Adaptive numerical integration of broken finite cells based on moment fitting applied to finite strain problems

Abstract

In the finite cell method (FCM) which is based on the fictitious domain approach the numerical integration of broken cells represents a major challenge. Commonly, an adaptive integration scheme is used which, usually, results in a large number of integration points and thus increases the numerical effort, especially for nonlinear applications. To reduce the number of integration points, we present an adaptive scheme which is based on moment fitting. Thereby, we introduce an approach based on Lagrange polynomials which avoids the necessity of solving the moment fitting equation system. We study the performance of this integration method considering two numerical examples for finite strain problems of the FCM. Adaptive integration based on moment fittingTo perform the numerical integration of broken cells more efficiently, recently, within the framework of the finite cell method [1] an integration method based on moment fitting was introduced [2, 3]. Thereby, for every broken finite cell an individual quadrature rule is generated by solving the moment fitting equation system. In [3], we showed that the moment fitting results in good conditioned quadrature rules when choosing the Gauss-Legendre points and using the Legendre polynomials as basis functions. However, this approach still requires the solution of the moment fitting equation system. In this contribution, we replace the Legendre basis by an equivalent Lagrange polynomial basis through the Gauss-Legendre points. For the onedimensional case, then, the moment fitting readsIn Eq.(1) f j define the moment fitting basis functions, l j (ξ ) the Lagrange polynomials, Ω phy is the physical domain, and the upper index GL indicates the Gauss-Legendre points. Following this approach, we avoid to solve the moment fitting equation system by taking advantage of the Kronecker delta property of the Lagrange polynomials f j (ξ GL i ) = δ ji , thus, the moment fitting weights can be computed on the fly as w i = Ω phy f j (ξ )dΩ. For the multidimensional case, the extension is straightforward applying the tensor product of the one-dimensional basis functions and points.For linear applications the presented moment fitting method works very well. However, for nonlinear applications this method performs less stable than the adaptive integration in which broken cells are subdivided applying spacetrees and performing a standard Gauss quadrature on subcell level. Due to this reason, we present an adaptive moment fitting approach. In this approach, we subdivide broken cells and subcells if their volume fraction of the physical domain is smaller than a predefined tolerance. Thus, the moment fitting is performed on cell or subcell level depending on the material distribution within the cells. Numerical examplesTo demonstrate the efficiency of the presented integration methods, we investigate two three-dimensional FCM examples for finite strain J 2 flow theory of plasticity [4]. In doing so, we compare the results of the moment fitting (MF) and the adaptive moment fitting (AMF) with th...

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

“…Since its introduction, the FCM has been successfully applied in various fields, e.g. applications to elastic and plastic problems in small and large strain [19,20,[23][24][25][26][27][28][29][30][31][32][33], homogenization of heterogeneous and cellular materials as well as foams [34][35][36][37][38][39][40], topology optimization [41,42], problems including material interfaces [43][44][45][46][47], contact problems [40,[48][49][50][51][52][53][54], multi-physic problems [55][56][57][58][59][60][61][62], fracture simulation [63,64], or simulation of wave propagation [65][66][67]…”

“…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.…”

“…In the following, this moment fitting approach is referred to as adaptive moment fitting. Further, to reduce the overhead in the generation of the moment fitting quadratures, in [31,33], we introduced an approach that avoids the necessity to solve the moment fitting equations. This moment fitting approach is explained in more detail in Sec.…”

Spline‐ and hp‐basis functions of higher differentiability in the finite cell method

“…Since the FCM can suffer from ill‐conditioning, due to badly broken finite cells, we follow the same approach presented by Taghipour et al and Hubrich and Düster that has been applied for stabilization of the FCM for problems in finite strain elastoplasticity. In doing so, we distribute additional integration points within the fictitious domain of broken finite cells.…”

“…The whole model is depicted in Figure . The geometry of a single cheese block is defined using the level set function where the center coordinates x c , y c , and z c as well as the inner radius r , the outer radius R , and the parameter d of each of the cheese blocks are listed in Table . Using the FCM, a Cartesian grid of 20 × 20 × 20 cells is created with BCs as described in Figure B.…”