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

Abstract

Higher-order displacement-based finite element methods are useful for simulating bending problems and potentially addressing mesh-locking associated with nearly-incompressible elasticity, yet are computationally expensive. To address the computational expense, the paper presents a matrix-free, displacement-based, higher-order, hexahedral finite element implementation of compressible and nearly-compressible (ν → 0.5) linear isotropic elasticity at small strain with p-multigrid preconditioning. The cost, solve time, and scalability of the implementation with respect to strain energy error are investigated for polynomial order p = 1, 2, 3, 4 for compressible elasticity, and p = 2, 3, 4 for nearly-incompressible elasticity, on different number of CPU cores for a tube bending problem. In the context of this matrix-free implementation, higher-order polynomials (p = 3, 4) generally are faster in achieving better accuracy in the solution than lower-order polynomials (p = 1, 2). However, for a beam bending simulation with stress concentration (singularity), it is demonstrated that higher-order finite elements do not improve the spatial order of convergence, even though accuracy is improved.

An 8-Node Plane Hybrid Element for Structural Mechanics Problems Based on the Hellinger-Reissner Variational Principle

“…The FEM is a computational tool for engineering design analysis; it fundamentally revolutionized the way we perform scientific modeling and engineering design, including automobiles, aircraft, marine structures, bridges, highways, and high-rise buildings [1][2][3][4][5][6][7][8][9]. Traditional isoparametric elements are often limited by various numerical problems, such as limited computational accuracy and sensitivity to mesh distortion.…”