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An application of a boundary element method to the solution of static field problems in closed domains is presented in this paper. The fully populated system matrix of the boundary element method is compressed with the fast multipole method. Two approaches of modified transformation techniques are compared and discussed in the context of boundary element methods to further reduce the computational costs of the fast multipole method. The efficiency of the fast multipole method with modified transformations is shown in two numerical examples.

The application of the fast multipole method reduces the computational costs and the memory requirements of the boundary element method from O(N²) to approximately O(N). In this paper we present that the computational costs can be strongly shortened, when the multipole method is not only used for the solution of the system of linear equations but also for the field computation in arbitrary points.

The purpose of this paper is to develop an easy-to-implement and accurate fast boundary element method (BEM) for solving large-scale elastodynamic problems in frequency and time domains.

A newly developed kernel-independent fast multipole method (KIFMM) is applied to accelerating the evaluation of displacements, strains and stresses in frequency domain elastodynamic BEM analysis, in which the far-field interactions are evaluated efficiently utilizing equivalent densities and check potentials. Although there are six boundary integrals with unique kernel functions, by using the elastic theory, the authors managed to accelerate these six boundary integrals by KIFMM with the same kind of equivalent densities and check potentials. The boundary integral equations are discretized by Nyström method with curved quadratic elements. The method is further used to conduct the time-domain analysis by using the frequency-domain approach.

Numerical results show that by the fast BEM, high accuracy can be achieved and the computational complexity is brought down to linear. The performance of the present method is further demonstrated by large-scale simulations with more than two millions of unknowns in the frequency domain and one million of unknowns in the time domain. Besides, the method is applied to the topological derivatives for solving elastodynamic inverse problems.

An efficient KIFMM is implemented in the acceleration of the elastodynamic BEM. Combining with the Nyström discretization based on quadratic elements and the frequency-domain approach, an accurate and highly efficient fast BEM is achieved for large-scale elastodynamic frequency domain analysis and time-domain analysis.

The purpose of this study is the formulation of an efficient method to compute and analyse the scattering characteristics of cracks or grooves in a conducting object, where the size of the crack is significantly larger than the wavelength of an incident plane wave.

A hybrid finite element-boundary element procedure is formulated for the computation of the scattering properties of the object, where the fast multipole method is used in the boundary integral formulation. The basic fast multipole procedure is enhanced by utilising a fast Fourier transform-based convolution algorithm for the computation of the interactions between groups of source and field elements.

The algorithm accelerates the evaluation of the group interactions and enables the reduction of the memory requirements without introducing an additional approximation into the procedure.

The fast multipole method with convolution algorithm shows to be more efficient for the computation of scattering problems with a large number of unknowns than the conventional procedure.

Domain integrals, known as volume potentials in 3D elasticity problems, exist in many boundary-type methods, such as the boundary element method (BEM) for inhomogeneous partial differential equations. The purpose of this paper is to develop an accurate and reliable technique to effectively evaluate the volume potentials in 3D elasticity problems.

An adaptive background cell-based domain integration method is proposed for treatment of volume potentials in 3D elasticity problems. The background cells are constructed from the information of the boundary elements based on an oct-tree structure, and the domain integrals are evaluated over the cells rather than volume elements. The cells that contain the boundary elements can be subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements. The fast multipole method (FMM) is further applied in the proposed method to reduce the time complexity of large-scale computation.

The method is a boundary-only discretization method, and it can be applied in the BEM easily. Much computational time is saved by coupling with the FMM. Numerical examples demonstrate the accuracy and efficiency of the proposed method..

Boundary elements are used to create adaptive background cells, and domain integrals are evaluated over the cells rather than volume elements. Large-scale computation is made possible by coupling with the FMM.

The purpose of this study is to calculate the frequency electric field in substation.

The paper proposes a novel fast multipole method (FMM) called Super-FMM to solve the PFEF problems in substations. The paper substitutes the original approaches for analytic expansions and translations through equivalent density representations.

The paper shows that the Super-FMM is more efficient in terms of the complexity of its storage spaces and computational costs compared with the best-known FMM when placed under scenarios with exactly the same error rates.

Using the fast Fourier transform algorithm can further improve the optimization algorithm and computational efficiency.

A novel FMM called Super-FMM is proposed, which has a structure similar to that of the adaptive FMM algorithm, but the paper substitutes the original approaches for analytic expansions and translations through equivalent density representations.

The present paper deals with the fast multipole acceleration of the 2D finite element‐boundary element modelling of electromechanical devices. It is shown that the fast multipole method, usually applied to large 3D problems, can also lead to a reduction in computational time when dealing with relatively small 2D problems, provided that an adaptive truncation scheme for the expansion of the 2D Laplace Green function is used. As an application example, the 2D hybrid modelling of a linear actuator is studied, taking into account saturation, the voltage supply and the mechanical equation. The computational cost without and with fast multipole acceleration is discussed for both the linear and nonlinear case.

Various parallelization strategies are investigated to mainly reduce the computational costs in the context of boundary element methods and a compressed system matrix.

Electrostatic field problems are solved numerically by an indirect boundary element method. The fully dense system matrix is compressed by an application of the fast multipole method. Various parallelization techniques such as vectorization, multiple threads, and multiple processes are applied to reduce the computational costs.

It is shown that in total a good speedup is achieved by a parallelization approach which is relatively easy to implement. Furthermore, a detailed discussion on the influence of problem oriented meshes to the different parts of the method is presented. On the one hand the application of problem oriented meshes leads to relatively small linear systems of equations along with a high accuracy of the solution, but on the other hand the efficiency of parallelization itself is diminished.

The presented parallelization approach has been tested on a small PC cluster only. Additionally, the main focus has been laid on a reduction of computing time.

Typical properties of general static field problems are comprised in the investigated numerical example. Hence, the results and conclusions are rather general.

Implementation details of a parallelization of existing fast and efficient boundary element method solvers are discussed. The presented approach is relatively easy to implement and takes special properties of fast methods in combination with parallelization into account.

3D steady heat conduction analysis considering heat source is conducted on the fundamental of the fast multipole method (FMM)-accelerated line integration boundary element method (LIBEM).

Due to considering the heat source, domain integral is generated in the traditional heat conduction boundary integral equation (BIE), which will counteract the well-known merit of the BEM, namely, boundary-only discretization. To avoid volume discretization, the enhanced BEM, the LIBEM with dimension reduction property is introduced to transfer the domain integral into line integrals. Besides, owing to the unsatisfactory performance of the LIBEM when it comes to large-scale structures requiring massive computation, the FMM-accelerated LIBEM (FM-LIBEM) is proposed to improve the computation efficiency further.

Assuming N and M are the numbers of nodes and integral lines, respectively, the FM-LIBEM can reduce the time complexity from O(NM) to about O(N+ M), and a full discussion and verification of the advantage are done based on numerical examples under heat conduction.

(1) The LIBEM is applied to 3D heat conduction analysis with heat source. (2) The domain integrals can be transformed into boundary integrals with straight line integrals by the LIM. (3) A FM-LIBEM is proposed and can reduce the time complexity from O(NM) to O(N+ M). (4) The FM-LIBEM with high computational efficiency is exerted to solve 3D heat conduction analysis with heat source in massive computation successfully.

If the fast multipole method (FMM) is applied in the context of the boundary element method, the efficiency and accuracy of the FMM is significantly influenced by the used hierarchical grouping scheme. Hence, in this paper, a new approach to the grouping scheme is presented to solve numerical examples with problem‐oriented meshes and higher order elements accurately and efficiently. Furthermore, with the proposed meshing strategies the efficiency of the FMM can be additionally controlled.