Page 191 - Numerical Analysis and Modelling in Geomechanics

172 ANNAMARIA CIVIDINI AND GIANCARLO GIODA
introduced into a direct search algorithm by means of a penalisation procedure.
When a point in the space of the free variables is reached outside the feasible
domain, the error function is assigned a large value so that the minimisation
algorithm automatically drives back the optimisation path into the feasible
region.
This penalty approach is general and simple to implement. In fact, no
assumptions are required on the characteristics of the constraints and the
computer program for constrained minimisation can be easily obtained with few
modifications of the code for unconstrained minimisation.
From the computational viewpoint the back analysis approach requiring the
minimisation of the error function expressed by eq. (6.11) presents non-
negligible differences with respect to that based on the least square method (eq.
(6.10)). In fact, the inverse technique, specifically developed for the calibration of
elasticity parameters, requires a number of iterations in general smaller than that
of the direct method.
On the other hand, each iteration of the direct method involves the inversion
of a partitioned stiffness matrix, which is in general computationally more
cumber some that the solution of the equation system required by the direct
procedure.
It is not straightforward to work out a general criterion for choosing the most
convenient algorithm for back analysis. However, it should be observed that
inverse techniques are particularly convenient when dealing with a relatively
large number of unknown parameters and when the finite element mesh has a small
number of nodal variables. On the contrary, the direct procedures are preferable
when a few parameters are back analysed using large finite element meshes.
Another point to be taken into account concerns the programming effort for
implementing these techniques. In this respect the back analysis methods based
on direct search algorithms present a non-negligible advantage compared with
the inverse procedures. In fact, while the inverse procedures require the
implementation of an ad hoc computer program, the direct approach can be
implemented on the basis of standard computer codes for non-linear function
minimisation in which the finite element program for stress analysis is
introduced as a subroutine. This requires some simple changes to the original
finite element code, and a limited programming effort. In addition, the same
stress analysis and minimisation programs can be used for various back analysis
problems, merely by considering the calculated quantities as functions of the
current unknown parameters, regardless of their physical nature.

Probabilistic back analysis
The field measurements that represent the data of the back analysis are in general
affected by errors that depend on the nature of the measured quantities, on the
characteristics of the adopted devices, on the field conditions, etc. Various
techniques have been proposed in the literature to evaluate the influence of these