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To provide first insight onto the application of hierarchical matrices and adaptive cross approximation (ACA) techniques for electromagnetic scattering problems.

The shielding effectiveness of metallic casings with apertures is analyzed via an electric field integral equation. To reduce the storage needs and the complexity of matrix equation solution, a technique combining the use of hierarchical matrices (H‐matrix) in conjunction with the ACA technique is used.

Provides first results for compression of a matrix resulting from a Helmholtz problem by means of hierarchical matrices and ACA techniques. Gives insight into the importance of obtaining a “cheap” preconditioner.

The technique resides on the smotheness of kernel functions – which is no longer valid for big wave numbers.

Gives means of solving problems of big dimensions in terms of number of unknowns – without the need to tailor the approach for the specific kernel function. The original integration functions used to fill the full matrix can be used here.

The paper represents one of the first attempts to use the above‐mentioned techniques for the high frequency domain.

The purpose of this paper is to present a novel strategy used for acceleration of free-vibration analysis, in which the hierarchical matrices structure and Compute Unified Device Architecture (CUDA) platform is applied to improve the performance of the traditional dual reciprocity boundary element method (DRBEM).

The DRBEM is applied in forming integral equation to reduce complexity. In the procedure of optimization computation, ℋ-Matrices are introduced by applying adaptive cross-approximation method. At the same time, this paper proposes a high-efficiency parallel algorithm using CUDA and the counterpart of the serial effective algorithm in ℋ-Matrices for inverse arithmetic operation.

The analysis for free-vibration could achieve impressive time and space efficiency by introducing hierarchical matrices technique. Although the serial algorithm based on ℋ-Matrices could obtain fair performance for complex inversion operation, the CUDA parallel algorithm would further double the efficiency. Without much loss in accuracy according to the examination of the numerical example, the relative error appeared in approximation process can be fixed by increasing degrees of freedoms or introducing certain amount of internal points.

The paper proposes a novel effective strategy to improve computational efficiency and decrease memory consumption of free-vibration problems. ℋ-Matrices structure and parallel operation based on CUDA are introduced in traditional DRBEM.

This paper seeks to deal with the computation of time‐harmonic electric potentials, currents, and surface charge distributions inside self‐healing metallized film capacitors in three dimensions. A 50 Hz exciting voltage is applied at contacts.

Extreme aspect ratios warrant dimensional reduction: the metallization is modelled as a 2D shell. This greatly reduces computational costs and makes possible an excellent resolution of the geometry. An integro‐differential equation for the complex amplitudes of the electric potential and surface charge densities on this shell is derived and discretized by means of boundary elements.

Adaptive cross approximation and H‐matrix technology is employed for matrix compression and preconditioning of iterative solvers. This permits one to use fine surface meshes and achieve satisfactory accuracy as demonstrated in numerical experiments.

The model is based on an electroquasistatic approach; thus it is valid for low frequencies only.

Numerical experiments of sophisticated real‐life capacitor‐designs show the efficacy of the method for industrial applications.

A novel model was developed and implemented for the 3D electric field computation inside metallized film capacitors in the frequency domain.

Based on the error analysis, the authors proposed a new kind of high accuracy boundary element method (BEM) (HABEM), and for the large-scale problems, the fast algorithm, such as adaptive cross approximation (ACA) with generalized minimal residual (GMRES) is introduced to develop the high performance BEM (HPBEM). It is found that for slender beams, the stress analysis using iterative solver GMRES will difficult to converge. For the analysis of slender beams and thin structures, to enhance the efficiency of GMRES solver becomes a key problem in the development of the HPBEM. The purpose of this paper is study on the preconditioning method to solve this convergence problem, and it is started from the 2D BE analysis of slender beams.

The conventional sparse approximate inverse (SAI) based on adjacent nodes is modified to that based on adjacent nodes along the boundary line. In addition, the authors proposed a dual node variable merging (DNVM) preprocessing for slender thin-plate beams. As benchmark problems, the pure bending of thin-plate beam and the local stress analysis (LSA) of real thin-plate cantilever beam are applied to verify the effect of these two preconditioning method.

For the LSA of real thin-plate cantilever beams, as GMRES (m) without preconditioning applied, it is difficult to converge provided the length to height ratio greater than 50. Even with the preconditioner SAI or DNVM, it is also difficult to obtain the converged results. For the slender real beams, the iteration of GMRES (m) with SAI or DNVM stopped at wrong deformation state, and the computation failed. By changing zero initial solution to the analytical displacement solution of conventional beam theory, GMRES (m) with SAI or DNVM will not be stopped at wrong deformation state, but the stress error is still difficult to converge. However, by GMRES (m) combined with both SAI and DNVM preconditioning, the computation efficiency enhanced significantly.

