Error estimation and adaptivity for incompressible hyperelasticity

Abstract

SUMMARYA Galerkin FEM is developed for nonlinear, incompressible (hyper) elasticity that takes account of nonlinearities in both the strain tensor and the relationship between the strain tensor and the stress tensor. By using suitably defined linearised dual problems with appropriate boundary conditions, a posteriori error estimates are then derived for both linear functionals of the solution and linear functionals of the stress on a boundary, where Dirichlet boundary conditions are applied. A second, higher order method for calculating a linear functional of the stress on a Dirichlet boundary is also presented together with an a posteriori error estimator for this approach. An implementation for a 2D model problem with known solution, where the entries of the strain tensor exhibit large, rapid variations, demonstrates the accuracy and sharpness of the error estimators. Finally, using a selection of model problems, the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity.

A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations

“…We will calculate a numerical solution of the governing equations using the finite element method described by Whiteley and Tavener [13]. In this section we first briefly summarise this scheme.…”

“…(7) and (8) differs very slightly from that used in [13] through q being replaced by −q. This has the effect of multiplying the discretisation of the constraint Eq.…”

“…This has the effect of multiplying the discretisation of the constraint Eq. (2) used in [13] by a factor of −1, and will allow us to write the Jacobian matrices arising from Newton's method in a more convenient form. The finite element solution follows a similar pattern to the weak solution.…”

“…Our first set of simulations uses a model problem described in [13], specified on the unit square 0 < X, Y < 1. For constant a > 0 we define…”

“…The system of partial differential equations describing this model are often solved numerically using the finite element method [7][8][9][10][11][12][13]. One step in the practical computation of the finite element approximation is B J. P. Whiteley jonathan.whiteley@cs.ox.ac.uk 1 Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK the solution of a system of nonlinear algebraic equations, often containing a large number of degrees of freedom.…”