A virtual element formulation for general element shapes
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ORIGINAL PAPER
A virtual element formulation for general element shapes P. Wriggers1 · B. Hudobivnik1 · F. Aldakheel1 Received: 16 May 2020 / Accepted: 27 July 2020 © The Author(s) 2020
Abstract The virtual element method is a lively field of research, in which considerable progress has been made during the last decade and applied to many problems in physics and engineering. The method allows ansatz function of arbitrary polynomial degree. However, one of the prerequisite of the formulation is that the element edges have to be straight. In the literature there are several new formulations that introduce curved element edges. These virtual elements allow for specific geometrical forms of the course of the curve at the edges. In this contribution a new methodology is proposed that allows to use general mappings for virtual elements which can model arbitrary geometries. Keywords Virtual element method · Stabilization · Bezier splines · Isoparametric maps
1 Introduction The many different approaches to the approximate solution of problems involving partial differential equations include finite difference schemes, finite elements, finite volume techniques, boundary elements, and particle methods. Within the finite element method there have been various significant developments, including for example classical isoparametric mapping, see e.g. Hughes [15] and Zienkiewicz and Taylor [35], but also isogeometric schemes, see Cottrell et al. [10]. Research continues to be motivated by the goal of developing stable, efficient and robust discretization schemes for finite deformation applications in solid mechanics. Within this line also the virtual element technology is further refined and applied to nonlinear problems in mechanics, see e.g. Beirão da Veiga et al. [6], Chi et al. [9], Wriggers et al. [34], Artioli et al. [3], De Bellis et al. [11], Wriggers and Hudobivnik [33], Aldakheel et al. [1], Hussein et al. [17] and De Bellis et al. [12]. So far the assumptions—even for higher order virtual elements—contain the restriction to straight edges of the elements that are directly defined in the physical space, see e.g. Beirão da Veiga et al. [5]. This makes the definition of virtual elements having a general geometric shape more complicated which is due to the fact that mappings like the isoparametric map for finite elements or NURBS type
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P. Wriggers [email protected] Institute for Continuum Mechanics, Leibniz Universität, Hannover, Germany
maps in isogeometric analysis are not easily applicable. New formulations that introduce curved element edges allow specific geometrical forms of the course of the curve at the edges were discussed in Beirão da Veiga et al. [7], Artioli et al. [4] and Aldakheel et al. [2]. This paper introduces a possibility to employ general mappings also within the virtual element formulation. The idea is to map a virtual element, defined at a reference configuration, to a general shape in the initial (physical) configuration. With such a map general shapes of virtual elements can be c
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