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# A Note on the Effect of the Choice of Weak Form on GMRES Convergence for Incompressible Nonlinear Elasticity Problems

[+] Author and Article Information
Pras Pathmanathan1

University of Oxford Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UKpras@comlab.ox.ac.uk

Jonathan P. Whiteley

University of Oxford Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UKjonathan.whiteley@comlab.ox.ac.uk

S. Jonathan Chapman

University of Oxford Mathematical Institute, Oxford OX1 3LB, UKchapman@maths.ox.ac.uk

David J. Gavaghan

University of Oxford Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UKgavaghan@comlab.ox.ac.uk

This is a mathematical inverse problem, but we use the terminology ”backward problem” rather than ”inverse problem,” as the latter is generally taken to mean the problem of determining material parameters given a deformation.

The zero block occurs because, if $i$ corresponds to an index of a pressure unknown in $a$, then (considering the forward problem only) $Fi=∫Ω0ψ(det F−1)dV0$ (for some basis function $ψ$), which is independent of the pressure $p$.

In this paper, we only consider the full Newton method (i.e., where the full Jacobian defined by Eq. 4 is used in each iteration).

1

Corresponding author.

J. Appl. Mech 77(3), 034501 (Feb 01, 2010) (6 pages) doi:10.1115/1.4000414 History: Received August 05, 2008; Revised July 15, 2009; Published February 01, 2010; Online February 01, 2010

## Abstract

The generalized minimal residual (GMRES) method is a common choice for solving the large nonsymmetric linear systems that arise when numerically computing solutions of incompressible nonlinear elasticity problems using the finite element method. Analytic results on the performance of GMRES are available on linear problems such as linear elasticity or Stokes’ flow (where the matrices in the corresponding linear systems are symmetric), or on the nonlinear problem of the Navier–Stokes flow (where the matrix is block-symmetric/block-skew-symmetric); however, there has been very little investigation into the GMRES performance in incompressible nonlinear elasticity problems, where the nonlinearity of the incompressibility constraint means the matrix is not block-symmetric/block-skew-symmetric. In this short paper, we identify one feature of the problem formulation, which has a huge impact on unpreconditioned GMRES convergence. We explain that it is important to ensure that the matrices are perturbations of a block-skew-symmetric matrix rather than a perturbation of a block-symmetric matrix. This relates to the choice of sign before the incompressibility constraint integral in the weak formulation (with both choices being mathematically equivalent). The incorrect choice is shown to have a hugely detrimental effect on the total computation time.

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## Figures

Figure 1

Solving the backward problem on a sheared cube: (a) shape taken as the deformed starting state for a backward problem computation (obtained by solving a forward problem on a cube), (b) result of the backward computation

Figure 2

Convergence histories for GMRES(50) on linear systems arising from a forward and backward problem

Figure 3

Spectra of some matrices: (a) forward problem; (b) backward problem, first Newton iteration; (c) backward problem, second Newton iteration; (d) modified backward problem, second Newton iteration

Figure 4

Convergence histories for GMRES(50) on the forward problem and two versions of a backward problem matrix. “Backward problem +” represents the original backward problem matrix: a perturbation of A+, and “backward problem −” represents the modified backward problem matrix: a perturbation of A−.

Figure 5

The average number of outer GMRES iterations per Newton iteration in various forward simulations. Black bars: positive choice of sign in incompressibility term. White bars: negative choice of sign. Simulations 1–3 are on the unit square, with solution x=(X+αX2/2,Y(1+αX)−1)T: (1) α=0.2, 800 triangular elements; (2) α=0.4, 800 elements; (3) α=1, 200 elements. Simulation (4) is on the unit cube, with solution x=(X+αX2/2,Y+αY2/2,Z(1+αX)−1(1+αY)−1)T, with α=0.1 and 750 tetrahedral elements.

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