Conventional Boundary Element Method Research Articles

A novel numerical method for solving the Kirchhoff-Helmholtz integral equation based on the sound field representation using the reproducing kernel in the frequency domain

The paper introduces a unique numerical solution method for the Kirchhoff-Helmholtz integral equation. This method is based on the representation of the sound field using the reproducing kernel in the solution space of the homogeneous Helmholtz equation. When compared to the conventional boundary element method, the proposed method holds numerous advantages. This paper presents the analysis theory of the proposed method and demonstrates its validity by calculating bounded and unbounded sound fields. The accuracy of the calculation is assessed by computing the sound field in a rectangular room under various conditions. Furthermore, although some theoretical issues remain, the proposed method effectively avoids the non-uniqueness of solutions in unbounded domain problems.

A novel hybrid boundary element for polygonal holes with rounded corners in two-dimensional anisotropic elastic solids

A novel hybrid boundary element is developed for polygonal holes in finite anisotropic elastic plates based on two different special fundamental solutions for holes. Since these special fundamental solutions satisfy traction-free condition along the hole's boundary, there is no mesh required on the boundary of polygonal holes. Various types of polygonal holes with rounded corners, such as triangles, rhombuses, ovals, pentagons, are considered by adding proper perturbation to an elliptical hole. The developed hybrid element is a mixture of two special boundary elements: one is based on the special fundamental solution derived through nonconformal mapping and the other is based on the solution derived through perturbation technique with conformal mapping. The special boundary element methods are combined through submodeling technique. First, the global model is solved using the perturbation solution. Then, using the displacements obtained from global model, an auxiliary submodel is set up and the results are evaluated with the nonconformal solution. The present method is compared and validated with conventional boundary element method and finite element method. The effect of hole curvature, material anisotropy, and loading condition on the stress distribution around the hole is presented.

Efficiently solving large-scale electrostatic lightning problems by integrating characteristic basis functions into boundary element method

The evaluation of the electric field near objects of complex geometries is critical in the pre-bridging regime of lightning strikes in high-voltage transmission systems, which is subject to an open domain problem and can be computationally expensive. In this paper, the characteristic basis function method (CBFM) is incorporated into the conventional boundary element method (BEM) to efficiently handle large-scale electrostatic problems. Within the CBFM, low-level basis functions have been replaced by high-level macro characteristic basis functions, resulting in a significant reduction in the number of degrees of freedom. This reduction in unknowns allows iteration-free direct solvers to approximate the matrices, further reducing the computational complexity of the problem. Numerical examples of the surface charge distribution of an actual 500 kV tower-transmission-line system and a three MW wind turbine in the lightning environment demonstrate the accuracy of the proposed CBFM, while requiring considerably much fewer computational resources compared with conventional BEM. Furthermore, CBFM is highly parallelizable, implying the feasibility of handling large electrostatic lightning problems with complex geometries.

Band structure calculations of three-dimensional solid-fluid coupling phononic crystals using dual reciprocity boundary element method and wavelet compression method

The algorithm for calculating the band structures of two-dimensional phononic crystals (PCs) using the boundary element method (BEM) has been proposed for many years. However, it has not yet been extended to three-dimensional (3D) PCs because the fundamental solutions of 3D dynamics are complex and are related to angular frequency. In this study, the BEM is applied to calculate the band structures of 3D solid-fluid coupling PCs by combining the dual reciprocity method. The use of the dual reciprocity method can help avoid nonlinear eigenvalue problems. To address the limitation of fully populated matrices in BEM, the wavelet compression method based on B-spline wavelet on the interval is adopted. Some small matrix entries, which are generated by the vanishing moment and local support characteristics, are set to zero using the provided truncation technique. This process results in the production of sparse matrices. The constructed generalized linear eigenvalue equations are modified to tackle the ill-conditioned matrix issues arising from the distinct fundamental solutions of solids and fluids. The results show that this method is superior to the finite element method in terms of calculation efficiency. Compared to conventional BEM, the current wavelet BEM can not only generate sparse matrices but also reduce the integration calculation time when handling large-scale problems.

