Fundamentals - Preventive Biomechanics - page 123

Fig. 3.30 Concept of the
inverse problem with material
parameters as output signal

boundary of a structural component are defined and the constants of integration of
the set of differential equations can be derived (Boundary Value Problem). Using
the above example of ( 3.382 ), the solution of such problems where F and p are
known and S is to be found, is referred to as a direct problem or a forward
problem, see (Aster et al. 2005; Tarantola 2004).
The solution of a direct problem, thus, involves finding effects based on a
complete description of their causes.
Referring to the terminology introduced in Fig. 3.28 and using the above ter-
minology, S is the output signal whereby F and external loading, as well as p, are
input signals of the system. The direct problem is illustrated in Fig. 3.29 as
follows.
Furthermore, the direct problem can be expressed mathematically making use
of an implicit definition of ( 3.382 )

Find S ð F ; p Þ such that GF ; p ; S h i¼ 0 for given F and p :

3.4.6 The Inverse Problem

In contrast to the corresponding direct problem, the solution of an inverse problem
entails ''determining unknown causes based on observation of their effects''
(Alifanov 1994; Turchin et al. 1971). Inverse modeling makes use of the actual
results of the system response to infer values of the material parameters. Using
again the example introduced in ( 3.382 ), the inverse problem can be formulated as
follows. Find an appropriate material parameter vector p to provide the least
possible deviation of simulation data S from the experimental data S Exp

! min

S ð p Þ S Exp with S ð p Þ¼ U F ; h i: ð 3 : 383 Þ

This leads to the following expression to be minimized by an adequate choice
of p

! min :

U F ; h i S Exp ð 3 : 384 Þ

The inverse problem can thus be illustrated in Fig. 3.30 as follows.