INTEGRATING A LINEAR INTERPOLATION FUNCTION ACROSS TRIANGULAR CELL BOUNDARIES Page: 4 of 19
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While multi-cell Taylor's series of interpolation functions are generally not valid for
numerical analyses, modified versions of these expansions are used for particle localization. For
spatial-transformation, the arguments of the interpolation function are logical-coordinate and cell-
vertex coordinate vectors. Existing logical-coordinate evaluation methods, generalized in
Reference  for various cell geometries, were developed from a truncated, single-variable
Taylor's series expansion of the interpolation function [1,3,5-7]. The modified Taylor's series
avoids discontinuous interpolation derivatives across cell boundaries by ignoring the function's
dependence on cell-vertex coordinates. Furthermore, non-linear spatial-transformation problems
are linearized by only considering the interpolation function's first-order dependence on logical
coordinates. The iterative solution of the resulting system of equations is, however, neither
algorithmically robust nor computationally efficient. An alternative expansion for linear
interpolation functions is required for robust and efficient particle localization methods.
An alternative type of expansion, a discrete-expansion, was recently proposed and
validated for multi-linear interpolation functions [8-10]. Discrete-expansions are similar to multi-
variable expansions but, unlike a Taylor's series, they are valid throughout a discretized domain.
Discrete-expansions are valid for numerical analyses since they acknowledge the full functional
dependence of interpolation and account for discontinuous derivatives across cell boundaries.
Furthermore, the solution of discrete-expansions for logical-coordinate evaluation is both
algorithmically robust and computationally efficient. Using a simple finite-difference technique, a
single discrete-expansion was developed for trilinear interpolation defined within three-
dimensional hexahedral cells [8,9]. Multiple discrete-expansions were recently developed for
bilinear interpolation defined within quadrilateral cells . These two-dimensional discrete-
expansions were developed using a general total-differential technique.
This report presents the development of discrete-expansions for linear interpolation
defined within two-dimensional triangular cell geometries. This report serves as a companion
paper to Reference  where the bilinear discrete-expansions were presented. This report will
show that the new discrete-expansions are a simplification of the trilinear and bilinear expansions;
linear interpolation is more simple than a multi-linear function. The unique formulations of
discrete-expansions for linear interpolation, however, will also be identified. This report continues
by parametrically integrating the linear interpolation function's total-differential between two
particles located in separate, non-contiguous grid cells. Application of the new linear
interpolation expansions for numerical analysis or localization within particle methods is beyond
the scope of this report. The utility of discrete-expansions for these purposes, however, is outlined
and discussed in Reference . A summary concludes this report, and then an appendix presents
a test problem, which clearly demonstrates the validity of linear discrete-expansions.
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WISEMAN, J. R. & BROCK, J. S. INTEGRATING A LINEAR INTERPOLATION FUNCTION ACROSS TRIANGULAR CELL BOUNDARIES, article, April 1, 2000; New Mexico. (digital.library.unt.edu/ark:/67531/metadc719526/m1/4/: accessed December 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.