Enhanced numerical integration scheme based on image-compression techniques: application to fictitious domain methods

Abstract

In the present work, we propose a new approach, the so-called compressed adaptive integration scheme (C-AIS), for the computation of the stiffness and mass matrices in fictitious domain methods requiring the integration of discontinuous functions. The novel approach extends the conventional quadtree-decomposition-based adaptive integration scheme (AIS) by an additional step, in which established image-compression techniques are exploited to decrease the number of integration sub-cells. The benefits of the C-AIS are manifold: First, the compression of the sub-cells inevitably leads to significant savings in terms of computational time required by the numerical integration. Second, the compression procedure, which is executed directly after the quadtree-decomposition algorithm, can be easily included in existing codes. Third, if applied to polynomial integrands, the C-AIS yields exactly the same accuracy as the conventional AIS. Finally, the fourth advantage is seen in the fact that the C-AIS can readily be combined with other approaches seeking a reduction of the number of integration points such as the Boolean-FCM. The efficiency of the C-AIS approach is presented in the context of the FCM based on Cartesian meshes applied to problems of linear elastostatics and modal analysis, while it is also suitable for the quadrature in other fictitious domain approaches, e.g., CutFEM and cgFEM.

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

“…(1) over the intersected elements. As an extension to their approach, we implement and investigate the novel numerical integration scheme proposed by Petö et al [38] that compresses the integration sub-cells resulting from the quadtree-decomposition (QTD) to achieve reduced computational times. Since this latter approach leads to a lower integration point density in the polygonal elements, a special focus is placed on the integration accuracy when computing Eq.…”

“…4 the different integration schemes over polygonal elements are elaborated with special attention to discontinuous integrands (see Ref. [38]). Finally, in Sects.…”

“…In the remainder, the focus is kept on the partitioning of the polygonal elements into quadrilateral sub-domains for which the compression algorithms are readily available (Sect. 4.2.2) [38]. The QTD is performed in the local-space of the cut quadrilateral sub-domains Ω…”

“…This leads to high numerical costs both for performing the QTD itself and for computing the integrals over the sub-cells [82], especially for nonlinear problems [83]. As a solution to this problem, Petö et al [38] introduced a novel approach based on image compression techniques for reducing the number of integration points and consequently the computational time. In the following, the main idea of this method is briefly discussed; for further information, we refer to the original article [38].…”

“…As a solution to this problem, Petö et al [38] introduced a novel approach based on image compression techniques for reducing the number of integration points and consequently the computational time. In the following, the main idea of this method is briefly discussed; for further information, we refer to the original article [38]. The compression algorithm is inserted directly between the QTD and the numerical integration stages as an intermediate step and requires only minor modification in an already existing code.…”