Performance of Different Integration Schemes in Facing Discontinuities in the Finite Cell Method

Abstract

In many extended versions of the finite element method (FEM) the mesh does not conform to the physical domain. Therefore, discontinuity of variables is expected when some elements are cut by the boundary. Thus, the integrands are not continuous over the whole integration domain. Apparently, none of the well developed integration schemes such as Gauss quadrature can be used readily. This paper investigates several modifications of the Gauss quadrature to capture the discontinuity within an element and to perform a more precise integration. The extended method used here is the finite cell method (FCM), an extension of a high-order approximation space with the aim of simple meshing. Several examples are included to evaluate different modifications.

Theoretical and Numerical Investigation of the Finite Cell Method

“…We set α = 10 −14 inside the hole and the quadtree is refined towards the boundary of the circle. The leafs of the quadtree correspond to the sub-cells that are used for the adaptive quadrature [2]. On each of the sub-cells a Gaussian quadrature is performed to accurately compute the stiffness matrix of the cell that is cut by the circle.…”

“…Since the quality of the finite cell approximation strongly depends on the accuracy of the numerical integration, an adaptive quadrature scheme is applied to compute the stiffness and mass matrices of cells that are cut by the boundary of the domain or include holes. The adaptive integration can be carried out very generally by applying quadtree (in 2D) and octree (in 3D) space partitioning schemes in a fully automatic, error-controlled fashion [2]. Summarizing, the finite cell method is based on three important ingredients: a fictitious domain approach, high order shape functions and an adaptive integration of the cell matrices.…”

Efficient and accurate numerical quadrature for immersed boundary methods

“…Due to the discontinuity that is introduced by penalizing the constitutive matrix C (Eq. 3), the Gaussian quadrature loses its accuracy in cut cells [5,6,32]. In order to improve the precision of the numerical integration, the FCM uses a composed Gaussian quadrature that is based on a spacetree decomposition of the cells that are cut by the domain boundaries.…”

Local enrichment of the finite cell method for problems with material interfaces

“…At the end of the partitioning, a standard Gauss quadrature rule is applied to the subcells. The quad/octree integration scheme is automatic and allows full control of the integration error [34].…”

“…This is due to the fact that the approximation of the displacement field is defined as a C 0 continuous Ansatz. Proper integration is vital if an optimal rate of convergence is to be achieved [34,47].…”