A mixed virtual element method for a pseudostress-based formulation of linear elasticity

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 computational implementation of the mixed-VEM method for the brinkman problem in 2D

“…donde P K k : L 2 (K) → P k (K) es el proyector L 2 (K)-ortogonal (ver (20)). Para ello, se emplea la propiedad (20) con la base {φ K i } M i=1 , donde es requerida una cuadratura adecuada para dominios poligonales.…”

“…Respecto a la validación del código, esta se presenta en la Sección 6 de [18], donde se demuestra que los resultados coinciden con el análisis desarrollado a lo largo de [18]. Además, en [20] se sigue el mismo paradigma de programación para el problema de elasticidad lineal, mientras que en [19] se extiende a un problema no lineal. En este sentido, en [26] se emplearon todos los operadores discretos presentados en la Sección 5 para una versión no lineal del problema de Brinkman estudiado en este trabajo.…”

“…Más aún, el análisis expuesto en [17] permite estudiar problemas de la misma naturaleza que Stokes, como ocurre con el problema de Brinkman, donde en [18] se introdujo y se analizó un método mixto de elementos virtuales. Para otras contribuciones sobre los métodos VEM, se considera, por ejemplo, a [4,7,12,15,19,20,22,25,26].…”

A low-order nonconforming method for linear elasticity on general meshes

“…In the context of solid-mechanics, this feature can be useful for several reasons including, e.g., improved robustness to mesh distortion and fracture, local mesh refinement, or the use of hanging nodes for contact and interface problems. A nonexahustive list of contributions in the context of elasticity problems includes [26,30,4,5,23,24,20,18,6,3,28,11,29,10]; see also references therein.…”