Numerical recipes for elastodynamic virtual element methods with explicit time integration

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.

Computational Morphogenesis: Morphologic constructions using polygonal discretizations

“…More advanced stabilization schemes 67 do not improve the trend of error evolution. For example, even if a matrix-based stabilization scheme 43 is used, the error evolution trend does not change in this study. Basically, the edge straightening in the morphogenesis improves the element quality, which reduces the estimated errors of coarsened elements.…”

“…To approximate the solution space, projection operators are employed, which can be exactly computed, 26,39 and the evaluation of a discrete bilinear form consists of consistency and stability terms. The VEM has been utilized to solve various engineering problems such as linear elasticity, 40,41 linear elastodynamics, 37,42,43 inelasticity problems, [44][45][46] fracture problems, [47][48][49][50] Stokes problems, 51 and topology optimization. [52][53][54] Alternatively, one should note that polygonal and polyhedral elements were employed using harmonic shape functions, 55,56 shape functions from a constrained minimization process, 57,58 and maximum-entropy shape functions, 59,60 while numerical integration should be carefully performed for the construction of element stiffness matrices.…”

The virtual element method for linear elastodynamics models: Design, analysis, and implementation

“…In the most recent years, a great amount of work has also been devoted to the development of approximation methods for the numerical modeling of linear and nonlinear elasticity problems and materials. VEM for plate bending problems [21,49] and stress/displacement VEM for plane elasticity problems [16], plane elasticity problems based on the Hellinger-Reissner principle [17], two-dimensional mixed weakly symmetric formulation of linear elasticity [119], mixed virtual element method for a pseudostress-based formulation of linear elasticity [50] nonconforming virtual element method for elasticity problems [120], linear [76] and nonlinear elasticity [66], contact problems [117] and frictional contact problems including large deformations [116], elastic and inelastic problems on polytope meshes [31], compressible and incompressible finite deformations [115], finite elasto-plastic deformations [59,78,114], linear elastic fracture analysis [96], phase-field modeling of brittle fracture using an efficient virtual element scheme [6] and ductile fracture [7], crack propagation [80], brittle crack-propagation [79], large strain anisotropic material with inextensive fibers [108], isotropic damage [67], computational homogenization of polycrystalline materials [90], gradient recovery scheme [60], topology optimization [62], nonconvex meshes for elastodynamics [98,99], acoustic vibration problem [37], virtual element method for coupled thermo-elasticity in Abaqus [69], a priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations [93], virtual element method for transversely isotropic elasticity [105].…”

“…We also mention a first work in thius direc¡tion, see Ref. [98,99], where the low-order virtual element method is applied to nonconvex polygonal meshes.…”

The arbitrary‐order virtual element method for linear elastodynamics models: convergence, stability and dispersion‐dissipation analysis

“…High‐order DG methods for elastic and elastoacoustic wave propagation problems have been extended to arbitrarily shaped polygonal/polyhedral grids 7,8 to further enhance the geometrical flexibility of the DG approach while guaranteeing low dissipation and dispersion errors. Recently, the lowest‐order virtual element method (VEM) has been applied for the solution of the elastodynamics equation on nonconvex polygonal meshes 9,10 …”