A finite volume format for structural mechanics

SUMMARY A general Finite Volume Method (FVM) for the analysis of structural problems is presented. It is shown that the FVM can be considered to be a particular case of finite elements with a non-Galerkin weighting. For structural analysis this can readily be interpreted as equivalent to the unit displacement method which involves mainly surface integrals. Both displacement and mixed FV formulations are presented for static and dynamic problems. The Finite Volume Method (FVM) evolved in the early seventies via finite difference approxima- tions on non-orthogonal grids. The popularity of the FVM has been extensive in the field of Computational Fluid Dynamics (CFD) and heat transfer.'-' On the contrary, in the field of Computational Solid Mechanics (CSM) the use of the FVM has never achieved such acceptance. An early attempt to use FV concepts in CSM is due to Wilkins6 as an alternative approximation to derivatives in a cell. In this he defines the average gradient of a function u in a volume l2 as using the well-known divergence theorem. Such a definition of gradients can be written entirely in terms of function values at the boundary of a volume and has been used in the early 'hydrocodes' of the Lawrence Livermore Laboratory. The reasons for the unpopularity of the FVM in structural mechanics is understandable, as finite volumes are well known to be less accurate than Galerkin-based finite elements for self-adjoint (elliptic) problems. A comparison between FVM and FEM has been recently pre- sented by Zienkiewicz and Oiiate.' Here the authors show that FVM and FEM share concepts such as mesh discretization and interpolation, giving precisely the same discretized systems of equations for some particular cases. This coincidence also shows clearly for 2-D and 3-D structural problems as detailed in this work. Here, however, surface integrals are mostly involved and the number of computations can be shown to be proportional to the number of 'sides' in the mesh. This leads to an overall solution cost very similar to that of FE computations (note that for a refined mesh of three-node triangles the number of sides is 1.5 times that of elements). This fact suggests that computational speed is not 'a prior? one of the keys to the possible success of FVM in structural problems. However, the possibility of obtaining the element matrices and vectors in terms of computations along the element sides opens new possibilities for the solution of some structural problems which, in the authors' opinion, may be worth exploring in detail. Indeed, it