The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes. In the VEM, the basis functions are implicit (virtual) | they are not known nor do they need to be computed within the problem domain. Suitable projection operators are used to decompose the bilinear form at the element-level into two parts: a consistent term that reproduces a given polynomial space and a correction term that ensures stability. In this study, we consider a low-order extended virtual element method (X-VEM) that is in the spirit of the extended nite element method for crack problems. Herein, we focus on the two-dimensional Laplace crack problem. In the X-VEM, we enrich the standard virtual element space with additional discontinuous functions through the framework of partition-of-unity. The nodal basis functions in the VEM are chosen as the partition-of-unity functions, and we study means to stabilize the standard and enriched sub-matrices that constitute the element stiness matrix. Numerical experiments are performed on the problem of a cracked membrane under mode III loading to arm the accuracy, and to establish the optimal convergence in energy of the method.
Numerical experiments with the extended virtual element method for the Laplace problem with strong discontinuities
E. BenvenutiConceptualization
;A. ChiozziSoftware
;
2018
Abstract
The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes. In the VEM, the basis functions are implicit (virtual) | they are not known nor do they need to be computed within the problem domain. Suitable projection operators are used to decompose the bilinear form at the element-level into two parts: a consistent term that reproduces a given polynomial space and a correction term that ensures stability. In this study, we consider a low-order extended virtual element method (X-VEM) that is in the spirit of the extended nite element method for crack problems. Herein, we focus on the two-dimensional Laplace crack problem. In the X-VEM, we enrich the standard virtual element space with additional discontinuous functions through the framework of partition-of-unity. The nodal basis functions in the VEM are chosen as the partition-of-unity functions, and we study means to stabilize the standard and enriched sub-matrices that constitute the element stiness matrix. Numerical experiments are performed on the problem of a cracked membrane under mode III loading to arm the accuracy, and to establish the optimal convergence in energy of the method.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
