Efficient numerical integration of arbitrarily broken cells using the moment fitting approach

Abstract

The finite cell method is based on a fictitious domain approach, providing a simple and fast mesh generation of structures with complex geometries. However, this simplification leads to intersected cells where the standard Gauss quadrature does not perform well. To perform the numerical integration of these cells, we use the moment fitting approach that generates an individual quadrature rule for every broken cell. In this paper, we will perform a non-linear optimization approach to find the optimal position and number of the integration points. The findings show that the proposed method leads to efficient quadrature rules that require less integration points than other existing integration methods.

Geometry smoothing and local enrichment of the finite cell method with application to cemented granular materials

“…( 2) are discontinuous-due to α(x)-the standard Gaussian quadrature scheme is not sufficient anymore, which is why the octree integration scheme [3,27] is deployed to capture the discontinuous integrands in the numerical integration process. There exist further improvements to reduce the integration costs, such as moment fitting [24,26,[44][45][46], smart octree [43,61], enhanced octree based on image-compression techniques [72,75], divergence theorem [22], Boolean FCM [1,73], Equivalent Legendre polynomials [2,88], and curve mapping based on Bézier approximation [41,42]. The recently developed non-negative moment fitting [32,63] has been proven to be well suited for complex nonlinear problems.…”

The hierarchical finite cell method for nonlinear problems: Moment fitting quadratures, basis function removel, and remeshing

“…In this chapter, we propose different versions of the moment fitting method and study their performance in terms of accuracy and robustness for linear and nonlinear applications of the finite cell method [31,33,122,[126][127][128][129][130]. To this end, in the first moment fitting method, we follow the approach suggested by Mousavi and Sukumar [121] and predefine the position of the quadrature points a priori -which turns the nonlinear moment fitting equation system into a linear one.…”

“…The moment fitting approach based on the GLP shows an excellent performance for linear applications in structural mechanics, see [128]. Considering nonlinear applications, however, there exist only a few examples in which the moment fitting performs as robust as the adaptive Gaussian quadrature scheme -where the numerical integration is performed on a quadtree (2D) or an octree (3D) mesh.…”