We examine a residual and matrix-free Jacobian formulation of compressible and nearly incompressible (v → 0.5) displacement-only linear isotropic elasticity with high-order hexahedral finite elements. A matrix-free p-multigrid method is combined with algebraic multigrid on the assembled sparse coarse grid matrix to provide an effective preconditioner. The software is verified with the method of manufactured solutions. We explore convergence to a predetermined L2 error of 10-4, 10-5 and 10-6 for the compressible case and 10-4, 10-5 for the nearly-incompressible cases, as the Poisson's ratio approaches 0.5, based upon grid resolution and polynomial order. We compare our results against results obtained from C3D20H mixed/hybrid element available in the commercial finite element software ABAQUS that is quadratic in displacement and linear in pressure. We determine, for the same problem size, that our matrix-free approach for displacement-only implementation is faster and more efficient for quadratic elements compared to the C3D20H element from ABAQUS that is specially designed to handle nearly-incompressible and incompressible elasticity problems. However, as we approach the near incompressibility limit, the number of Conjugate Gradient iterations required to achieve the desired solution increases significantly.
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