Efficient Block Preconditioners for the Numerical Modelling of Geological Faults

M. Ferronato, G. Gambolati, C. Janna
Dept. of Mathematical Methods and Models for Scientific Applications, University of Padova, Italy
Keywords: interface finite elements, iterative solver, preconditioning
ABSTRACT: The Finite Element modelling of geological faults by penalty contact elements may give rise to illconditioned
stiffness matrices with the convergence of iterative solvers accelerated, or even allowed for, by the
development and implementation of appropriate preconditioners. The present communication investigates the
performance of three different block preconditioners in a realistic geomechanical problem of large size. The
results show that the proposed approach may represent an easy to implement, robust and effective alternative to
well-established preconditioners such as the incomplete Cholesky factorization with partial controlled fill-in.
1 Introduction
The Finite Element (FE) discretization of faulted or fractured rocks requires the use of ad hoc contact Interface
Elements (IE) to model a discontinuity within a solid continuum. Several different formulations have been
proposed in the literature, e.g. Goodman et al. (1968), Beer (1985), Cescotto & Charlier (1993). A major issue to
be addressed is the prescription of the no-penetration constraint for the contact surfaces. A frequent approach is
based on a penalty formulation which introduces a high cost for the constraint violation so as to satisfy all the
contact conditions to get the minimum of the total potential energy (Ferronato et al., 2007a). A distinct advantage
of the penalty formulation is that it leads to a symmetric positive definite (SPD) set of linearized equilibrium
equations which can be efficiently solved by the Preconditioned Conjugate Gradient (PCG) algorithm. A major
drawback, however, is that large penalty parameters are often needed to enforce accurately the constraints, thus
producing a global ill-conditioned stiffness matrix requiring effective preconditioners to accelerate, or even to
allow for, convergence.
The stiffness matrix A resulting from the FE-IE discretization of faulted porous media takes on the following block
form (Ferronato et al., 2007b):
⎥⎦
⎤
⎢⎣
⎡
=
B C
K B
A T (1)
where K and C are both SPD matrices, with C containing all the entries of the penalty contribution from the nodes
lying on the fault surfaces, and B is a rectangular matrix connecting the standard FE to the IE. Efficient
preconditioners for A are based on the preservation of the A block structure using proper approximations of K and
C. For example, several variants in this class, corresponding to different choices for the approximations of K and
C, have been recently developed and implemented in constrained optimization, e.g. Keller et al. (2000),
Bergamaschi et al. (2004), and in the numerical solution to both FE-discretized Navier-Stokes, e.g. Elman et al.
(2002), and coupled consolidation equations, e.g. Bergamaschi et al. (2007). Numerical experiments in the
simulation of fault mechanics suggest that this preconditioning strategy may be a quite promising alternative to
more traditional and well-established preconditioners such as the incomplete factorization with partial controlled
fill-in (Ferronato et al., 2007b).
The present communication focuses on a few variants of the mixed constraint approach for contact problems as
proposed by Ferronato et al. (2007b). The variants rely on neglecting, partially or fully, the extra-diagonal blocks
of the preconditioner. The eigenvalue distribution of the preconditioned matrix indicates that the approximated
variants are less efficient in terms of convergence rate than the full block preconditioner, however they can take
advantage from their cheaper application thus providing overall a more cost-effective scheme. The paper is
organized as follows. The mixed constraint approach and its variants are described, with the main properties of
the eigenspectrum of the preconditioned matrix outlined. The performance of the variants is discussed with a
large size real 3D FE-IE simulation and compared with the full Mixed Constraint Preconditioner (MCP, Ferronato

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Efficient Block Preconditioners for the Numerical Modelling of Geological Faults 2011-04-12T01:47:00-07:00 Rating: 4.5 Diposkan Oleh: Andre S