Efficient integration method for fictitious domain approaches

In the current article, we present an efficient and accurate numerical method for the integration of the system matrices in fictitious domain approaches such as the finite cell method (FCM). In the framework of the FCM, the physical domain is embedded in a geometrically larger domain of simple shape which is discretized using a regular Cartesian grid of cells. Therefore, a spacetree-based adaptive quadrature technique is normally deployed to resolve the geometry of the structure. Depending on the complexity of the structure under investigation this method accounts for most of the computational effort. To reduce the computational costs for computing the system matrices an efficient quadrature scheme based on the divergence theorem (Gau-Ostrogradsky theorem) is proposed. Using this theorem the dimension of the integral is reduced by one, i.e. instead of solving the integral for the whole domain only its contour needs to be considered. In the current paper, we present the general principles of the integration method and its implementation. The results to several two-dimensional benchmark problems highlight its properties. The efficiency of the proposed method is compared to conventional spacetree-based integration techniques.