inite element analyses of 3D rubber problems generate large system on nonlinear equations. The solution of these systems, using the Newton-Raphson technique, requires the solution of several large systems of linear equations. For large 3D problems, this becomes the dominant factor in CPU time and storage spent.
For reasons of economy, it would be desirable to solve these large systems with iterative solvers instead of direct solvers such as LU decomposition methods. The rate of convergence of standard iterative techniques such as the conjugate gradient method deteriorates when the condition number of the matrix is large. A large condition number may be due to mesh refinement, mesh distortion or large variation of material coefficients. In this work, we analyze the bad condition number of the stiffness matrix due to a large bulk modulus (near incompressible materials). To improve the convergence rate, a preconditioner for the gradient method is formulated.
The preconditioner is based on projections onto a subspace associated with bulk mode of deformations. As a first step towards the solution of the nonlinear 3D rubber problems, the method has been applied to model problems in linear near-incompressible 2D elasticity. The performance of the iterative method compares favorable with direct solvers as well as with the standard conjugate gradient method, in terms of operation count and CPU time. The method is stable for the Stokes equations (complete incompressibility) as well.