Computational homogenization of polycrystalline materials with the Virtual Element Method

The Virtual Element Method for the numerical homogenization of electro‐mechanical responses

“…The virtual element approach employs a split of primary fields U = {u, φ} into projected part and a remainder U = U h +(U − U h ). The latter induces a split of the potential into a consistency and a stabilization part [4], with a control-parameter β ∈ [0, 1] for stabilization-influence. Following results in [4], the latter parameter is chosen equal to β = 0.1.…”

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].…”

Statistical homogenization of polycrystal composite materials with thin interfaces using virtual element method

“…In this paper, low order virtual elements are used: the adoption of virtual elements of order one is particularly suitable for the homogenization procedure, as shown in [36,37,58]. Moreover, stress/strain is constant over the elements, but this approximation do not affect homogenization results.…”