An iterative algorithm for the finite element analysis of near-incompressible materials

A new preconditioner for the conjugate gradient method applicable to the finite element analysis of near-incompressible finite elasticity has been developed. The boundary value problem is formulated with a displacement/pressure mixed variational statement. The pressure degrees of freedom are eliminated at the element level. The problem is linearized using a Newton-Raphson technique. Each linear system is solved using the preconditioned conjugate gradient.;The new preconditioner is based on projections onto a subspace associated with the bulk modes of deformation. The condition number of the preconditioned stiffness matrix and consequently the rate of convergence of the preconditioned conjugate gradient are totally insensitive to increasing bulk modulus.;The method has been applied to model problems in linear and nonlinear 2D and 3D using a near-incompressible hyperelastic material model. The performance of the iterative method compares favorable with direct solvers as well as with conjugate gradient methods for indefinite systems, in terms of operation count and CPU time.