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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.

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 paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).

The elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.

As the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.

The authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

In this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.

From the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.

As the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.

The authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

This study aims to propose and numerically assess different ways of discretising a very weak formulation of the Poisson problem.

We use integration by parts twice to shift smoothness requirements to the test functions, thereby allowing low-regularity data and solutions.

Various conforming discretisations are presented and tested, with numerical results indicating good accuracy and stability in different types of problems.

This is one of the first articles to propose and test concrete discretisations for very weak variational formulations in primal form. The numerical results, which include a problem based on real MRI data, indicate the potential of very weak finite element methods for tackling problems with low regularity.

Thin conducting sheets are used in many electric and electronic devices. Solving numerically the eddy current problems in presence of these thin conductive sheets requires a very fine mesh which leads to a large system of equations, and it becomes more problematic in case of higher frequencies. The purpose of this paper is to show the numerical pertinence of equivalent models for 3D eddy current problems with a conductive thin layer of small thickness e based on the replacement of the thin layer by its mid-surface with equivalent transmission conditions that satisfy the shielding purpose, and by using an efficient discretization using the boundary element method (BEM) to reduce the computational work.

These models are solved numerically using the BEM and some numerical experiments are performed to assess the accuracy of the proposed models. The results are validated by comparison with an analytical solution and a numerical solution by the commercial software Comsol.

The error between the equivalent models and analytical and numerical solutions confirms the theoretical approach. In addition to this accuracy, the computational work is reduced by considering a discretization method that requires only a surface mesh.

Based on a hybrid formulation, the authors present briefly a formal derivation of impedance transmission conditions for 3D thin layers in eddy current problems where non-conductive materials are considered in the interior and the exterior domain of the sheet. BEM is adopted to discretize the problem as there is no need for volume discretization.

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.

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.

The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential equations (PDEs).

The original version of MLEBF method is limited to linear, constant coefficient PDEs. The reason is that exponential bases which satisfy the homogeneous operator can only be determined for this class of problems. To extend this method to the general case of linear PDEs, the variable coefficients along with all involved derivatives are first expanded. This expanded form is evaluated at the center of each cloud, and is assumed to be constant over the entire cloud. The solution procedure is followed as in the former version. Nonlinear problems are first converted to a succession of linear, variable coefficient PDEs using the Newton-Kantorovich scheme and are subsequently solved using the aforementioned approach until convergence is achieved.

The results obtained show good performance of the method as solution to a wide range of problems. The results are compared with the well-known methods in the literature such as the finite element method, high-order finite difference method or variants of the boundary element method.

The MLEBF method is a simple yet effective tool for analyzing various kinds of problems. It is easy to implement with high parallelization potential. The proposed method addresses the biggest limitation of the method, and extends it to linear, variable coefficient PDEs as well as nonlinear ones.

Finite element analysis of underwater transducers typically requires a high level of expertise, and the iterative process of testing various sizes, material parameters and other factors is often inefficient. To address this challenge, this paper aims to introduce underwater transducer parametric simulation (UTPS) software to streamline the design and optimization process.

The design methodology integrates the strengths of ANSYS Parametric Design Language (APDL) for parametric design with the Qt Creator framework for developing a visual interface. C++ is used to encapsulate complex, hard-to-master APDL macros and interact with ANSYS software to execute the relevant APDL macros, performing finite element analysis on the underwater transducer in the background. The results are then processed and displayed on the visual interface.

UTPS enables parametric modeling, modal analysis, harmonic response analysis and directivity analysis of underwater transducers. Users only need to input parameters into the software interface to obtain the transducer’s performance, significantly improving work efficiency and lowering the professional threshold. A prototype transducer was fabricated and tested based on UTPS results, which confirmed the accuracy of the software.

This paper presents an innovative parametric simulation tool for underwater transducers, combining finite element analysis and APDL to simplify and expedite the design process. UTPS reduces the need for specialized knowledge, cutting down on training costs, while its parametric design capabilities accelerate the design process, saving resources.