Biomedical Engineering Reference
In-Depth Information
finite element method. The major task after modelling, is that the required material
parameters of the constitutive equations to solve the boundary value problem are still
unknown and must still be determined! To approach this problem, the ''inverse finite
element method (iFEM)'' is used where the quality functional (
3.364
) can be derived
based on an initial ''guess'' of the material parameters. This is done by means of the
finite element solver, and with respect to the quality functional, it can be determined
whether additional iteration loops are needed. Thus, for each variation of the
parameter vector p, a complete finite element simulation of the particular boundary
value problem is required, possibly including simulation run time. The agreement of
the simulation output with the test data, e.g. force-displacement data, represents the
quality criterion to be checked for each iteration.
A further difficulty that arises in conjunction with in vivo experiments on the
human body is that the anatomical data to be modelled (which represents the
''material sample'') by finite elements must be digitalized. A first step in digita-
lization is using imaging techniques such as MRI and then three-dimensional
reconstruction, cf.
Sect. 5.3.3.1
.
Stability Criteria. The success of the optimization process outlined previously to
determine an optimal parameter vector p strongly depends on the stability of the finite
element simulations. An important contribution is provided by choosing material
parameters satisfying certain restrictions which often arise in the form of inequali-
ties. A more detailed discussion on material stability is provided in
Sect. 3.4.8
.
3.4.2 Overview and Classification
The search for an optimal state is one of the most fundamental principles. Natural
physical systems are optimized in the sense that they tend towards a state of
minimum energy. Man-made solutions are optimized to minimize costs, while
striving for maximum efficiency in production processes or maximum strength of a
mechanical structure at minimum weight. Optimization is a process to obtain the
best of all possible solutions. This process entails criteria to define whether a
solution is good or bad.
Though the desire for optimization (perfection) seems inherent in humans, nature
cannot be credited with cognizant optimization. Mutation, cross-over and natural
selection are the optimizers with the biological 'constructions' adapting themselves
to the particular (dynamic) environmental conditions in the struggle for survival.
Basic terminology is introduced and serves as the background in subsequent
chapters. More detailed overviews about taxonomies of optimization algorithms
are given in Luenberger (1984); Nash (1990); Nocedal and Wright (1999); Rao
(2009); Schwefel (1994) and Spall (2003).
Optimization, in general, requires the existence of more than one possible
solution of conditions for a system to work. If more than a single solution exists,
the question arises as to which solution works best in a particular situation. To