Three-dimensional broadband simulation of near-fault seismic ground motion considering complete source-path-site effect: Modeled by fast multipole indirect boundary element method

Efficient and accurate numerical methods are essential for analyzing seismic wave propagation and amplification in near-fault complex sites. Understanding the complete process of ground motion simulation, including fault rupture, path propagation, and near-surface complex site response, is crucial for studying earthquake damage mechanisms, seismic zoning, and designing large-scale engineering structures. In this study, we propose a fast multipole indirect boundary element method (FMIBEM) to achieve broadband and high-efficiency simulation, enabling a comprehensive analysis of the complete process involving seismic ground motion. The FMIBEM significantly reduces the computational and storage costs associated with 3D near-fault complex site seismic wave scattering problems to O(N). We validate the accuracy of the method by comparing it with the analytical solution. Compared to the conventional indirect boundary element method (IBEM), our proposed method reduces computational time by over 95 % and storage costs by nearly 90 %. FMIBEM greatly enhances the efficiency of the boundary element method for simulating seismic wave scattering in near-fault complex sites. To demonstrate the effectiveness of our approach, we apply the FMIBEM method to two typical 3D near-fault complex site examples of seismic wave scattering. The simulation results successfully capture both the local site amplification effect and the characteristic near-fault seismic features, such as the hanging wall effect, permanent displacement effect, and large velocity pulse.

The equivalent source method is an indirect boundary element method with an implicit Voronoi mesh

The Equivalent Source Method (ESM) consists of modeling the acoustical field radiated or scattered from an object by a set of radiating monopole sources located below its surface. While highly efficient for modeling the scattered field by a smooth object, ESM becomes less effective in representing acoustic diffraction from sharp edges. In this article, ESM is understood as an approximation to the single layer integral equation. In this framework, ESM belongs to the family of Indirect Boundary Element Methods (BEM) with an implicit surface mesh slightly below the object surface, and a zero order integration scheme on each element of the mesh. This implicit mesh is well approximated by the Voronoi diagram associated to the source distribution. This formalism establishes a strong link between these two widely used methods (ESM and BEM) and allows the construction of hybrid methods that can combine performances of both methods. On the first hand, a new hybrid method able to model edge diffraction while remaining as computationally efficient as ESM is developed. In this method, labeled “BEM Voronoi”, equivalent sources are positioned on the surface’s object rather than below it and singularities are handled with the integral formulation. On the other hand, a set of new hybrid methods which combines ESM with higher order integration scheme is constructed. Performances of the new methods developed in this paper are numerically compared to each other and to Finite Element simulations (used as a reference) for modeling scattering from an object excited by a plane wave. Each hybrid model is tested on multiple configurations involving two sets of boundary conditions (rigid and elastic objects respectively) successively immersed in two different external environments (water and air) and for both smooth and a sharp-edged objects. The results show that the hybrid method “BEM Voronoi” better handles edge diffraction than conventional ESM, at a much lower computational cost (implementation of BEM Voronoi is 50 times faster than conventional BEM implementation).

Isogeometric boundary element method for axisymmetric steady-state heat transfer

Isogeometric analysis (IGA) utilizes the Non-Uniform Rational B-Splines (NURBS) basis functions, which are commonly employed in Computer Aided Design (CAD), to both construct the problem’s geometry and approximate the field variables. In axisymmetric scenarios, the 3D problem can be simplified to 2D, requiring only the discretization of the 1D curve boundary for the boundary element method (BEM). This study proposes an isogeometric boundary element method (IGABEM) to simulate steady-state heat transfer in axisymmetric domains. Both the precise geometry of the boundary and the physical variables are approximated with the same NURBS basis functions, yielding higher-order continuity, superior accuracy per degree of freedom (DOF), and continuous nodal gradients with fewer DOFs than the traditional BEM. Additionally, a zoning method is introduced to handle heterogeneities. Five cases including a solid cylinder, a revolution hyperboloid with a coaxial cylindrical tube, an ellipsoid with a spherical cavity, a bimaterial ellipsoid and a complex geometry with a toroidal tube validate the accuracy, convergence, computational efficiency of IGABEM, and the applicability in addressing multiply-connected domains. The method outperforms Finite Element Method (FEM) and conventional BEM in accurately handling steady-state axisymmetric heat transfer issues under Dirichlet, Neumann, and Robin boundary conditions.

