Numerical recipes for elastodynamic virtual element methods with explicit time integration. (22nd October 2019)

• Record Type: Journal Article
• Authors: Park, Kyoungsoo; Chi, Heng; Paulino, Glaucio H.
• Is Part Of: International journal for numerical methods in engineering. Volume 121:Number 1(2020)
• Page Start: 1
• Page End: 31
• Publication Date: 2019-10-22
• Subjects: convex and nonconvex elements -- critical time step -- elastodynamics -- lumped mass matrix -- virtual element method (VEM); Numerical analysis -- Periodicals; Engineering mathematics -- Periodicals; 620.001518
• DOI: 10.1002/nme.6173
• Languages: English
• ISSNs: 0029-5981

Abstract

Summary: We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B‐bar method. We show that, to establish a B‐bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.