Integrating a Bilinear Interpolation Function Across Quadrilateral Cell Boundaries

Abstract

Computational models of particle dynamics often exchange solution data with discretized continuum-fields using interpolation functions. These particle methods require a series expansion of the interpolation function for two purposes: numerical analyses used to establish the models consistency and accuracy, and logical-coordinate evaluation used to locate particles within a grid.This report presents a new method of developing discrete-expansions for interpolation; they are similar to multi-variable expansions but, unlike a Taylor's series, discrete-expansions are valid throughout a discretized domain. Discrete-expansions are developed herein by parametrically integrating the interpolation function's total-differential between two particles located within separate, non-contiguous cells. Discrete-expansions are valid for numerical analyses since they acknowledge the functional dependence of interpolation and account for mapping discontinuities across cell boundaries. The use of discrete-expansions for logical-coordinate evaluation provides an algorithmically robust and computationally efficient particle localization method. Verification of this new method is demonstrated herein on a simple test problem.