A hybrid nearfield acoustic holography based on the mapping relationship and boundary element method

The nearfield acoustic holography (NAH) based on mapping relationship (MRS) had been developed. However, it is superior to the reconstructions on regular shapes by using the MRS-based NAH with spherical fundamental solutions (FS). In contrast, the boundary element method (BEM) based NAH is capable of handling complex shapes but requires relatively more measurement points. In this work, a hybrid method is developed by combing the MRS with spherical FS and the BEM. To circumvent the non-unique problems associated with the conventional BEM, the Burton-Miller formulation is adopted into the BEM based NAH. The merit of easy measurement setup, such as the determination of positions and number of microphones in the array is preserved in the hybrid method. Furthermore, it is possible to recast the reconstruction operation by the BEM-based NAH with sufficient measurements data on an artificial hologram that are implicitly interpolated by the MRS-based NAH. In conjunction with regularization techniques, results with improved accuracy are clearly observed in the velocity reconstructions for irregular shapes by the hybrid NAH. It is validated by numerical examples especially for an irregular box that the hybrid NAH is capable of producing reconstructions with significant improvement of accuracy even with small signal-to-noise ratio (SNR) as compared with the MRS-based NAH. An experiment is setup to further demonstrate the potential of the hybrid NAH for practical reconstruction of an irregular radiator. More robust and accurate reconstructions are achieved in contrast with the MRS-based NAH, which clearly illustrated advantages of the hybrid NAH.

Novel and efficient implementation of multi-level fast multipole indirect BEM for industrial Helmholtz problems

Fast multipole boundary element methods (FMBEM) have been successfully applied to solve ultra-large acoustic problems. In practice, there are still a large number of industrial applications where model sizes are well below a million Degrees of freedom. Thus, it is of practical importance to further improve the solver’s efficiency for such applications. This paper presents a novel formulation of the multi-level fast multipole indirect BEM, which relies on redesigning a conventional FMBEM with considerations from several aspects. A new flexible partition algorithm is proposed to generate a flexible and efficient multi-level structure for fast multipole calculations, which also allows the explicit calculation of the far field contributions. A hybrid and efficient regularization approach for hypersingular integral is investigated and implemented, which significantly reduces the iterative solving time. The use of efficient sparse approximate inverse preconditioners is key for fast convergence. Two strategies for constructing the sparsity pattern are implemented and evaluated. Numerical results show that by bringing the aforementioned techniques all together in one package, the new flexible multi-level fast multipole indirect BEM (f-FMBEM) provides significant improvements on actual simulation time over the conventional BEM and FMBEM solvers for industrial size problems (with a non-dimensional Helmholtz number up to kA∼323) on a modern workstation. To efficiently address larger problems, a more efficient multipole translation should be considered.

The Accuracy and Efficiency of a Single-Level Fast Multipole Boundary Element Model for Analyzing Cathodic Protection of Large Pipeline Networks

A single-level fast multipole boundary element method was developed for analyzing cathodic protection (CP) systems of large pipeline networks. This method was obtained by embedding far-field approximation within the traditional single-level fast multipole method. The far-field approximation was used for computing the coefficients for far elements within adjacent cells and determining the moments of the elements within far cells. This approximation reduced the difficulty of the procedures and programming leading to a significant decrease in computational time. The Newton-Raphson method and generalized minimal residual method were combined based on the proposed method to solve the nonlinear boundary conditions due to the polarization curve. Several CP problems were considered to verify and evaluate the method. The calculated potentials of this method were in good agreement with the conventional boundary element method, which was achieved by using pipe elements and quadrilateral elements to mesh the surfaces. Finally, the impressed current CP systems of a large network (more than 100,000 elements) and a complex urban gas network were investigated. The results indicated the capability, efficiency, and precision of the present method for solving large and complicated problems on a common desktop computer.

Steady State Heat Conduction in Exchanger Tubes by Using the Meshfree Boundary Integral Equation Method: Conduction Shape Factor and Degenerate Scale

AbstractRegarding the steady-state heat conduction problem in exchanger tubes, the meshfree boundary integral equation method is employed to determine the conduction shape factor in this paper. Different from the conventional boundary element method, the present method is free of mesh generation. After using the parametric function to represent the boundary contour and adopting the Gaussian quadrature, only collocating points on the boundary is required to obtain the linear algebraic equations. By introducing the local exact solution, the singular integral in the sense of the Cauchy principal value can be novelly determined. In addition, the boundary layer effect due to the nearly singular integral in the boundary integral equation can be dealt with. Two cases of different boundary conditions are considered. One is the isothermal condition on both the inner and outer surfaces. The other is the isothermal condition on the inner surface and the convection condition on the outer surface. Besides, numerical instability is found and the nonuniqueness solution due to the degenerate scale is examined by calculating the conduction shape factor and the temperature on the outer surface.

