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An Introduction to the Virtual Element Method with Focus on Mechanics
Title: AN INTRODUCTION TO THE VIRTUAL ELEMENT METHOD WITH FOCUS ON MECHANICS Prof. Lourenço Beirão da Veiga Dipartimento di Matematica e Applicazioni Università di Milano Bicocca, Italy Abstract: The Virtual Element Method (VEM), is a very recent technology introduced in [Beirão da Veiga, Brezzi, Cangiani, Manzini, Marini, Russo, 2013, M3AS] for the discretization of partial differential equations. The VEM can be interpreted as a novel approach that shares the same variational background of the Finite Element Method. By avoiding the explicit integration of the shape functions that span the discrete Galerkin space and introducing a novel construction of the associated stiffness matrix, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for general polygonal/polyhedral meshes (even non-conforming) with non-convex elements; it allows to build easily spaces of higher global regularity, defined on unstructured meshes; it can lead to methods that satisfy exactly given constraints. The present talk is an introduction to the VEM, focusing (among the many applications and problems studied in the recent literature) on the realm of structural mechanics. We first introduce the basics of Virtual Elements (such as the definition of the discrete space, its degrees of freedom, computable projection operators) and follow by showing its application to elastic and inelastic deformation problems in two dimensions. In the final part of the talk, we show recent results for large deformation problems, both in 2 and 3 dimensions, for the lowest order case.
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