A vertex‐based finite volume method applied to non‐linear material problems in computational solid mechanics

Abstract

INTRODUCTIONOver the last three decades the FE method has firmly established itself as the standard approach for problems in computational solid mechanics (CSM), especially with regard to deformation problems involving non-linear material analysis [1,2]. As a contemporary, the FV method has similarly established itself within the field of computational fluid dynamics (CFD) [3,4]. Both classes of methods integrate governing equations over pre-defined control volumes [3,5], which are associated with the elements making up the domain of interest. Additionally, both approaches can be classified as weighted residual methods where they differ with respect to the weighting functions that are adopted [6].Over the last decade a number of researchers have applied FV methods to problems in CSM (see [7] for a review) and it is now possible to classify these methods into two approaches, cell-centred [8,9,10,11,12] and vertex-based [13,6,14,7]. The first approach is based upon traditional FV methods [3] that have been widely applied inin the context of CFD [4]. Subsequently, in the last decade such techniques have been applied to CSM problems involving structured [8,9] and unstructured meshes [15,10,11,12]. With regard to these techniques, it should be noted that when solid bodies undergo deformation the application of mechanical boundary conditions is best affected if they can be set at the physical boundary. However, if the disretisation approach is cell-centred then displacements at the boundary, for example, have to be projected from the nearest node of discretisation. Therefore, cell-centred approximations may be problematic when considering complex geometries where displacements at the boundary are not prescribed and are determined as part of the simulation.The second approach is based on traditional FE methods [2] and employs shape functions to describe the variation of an independent variable, such as displacement, over an element and is therefore well suited to complex geometries [13,6,14]. In a more general sense the approach can be classified as a Both the above FV approaches apply strict conservation over a control volume and have demonstrated superiority over traditional FE methods with regard to accuracy [10,7]. Some researchers have attributed this to the local conservation of an independent variable as enforced by the control volumes employed [13,14] and others to the enforced continuity of the derivatives of the independent variables across cell boundaries [10]. The objectives of this paper are to describe the application of a vertex-based FV method to problems involving elasto-plastic deformation, to describe implementations and to provide a detailed comparison with a standard Galerkin FE method for an extended range of 3D elements, consisting of tetrahedral, pentahedral and hexahedral types. MATHEMATICAL FORMULATIONIn this section standard mathematical models that have been employed generally in computational solid mechanics are presented. The models are described in a general sense with regard to dimension...

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