can be due to the fact that effects on the sum of squared residuals, as described
subsequently, of changes in one parameter can possibly be compensated by
another parameter. This can especially be true if the number of parameters is large.
In particular, the problem of non-uniqueness arises when employing material
models such as the O GDEN -model (Ogden 1972a, b). Such models are purely
continuum-based and incorporate phenomenological material parameters, which
are not physically motivated. Using such a model to describe homogenous
material, the parameters are usually characterized by reproducing specific,
homogenous experiments, e.g. uniaxial-, biaxial-, planar- or volumetric com-
pression/tension and shear. In this context, often the evaluated model parameters,
using one specific experiment, do not validly predict a case of arbitrary defor-
mation, even if the particular experiment is accurately simulated. Then, it is
necessary to perform multiple experiments to employ more than a single objective
function in the parameter optimization process, to find an appropriate parameter
set minimizing this multi-objective problem.
Parameter identification using the inverse modelling technique requires a pro-
gram to solve the direct problem, Fig. 3.28 b, for each tested parameter set to
obtain comparable simulation output. A convenient computational method
regarding nonlinear analysis of mechanical problems involving complex material