An interpolation-based fast multipole method for higher order boundary elements on parametric surfaces

In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reduced by one order. This gain is independent of the application of either (mathcal{H})- or (mathcal{H}^2)-matrices. In fact, several simplifications in the construction of (mathcal{H}^2)-matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method. In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reducedby one order. This gain is independent of the application of either H - or H 2- matrices. In fact, several simplificationsin the construction of  H 2 -matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method

NiHu: A multi-purpose open source fast multipole solver

We introduce the Fast Extension of NiHu, an open source C++ toolbox for the formulation of various engineering BEM problems. The NiHu toolbox has been widely used to discretise boundary integrals in a generic way, using the conventional BEM. The recently developed Fast Multipole extension aims to provide an interface for the efficient and flexible implementation of different fast integration methods, such as the kernel-dependent Fast Multipole Method, it's black-box extensions, or algebraic approaches, such as ACA. Similar to the conventional BEM core, the Fast Extension massively exploits C++ template metaprogramming in order to generate efficient code for different formalisms, while maintaining a generic programming interface. Parallel processing is implemented using OpenMP. The flexibility of the toolbox is demonstrated by formulating different problems related to computational acoustics: 2D and 3D acoustic radiation-scattering, potential problems, as well as stochastic eigendecomposition. The open source implementation is validated using the EAA benchmark cases such as the Radiatterer, the Pac-man and the Cat's eye arrangements

Comparison of fast boundary element methods on parametric surfaces

We compare fast black-box boundary element methods on parametric surfaces in $\mathbb{R}^3$. These are the adaptive cross approximation, the multipole method based on interpolation, and the wavelet Galerkin scheme. The surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels. Nonetheless, parametric surface representations are easily accessible from Computer Aided Design (CAD) and are recently topic of the studies in isogeometric analysis. Especially, we can apply two-dimensional interpolation in the multipole method. A main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry. This results in a superior compression of the far field compared to other cluster methods