This paper presents two preconditioners: DNVM and a modified SAI based on adjacent nodes along the boundary line of slender thin-plate beam. In the LSA, by using GMRES (m) combined with both DNVM and SAI, the computation efficiency enhanced significantly. It provides a reference for the further development of the 3D HPBEM in the LSA of real beam, plate and shell structures.

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.

Inductances of complex coils, in the presence of linear materials only, can be computed by discretizing coils into simpler elements, whose magnetic behavior is analytically expressible, and suitably combining elementary contributions. Reliable results require high numbers of elements. In such cases, advantages can be taken from Graphic Processor Unit (GPU) capabilities of dealing efficiently with high numbers of repeated simple computational tasks. The purpose of this paper is to set up a fast and prompt numerical procedure to cope with the above described task.

The coils are first decomposed into current segments, taking into account accuracy, relative position and shape of coils to determine the number of segments. An analytical formula is then used to compute elementary contributions using GPUs to speed up the process, and finally superposition is used to recover the result.

The main advantages of the proposed approach are first demonstrated using simple examples, with analytical solutions, to validate the method accuracy and promptness, then more complex cases are taken to demonstrate its generality.

The method is intrinsically limited by the linearity assumption, excluding the presence of magnetic materials. The adopted formulas require in addition that coils must lie in free space.

The proposed method can help in the design of complex coils or coils systems, where the performance depends on total magnetic energy or magnetic forces among coils.

The paper presents an original implementation in GPU-based computational environment of a procedure to compute inductances, based on the superposition of a high number of current segments. The procedure includes an original method to self-adaptively define number and position of current segments used in the coils discretization.

This paper aims to present a domain decomposition method to overcome the challenges posed by multi-domain, multi-scale high frequency problems.

A hybrid finite element and boundary integral procedure is also presented that allows for domains to employ different solution methods in different subdomains.

By decomposing large electromagnetic regions into smaller domains, the finite element method can cope with the simulation of electrically large problems.

Real life examples demonstrate the accuracy and efficiency of the new method.

The Robin transmission condition (RTC) is applied to link the domains and preserve field continuity on interfaces.

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 paper is to present a computationally efficient approach for the stochastic modeling of an inhomogeneous reluctivity of magnetic materials. These materials can be part of electrical machines such as a single-phase transformer (a benchmark example that is considered in this paper). The approach is based on the Karhunen–Loève expansion (KLE). The stochastic model is further used to study the statistics of the self-inductance of the primary coil as a quantity of interest (QoI).

The computation of the KLE requires solving a generalized eigenvalue problem with dense matrices. The eigenvalues and the eigenfunction are computed by using the Lanczos method that needs only matrix vector multiplications. The complexity of performing matrix vector multiplications with dense matrices is reduced by using hierarchical matrices.

The suggested approach is used to study the impact of the spatial variability in the magnetic reluctivity on the QoI. The statistics of this parameter are influenced by the correlation lengths of the random reluctivity. Both, the mean value and the standard deviation increase as the correlation length of the random reluctivity increases.

The KLE, computed by using hierarchical matrices, is used for uncertainty quantification of low frequency electrical machines as a computationally efficient approach in terms of memory requirement, as well as computation time.

A wide range of micro‐electro‐mechanical‐systems are based on the electrostatic principle, and for their design the computation of the electric capacities is of great importance. The purpose of this paper is to efficiently compute the capacities as a function of all possible positions of the two electrode structures within the transducer by an enhanced boundary element method (BEM).

A Galerkin BEM is developed and the arising algebraic system of equations is efficiently solved by a CG‐method with a multilevel preconditioner and an appropriate fast multipole algorithm for the matrix‐vector operations within the CG‐iterations.

It can be demonstrated that the piecewise linear and discontinuous trial functions give an approximation, which is almost as good as the one of the piecewise constant trial functions on the refined mesh, at lower computational costs and at about the same memory requirements.

The paper can proof mathematically and demonstrate in practice, that a higher order of convergence is achieved by using piecewise linear, globally discontinuous basis functions instead of piecewise constant basis functions. In addition, an appropriate preconditioner (artificial multilevel boundary element preconditioner, which is based on the Bramble Pasciak Xu like preconditioner) has been developed for the fast iterative solution of the algebraic system of equations.