Numerical Integration of on-the-fly-computed Enrichment Functions in the PUM
The approximation power of a Partition of Unity Method stems from the use of problem-dependent enrichment functions which are, in general, non-smooth functions. However, these enrichments are not always known analytically. Even in the case of analytically known enrichments, their integration is usually challenging. The direct use of standard quadrature rules is, in general, not appropriate. But, effective numerical integration is possible by using all a priori information of the analytically known enrichment, i.e. the locations of discontinuities and singularities and the orders of the latter.In this study, we now consider the more involved case of numerical enrichments which are computed on-the-fly via an embedded particle method. Therefore, subdivision approaches using a priori information are not (directly) applicable. This study aims to investigate, whether the particularities of the underlying particle method and the construction of the enrichments might allow for better convergence than theory would suggest.
Meshfree Methods for Partial Differential Equations VII