Nitsche’s Method - An approach to imposing essential boundary conditions in Element Free Galerkin Implemented in INSANE

Abstract

Mesh-free methods use nodes to establish a system of algebraic equations. One of the advantages of mesh free methods is their independency of element connectivity, allowing some freedom in dealing with complex problems, such as large deformation, crack propagation, complex geometry, fluid flow, among others. The Element Free Galerkin is an example of such methods. As some mesh-free methods, its shape functions do not present the Kronecker Delta property, which is one of the reasons that the imposition of essential boundary conditions is not trivial as it is in FEM, for instance. There is a large effort to finding an efficient strategy for imposition of essential boundary conditions in mesh-free methods, besides the well known Lagrange multipliers, penalty and FEM coupling methods. As an alternative, Nitsche’s method presents a consistent variational formulation and renders a better conditioned system matrix as it requires a smaller scalar factor to be used, in comparison to the penalty method. It also maintains the size of the original algebraic system of equations as opposed to the Lagrange multiplier method. However, the generalization and implementation of this method is not straightforward and is problem dependent in contrast to the methods aforementioned. The aim of this paper is to show the results of an implementation of the Nitsche’s method in INSANE and compare the results of different methods for imposition of essential boundary conditions against it.