A multiscale virtual element method for the analysis of heterogeneous media

Abstract

Summary We introduce a novel heterogeneous multiscale method for the elastic analysis of two‐dimensional domains with a complex microstructure. To this end, the multiscale finite element method is revisited and originally upgraded by introducing virtual element discretizations at the microscale, hence allowing for generalized polygonal and nonconvex elements. The microscale is upscaled through the numerical evaluation of a set of multiscale basis functions. The solution of the equilibrium equations is performed at the coarse scale at a reduced computational cost. We discuss the computation of the multiscale basis functions and corresponding virtual projection operators. The performance of the method in terms of accuracy and computational efficiency is evaluated through a set of numerical examples.

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

“…In this section, the results of parametric analyses performed using the described computational FSHP in conjunction with VEM are provided. We restrict our investigation to polycrystalline material with thin interfaces, since the other type of composite has been already extensively studied in [67,58,68].…”

Resolving vibro‐acoustics in poroelastic media via a multiscale virtual element method

“…Mesh optimization requires, as a prerequisite, numerical methods that allow for more flexible mesh generation capabilities. To achieve this, the virtual element method (VEM) is employed to accurately and efficiently resolve the heterogeneities at the fine scale, as already done by the authors for elastostatic 44 and consolidation problems 45 …”

Multiscale VEM for the Biot consolidation analysis of complex and highly heterogeneous domains

“…This allows for extension to more generalized inter-element continuity and conformity requirements [73]. The authors have introduced a multiscale VEM formulation for elasto-statics, where the VEM has been introduced within a multiscale setting considering the case of regular coarse element domains only [74]. Very recently, the VEM has been employed within a mixed-formulation setting to address elliptic problems [43,75,76].…”

“…Contrary to the work of [63], we employ the VEM to resolve both the solid and pore-pressure governing equations. Further to the methodology provided in [74], the proposed CMsVEM is specifically designed to treat the generic case of arbitrary polygonal coarse element geometries. Using this novel approach, we derive multiscale basis functions to upscale highly heteregoneous porous domains and perform the solution procedure in the time domain at the macroscopic scale at a reduced computational cost.…”