Page 190 - Numerical Analysis and Modelling in Geomechanics

BACK ANALYSIS OF GEOTECHNICAL PROBLEMS 171
The solution of this equation system is reached through an iterative procedure.
Each iteration requires the inversion of the sub-matrix K 22 of the assembled
stiffness matrix (cf. eq. (6.5)). The assembled matrix is evaluated on the basis of
the parameter vector p determined at the end of the preceding iteration.
Other approaches for the back analysis of elastic parameters have been
proposed in the literature, still leading to a set of non-linear equations. The one
proposed in [4] could be mentioned; which offers the advantage of being
applicable also in the case of non-linear or time dependent material behaviour.

Direct solution technique

An alternative back analysis procedure can be based on the minimisation of the
discrepancy between the field measurements and the corresponding numerically
evaluated quantities. This approach presents the advantage of avoiding
the “inversion” of the stress analysis equations, which was required by the
technique discussed in the previous section.
The following error function E r can be adopted to define the discrepancy
between the measured displacements (denoted by a star) and those deriving from
a numerical stress analysis in which a given set of material parameters p is used,

(6.11)

Note that the error function depends, through the numerical results, on the
parameters being back calculated, which in this context has a rather general
meaning and may correspond to elasticity or shear strength properties, viscosity
coefficients, etc. Consequently, the back analysis reduces to determining the set
of parameters that minimises the error function, i.e. that leads to the best
approximation of the field observation through the chosen numerical model.
The error defined by eq. (6.11) is in general a complicated non-linear function
of the unknown quantities, and in most cases the analytical expression of its
gradient cannot be determined. This is particularly evident for non-linear or
elasto-plastic problems. Therefore, the adopted minimisation algorithm must
handle general non-linear functions and should not require the analytical
evaluation of the function gradient.
Methods of this kind, known in mathematical programming as direct search
methods, are iterative procedures that perform the minimisation process only by
successive evaluations of the error function [7, 8]. In the present contest, each
evaluation requires a stress analysis of the geotechnical problem on the basis of
the trial vector p chosen for that iteration.
In most practical cases some limiting values exist for the unknown
parameters. For instance, the modulus of elasticity or the cohesion cannot have
negative values. These limits, expressed by inequality constraints, can be easily