A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations

Abstract

We present a priori and a posteriori error analysis of a Virtual Element Method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyze a variational formulation relying only on the solid displacement and propose an H 1 (Ω)-conforming discretization by means of VEM. Under standard assumptions on the computational domain, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal order error estimate for the eigenfunctions and a double order for the eigenvalues. Since, the VEM has the advantage of using general polygonal meshes, which allows implementing efficiently mesh refinement strategies, we also introduce a residual-type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests that allow us to assess the performance of this approach.Key words: virtual element method, elasticity equations, eigenvalue problem, a priori error estimates, a posteriori error analysis, polygonal meshes 2000 MSC: 65N25, 65N30, 70J30, 76M25.Cτ := 2µ S τ + λ S tr(τ )I, where λ S and µ S are the Lamé coefficients, which we assume constant.We introduce the following bounded bilinear forms:Then, the eigenvalue problem above can be rewritten as follows:Problem 2. Find (λ, w) ∈ R × V, w = 0, such that a(w, v) = λb(w, v) ∀v ∈ V.

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].…”

A posteriori analysis for a mixed FEM discretization of the linear elasticity spectral problem

“…On this subject, we can mention as main references [8,9,10,15,16,19], whereas for the elasticity spectral problems, the literature available is scarce. In fact, there are three works where a posteriori error analysis for the elasticity eigenproblem is performed: [1,3,24].…”

A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems

“…For this reason, the authors in [24] present two kinds of lowest-order VEMs with consistent convergence, in which the first one is achieved by introducing a special stabilization term to ensure the discrete Korn's inequality, and the second one can be seen as an extension of the idea of Kouhia and Stenberg suggested in [23] to the virtual element method. Some other VEMs for elasticity problems in two and three dimensions can be found in [2,3,7,9,13,16,17,19,26,27,30].…”