Keywords: finite element, finite volume, spatial discretisation, structural mechanics, multi-physics.

Most real-life modelling problems involve a range of continuum physics phenomena, which render them as being essentially multi-physics (MP) in nature. However, in the past there has been a tendency to either ignore one of the phenomena involved, or to treat the problem as weakly or totally un-coupled, since this was both simpler and cheaper. In recent years, the need for multi-physics capability has emerged in numerous industries due to the demand for greater definition in analysis. MP simulation requires analysis tools that handle the full spectrum of continuum physics and all the relevant levels of coupling in both a time and space accurate manner.

Our approach to MP problems involved the use of a single software framework (SSF) using compatible approaches to mesh structure, discretisation and solver strategy as it was thought that this would enable the various physical phenomena and their interactions to be solved in a consistent time and space accurate manner. Our approach led to the development of a toolkit, PHYSICA [1,2], including structural dynamics, which employed finite volume (FV) methods on unstructured meshes for all phenomena. However, it has become apparent that, given the similarity between the control volumes used for FV and finite element (FE) spatial discretisation in PHYSICA, the important factors are the consistency of the mesh across all phenomena and the accuracy of the solution. For example, in fluid structure interaction it is the accuracy of the displacement and velocity of the structure and the implementation of the two-way boundary conditions which are important, rather than whether the structural dynamics is solved using FV methods, as for the fluid, or using FE methods.

Structural Dynamics

The equilibrium equations of structural mechanics, may be written as

where is the differential operator, is the stress tensor, are body forces, is density and is the displacement. Boundary conditions on the surface of the domain are described in terms of prescribed tractions on the boundary and prescribed displacements on the boundary of the structure.

The method of weighted residuals and Green's first theorem are applied to the equilibrium equation and together with standard Galerkin techniques [3], which approximate the unknown displacement by use of the basis functions, it is transformed into the following form:

The displacement boundary condition is satisfied by restricting the weighting function associated with the integral on the boundary to be zero. The essential difference between the FE and the FV Method is that for the former, the weighting functions for a node are equal to the shape functions for that node i.e. . For cell-vertex FV method, the weighting functions are unity within the control volume and zero elsewhere, i.e. within the control volume and elsewhere [4,5].

In this paper, we will focus upon an assessment of the essential differences between FE and FV formulations where all other aspects of the solution strategy are similar. It is demonstrated that these discretisation approaches perform very similarly and can be used effectively and inter-changeably within a multi-physics environment.

1

Bailey C., Taylor G. A., Cross M. and Chow P., "Discretisation procedures for multi-physics phenomena", Journal of Computational and Applied Mathematics, Volume 103, Issue 1, Pages 3-17, 1999. doi:10.1016/S0377-0427(98)00236-2

2

PHYSICA+, http://www.multi-physics.com

3

Zienkiewicz O.C. and Taylor R.L., The Finite Element Method: Volume 1: The Basis, Butterworth Heinemann, 2000.

4

Onate E., Cervera M., and Zienkiewicz O.C., "A Finite Volume Format for Structural Mechanics", International Journal for Numerical Methods in Engineering, Volume 37, Pages 181-201, 1994. doi:10.1002/nme.1620370202

5

Taylor G.A., Bailey C. and Cross M., "A Vertex Based Finite Volume Method Applied to Non-Linear Material Problems in Computational Solid Mechanics". International Journal for Numerical Methods in Engineering, Volume 56, Issue 4, Pages 507-529, 2003. doi:10.1002/nme.574

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