Meshing as the Choice of Basis Functions for Finite Element Analysis

Finite Element Analysis is seen as a particular technique for solving problems in Field Analysis in which some quantity which varies with position in a domain satisfies a Partial Differential Equation (PDE). Traditionally the approach to finite element analysis has been first to partition the domain into elements, sharing nodes. Each element then has a shape function associated with each node, and the shape functions associated with a given node together form a basis function. The coefficients of these basis functions are the field values at their nodes, and so the basis functions are interpolating. Gradually it became evident that some of these assumptions were unnecessary. This paper traces the development of the technology, and leads to the conclusion that in order to achieve significantly higher performance in our field analysis we need to regard the partitioning as a side-effect of the choice of basis functions instead of being the first step. Explicitly choosing the basis functions can give very significant enhancements in the cost/accuracy pareto tradeoff, which is of particular importance when the analyses are embedded in optimisation.