Methods for improving the computational performance of sequentially linear analsysis

Abstract

The numerical simulation of brittle failure with nonlinear finite element analysis (NLFEA) remains a challenge due to robustness issues. These problems are attributed to the softening material behaviour and the iterative nature of the Newton-Raphson type methods used in NLFEA. However, robust numerical simulations become increasingly important, for example due to recent developments in Groningen. 
To address these issues, sequentially linear analysis (SLA) was developed which exploits the
fact that a linear analysis is inherently stable. By assuming a stepwise material degradation
the nonlinear response of a structure can be approximated with a sequence of linear analyses.
Although this approach has been proven to be effective for several case studies, the numerical
performance is still a problem that has to be solved. After every linear analysis, a single
element is damaged resulting in incremental damage. As a result, the system of equations
only changes locally between these linear analyses. Traditional solution techniques do not
exploit this property and calculate a matrix factorisation every linear analysis, resulting in high
computational times per analysis step. Since SLA typically requires many linear analyses to
obtain the desired structural response, this leads to unacceptable analysis times.
The aim of this thesis is to improve the computational performance of SLA by developing
numerical solution techniques which exploit the incremental approach of SLA. To this extend,
the following methods have been developed.
A direct solution technique has been developed which is based on the Woodbury matrix
identity. This identity allows for the numerically cheap computation of the inverse of a
low-rank corrected matrix. In this approach, the expensive matrix factorisation does not
have to be calculated every linear analysis step. Instead, the old factorisation can be
reused along with some additional matrix- and vector multiplications and solving a significantly
smaller linear system of equations. An optimal strategy is derived to determine
at which point a new factorisation should be calculated.
An improved preconditioner for the conjugate gradient (CG) method has been developed.
Instead of an incomplete factorisation, the complete factorisation is used as a
preconditioner which reduces the number of required CG iterations significantly. The
point at which too many CG iterations are required and a new factorisation is necessary,
is determined using the same strategy as the first method.
From numerical experiments it follows that both methods perform significantly better than the
direct solution method, especially for large 3-dimensional problems. The best performance
is achieved using the Woodbury matrix identity resulting in the solver no longer being the
dominant factor in SLA. Furthermore, significantly larger problems are not solvable in time
frames in which previously only smaller problems were solved.