Reproducing kernel element method Part III: Generalized enrichment and applications

Abstract

In this part of the work, a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable. This is accomplished by either multiplying enrichment functions with the original global partition polynomials, or increasing the order of global partition polynomials in the same mesh. Without refining mesh, high order consistency in interpolation hierarchy with generalized Kronecker delta property can be straightforwardly achieved in quadrilateral and triangular mesh in 2D by the proposed scheme. Comparing with the traditional finite element methods, the construction proposed here has more flexibility and only needs minimal degrees of freedom. The optimal element with high reproducing capacity and overall minimal degrees of freedom can be constructed by the generalized enrichment procedure. Two optimal elements in two dimensional space have been constructed: T10P3I 4 3 triangular element satisfies third order consistency condition with only 10 degrees of freedom, and Q15P4I 4 3 quadrilateral element satisfies fourth order consistency condition with 15 degrees of freedom. The performance of interpolation hierarchy is evaluated through solving some bench-mark problems for thin (Kirchhoff) plates.