A spectral coupled boundary element method for the simulation of nonlinear surface gravity waves

The challenge of simulating the broad open sea with limited computational resources has long been of interest in ocean engineering research. In view of this issue, this paper proposes a fully nonlinear potential flow method named the spectral coupled boundary element method (SCBEM). By leveraging the approach of domain decomposition, SCBEM achieves significantly reduced computational cost and an order of magnitude increase in computational domain compared to the conventional boundary element method (BEM). The SCBEM encompasses the marine structure with only a tiny BEM domain and employs a high-order spectral layer to simulate the broad water outside the BEM domain. The performance of the SCBEM is evaluated through comparison with the wave damping approach and literature results for regular waves, modulated wave trains, focused waves, and diffraction of a vertical cylinder. The numerical results demonstrate the effectiveness and accuracy of the SCBEM in simulating a wide range of wavelengths and nonlinear wave interactions.

Enriched physics-informed neural networks for 2D in-plane crack analysis: Theory and MATLAB code

In this paper, a method based on the physics-informed neural networks (PINNs) is presented to analyze 2D in-plane crack problems in the linear elastic fracture mechanics. Instead of using a discretizing mesh, the PINNs-based method is meshless and can be trained on batches of discretely sampled collocation points. In order to capture the asymptotic behavior of the near-tip displacement and stress fields, the standard PINNs formulation is enriched here by including the crack-tip asymptotic functions such that the local behavior in the crack-tip region can be modeled accurately without a nodal refinement. The trainable parameters of the enriched PINNs are learned to satisfy the governing partial differential equations of the cracked elastic solid and the corresponding boundary conditions. It is found that the incorporation of the crack-tip enrichment functions in the PINNs is substantially simpler and more feasible than in the conventional finite element method (FEM) or boundary element method (BEM). The present PINNs-based algorithm is tested and verified by several representative benchmark examples for different loading and crack-mode types. Numerical results show that the present PINNs-based method can provide highly accurate stress intensity factors (SIFs) with only few degrees of freedom. For interested readers, a self-contained MATLAB code and data-sets supplementing this paper are also provided in the Supplementary material.

A local domain boundary element method for solving 2D incompressible fluid flow problems

The Boundary Element Method (BEM) is a well-established, accurate numerical method for solving engineering problems described by linear partial differential equations. However, the application of BEM to non-linear problems leads to the appearance of volume integrals in the integral formulation, and combined with the non-local character of the method the required computational effort is dramatically increased. In this work, a Local Domain BEM (LD-BEM) is proposed for solving incompressible fluid flow problems. The domain of interest is fragmented into small subdomains, in each of which the integral representation of the solution is considered separately. Eliminating the fluxes at all subdomain interfaces, the proposed LD-BEM leads to sparse linear system coefficient matrices and reduced computational complexity, overcoming some of the well-known disadvantages of the conventional BEM. The proposed method is applied to the solution of the well-known two-dimensional (2D) lid-driven cavity problem, for Reynolds numbers up to Re=125 × 103, with a discretization involving more than 10 million unknowns.

An isogeometric boundary element formulation for stress concentration problems in couple stress elasticity

An isogeometric boundary element method (IGABEM) is developed for the analysis of two-dimensional linear and isotropic elastic bodies governed by the couple stress theory. This theory is the simplest generalized continuum theory that can effectively model size effects in solids. The couple stress fundamental solutions are explicitly derived and used to construct the boundary integral equations. A new boundary integral equation arises to obtain the moments and rotations introduced by the couple stress formulation. A new analytical solution is also derived in the present work for an elliptical opening in an infinite sheet under uniaxial far-field stress. Several stress concentration problems are examined to illustrate and validate the application of the IGABEM in couple stress elasticity. It is shown that the IGABEM scheme exhibits advantageous convergence properties in comparison with the conventional BEM for boundary value problems within the framework of couple stress elasticity.

Isogeometric and NURBS-enhanced boundary element formulations for steady-state heat conduction with volumetric heat source and nonlinear boundary conditions

Boundary Element Method (BEM) is a numerical tool that is applied to many different types of engineering problems. BEM possesses several advantages over other numerical methods such as boundary-only discretization and its semi-analytical nature. Application of Isogeometric Analysis (IGA) to BEM is a recently proposed computational method where the main purpose is to use geometrical basis functions as the shape functions of field variables. Key features of combining these two techniques are the exact representation of the geometry, elimination or suppression of the meshing procedure, reduction of the problem dimension and obtaining highly accurate results especially for the flux values on the boundaries compared to conventional BEM. In this study, steady-state heat conduction problems with nonlinear boundary condition and surface heat source are analyzed where geometries are formed by NURBS, and field variables are described with Lagrangian (constant and quadratic) basis functions and NURBS basis functions. Convergence rates and CPU time of different types of element representations are discussed which eventually reveals the performance and capabilities of IGABEM.

