A framework for studying the rkem representation of discrete point sets

Abstract

The application of engineering analysis to new areas, such as nanomechanics and the life sciences, often involves geometric problem domains defined by discrete point sets as measured from diagnostic equipment. The development of a suitable mesh for finite element analysis can be a tedious task. One approach to simplifying the geometric description is to use a parametrized set of basis functions, and fit the parameters to the data set. In this paper, we discuss the problem of determining suitable parameters for the Reproducing Kernel Element Method representation of discrete point sets, and in particular the solution of the inverse problem of determining pre-image evaluation points in the parametric space that correspond to a given input point. We justify our solution by posing a theoretical framework and an error indicator.