Simulating Compressible and Nearly-Incompressible Linear Elasticity Using an Efficient Parallel Scalable Matrix-Free High-Order Finite Element Method

Abstract

We examine a residual and matrix-free Jacobian formulation of compressible and nearly incompressible (ν → 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 L 2 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. Notation Boldface denotes vectors and tensors in symbolic notation. Unless otherwise indicated, all vector and tensor products in symbolic form are assumed to be inner products, such as v v vv v v = v i v i , (a a ab b b) ik = a i j b jk and a a a : b b b = a i j b i j , where repeated indices denote a sum over those indices. Cartesian coordinates are assumed. The symbol tr(•) is the trace operator, such that tr(σ σ σ) = σ ii. The symbol I I I is the unit tensor,

Matrix-Free Higher-Order Finite Element Method for Parallel Simulation of Compressible and Nearly-Incompressible Linear Elasticity on Unstructured Meshes

“…Therefore, we conclude that for this problem the order of convergence of the FE solution on structured meshes with stress concentration (singularity) cannot be improved by using higher-order polynomials. Thus, the lack of optimal spatial convergence with regard to higherorder FE when there is a stress concentration is confirmed, as opposed to when there is no stress concentration (or singularity) [22]. The paper investigated the use of displacement-based higher-order finite elements for compressible and nearly-incompressible linear isotropic elasticity using a matrix-free implementation with p-multigrid preconditioning.…”

“…In this work, we apply a previously-developed matrix-free, higher-order, FEM for solving three-dimensional (3D) linear isotropic elasticity with p-multigrid preconditioning [22,23] to (i) a tube geometry subjected to method of manufactured solutions (MMS), (ii) a tube bending problem for compressible (ν = 0.3) and nearly-incompressible (ν = 0.499999) elasticity and polynomial order p = 1, 2, 3, 4, and (iii) a beam bending problem under body force loading for which the beam is clamped on both ends. Simulations (ii) and (iii)-but simulation (iii) in particular-cause a stress concentration (singularity) in the clamped regions, where the higher-order FEM may not converge optimally as expected [22]. Therefore, we investigate order of convergence of the matrixfree higher-order FEM for compressible elasticity when there is a stress concentration.…”