Accelerating boundary element methods in wideband frequency sweep analysis by matrix-free model order reduction

In frequency-domain acoustic simulations, a wideband frequency sweep analysis is often required. This becomes particularly challenging for boundary element methods (BEM), as the system matrix has a complex dependency on frequency. Previous attempts have been made to accelerate the frequency sweep analysis for the conventional BEM i.e., without fast multipole or ℋ-matrices accelerations, which restrains the applications to limited scale problems due to the dense system matrix. This paper investigates and incorporates a rational Krylov projection based model order reduction (MOR) technology i.e., matrix-free method into various BEM formulations to address the critical challenge of accelerating frequency sweep analysis. Operating on the transfer functions of the BEM system’s input and output, the matrix-free MOR avoids the direct interference with the BEM kernels, which makes it feasible and attractive to work with existing fast BEM techniques such as ℋ-matrices and fast multipole method. The combined holistic BEM solution retains the merits of the acceleration from both MOR and fast BEM techniques. Significant speed-ups have been achieved by the matrix-free MOR for wideband BEM simulations. Numerical examples are presented to demonstrate the capability of the presented holistic approach in simulating various acoustic problems, including interior–exterior models, complex geometries and frequency dependent boundary conditions.

Fast model order reduction boundary element method for large-scale acoustic systems involving surface impedance

This paper presents a newly developed fast model order reduction boundary element method (FMORBEM) for three-dimensional acoustic wave problems with the impedance boundary condition at relatively high frequencies. This method proceeds in two phases: a precomputation phase and a solution phase. In the precomputation phase, a simple yet powerful cutoff technique which benefits from the amplitude decay property of the kernel functions in the Burton–Miller formulation is proposed. Because it does not operate on the underlying boundary element full-order model (FOM), the computational cost and memory consumption in the process of offline constructing a proper subspace are nearly O(N) complexity with N being the number of degrees of freedom of the FOM. After that, the complete system matrices are computed and projected onto the spanned subspace column-by-column, yielding acceptable computational complexity of O(N2). In the solution phase, the resulting reduced-order model (ROM) allows for efficient storage and fast analyses, which merely requires O(n2) arithmetic operations by using direct solvers with n being the dimension of the ROM. Application of this method to the high-frequency scattering analyses of acoustic systems, i.e. simple sphere models and complex scatterers, shows the improved computational efficiency and reduced memory requirement as compared to the conventional boundary element method and further illustrates that it may be competitive with the fast multipole method based on iterative solvers.

Electromagnetic Modeling of Lossy Interconnects From DC to High Frequencies With a Potential-Based Boundary Element Formulation

The accurate electromagnetic modeling of both low- and high-frequency physics is crucial in the signal and power integrity analysis of electrical interconnects. The boundary element method (BEM) is appealing for lossy conductor modeling because it can capture the frequency-dependent variation of skin depth with only a surface-based discretization of the structure. Conventional BEM formulations rely on the mutual coupling of electric and magnetic fields, and can become inaccurate or unstable at low frequencies. We develop a new full-wave BEM formulation based on potentials which can accurately model lossy conductors from exactly DC to very high frequencies. A new set of simple boundary conditions is proposed along with a modified Lorenz gauge to ensure that the proposed formulation has a stable condition number down to DC. Moreover, coupling the potential-based integral equations to a circuit model allows the straightforward extraction of network parameters. Realistic numerical examples at both the chip and package level demonstrate the accuracy and stability of the proposed method from DC to high frequencies, beyond the capabilities of state-of-the-art BEM formulations based on fields.

A local domain BEM for solving transient convection-diffusion-reaction problems

The convection-diffusion-reaction (CDR) equation has been extensively used to simulate a variety of physical phenomena. A robust numerical method for solving linear CDR problems is the Boundary Element Method (ΒΕΜ). However, the conventional BEM leads to dense coefficient matrices and as a result the memory requirements grow quadratically with respect to the number of degrees of freedom. In this work, a Local Domain BEM (LD-BEM) for solving the transient CDR equation with a constant velocity field is presented. The domain of interest is fragmented into small subdomains and the integral representation of the solution is considered separately for each of the subdomains. Eliminating the fluxes at all subdomain interfaces, the proposed LD-BEM leads to sparse linear system coefficient matrices and a reduced number of degrees of freedom. Eight numerical examples are solved to assess the efficiency and accuracy of the proposed method.