Matrix-Free Higher-Order Finite Element Method for Parallel Simulation of Compressible and Nearly-Incompressible Linear Elasticity on Unstructured Meshes
  • Arash Mehraban1, Henry Tufo1, Stein Sture2, Richard Regueiro2,*
1 Department of Computer Science, University of Colorado Boulder, Boulder, CO, USA
2 Department of Civil, Environmental, and Architectural Engineering, University of Colorado Boulder, Boulder, CO, USA
* Corresponding Author: Richard Regueiro. Email: richard.regueiro@colorado.edu
(This article belongs to this Special Issue:Advances in Computational Mechanics and Optimization
To celebrate the 95th birthday of Professor Karl Stark Pister
)
Received 13 May 2021; Accepted 02 July 2021 ; Published online 31 August 2021
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.
Keywords
Matrix-free; higher-order; finite element; parallel; linear elasticity; multigrid solvers; unstructured meshes