Fast Boundary Element Method Research Articles

A fast and stable solver for acoustic scattering problems based on the nonuniform grid approach.

A fast and stable boundary element method (BEM) algorithm for solving external problems of acoustic scattering by impenetrable bodies is developed. The method employs the Burton-Miller integral equation, which provides stable convergence of iterative solvers, and a generalized multilevel nonuniform grid (MLNG) algorithm for fast evaluation of field integrals. The MLNG approach is used here for the removal of computational bottlenecks involved with repeated matrix-vector multiplications as well as for the low-order basis function regularization of the hyper-singular integral kernel. The method is used for calculating the fields scattered by large acoustic scatterers, including nonconvex bodies with piece-wise smooth surfaces. As a result, the algorithm is capable of accurately incorporating high-frequency effects such as creeping waves and multiple-edges diffractions. In all cases, stable convergence of the method is observed. High accuracy of the method is demonstrated by comparison with the traditional BEM solution. The computational complexity of the method in terms of both the computation time and storage is estimated in practical computations and shown to be close to the asymptotic O(N log N) dependence.

A kernel-independent fast multipole BEM for large-scale elastodynamic analysis

Purpose – 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. Design/methodology/approach – 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. Findings – 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. Originality/value – 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.

A fast boundary element method for the scattering analysis of high-intensity focused ultrasound.

High-intensity focused ultrasound (HIFU) techniques are promising modalities for the non-invasive treatment of cancer. For HIFU therapies of, e.g., liver cancer, one of the main challenges is the accurate focusing of the acoustic field inside a ribcage. Computational methods can play an important role in the patient-specific planning of these transcostal HIFU treatments. This requires the accurate modeling of acoustic scattering at ribcages. The use of a boundary element method (BEM) is an effective approach for this purpose because only the boundaries of the ribs have to be discretized instead of the standard approach to model the entire volume around the ribcage. This paper combines fast algorithms that improve the efficiency of BEM specifically for the high-frequency range necessary for transcostal HIFU applications. That is, a Galerkin discretized Burton-Miller formulation is used in combination with preconditioning and matrix compression techniques. In particular, quick convergence is achieved with the operator preconditioner that has been designed with on-surface radiation conditions for the high-frequency approximation of the Neumann-to-Dirichlet map. Realistic computations of acoustic scattering at 1 MHz on a human ribcage model demonstrate the effectiveness of this dedicated BEM algorithm for HIFU scattering analysis.

A parallel fast boundary element method using cyclic graph decompositions

We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is O((n/N)log(n/N))$O((n/\sqrt {N})\log (n/\sqrt {N}))$ in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n=2744832 and N=273 on a real-world geometry document that the method exhibits superior parallel scalability O((n/N)logn)$O((n/N)\,\log n)$ of the overall time, while the memory consumption scales accordingly to the theoretical estimate.

Meshfree Computation of Field Lines Across Multiple Domains Using Fast Boundary Element Methods

A meshfree postprocessing for the computation of field lines suggests itself in the case of fast boundary element methods. Field values are only calculated and stored in points, which are absolutely necessary for field line computations. Hence, the total amount of processed data is dramatically reduced in comparison to a mesh-based approach along with precomputed field values in all mesh nodes. An automatic and robust domain detection method enables reliable identifications of intersections of field lines with domain boundaries even in the case of complex-shaped surfaces. A combination of automatic domain detection and adaptive step size control of the underlying Runge–Kutta–Fehlberg method results in correctly and efficiently computed field lines, including sharp bends at domain boundaries.

A fast boundary element method for the solution of periodic many-inclusion problems via hierarchical matrix techniques

Our work is motivated by numerical homogenization of materials such as concrete, modeled as composites structured as randomly distributed inclusions imbedded in a matrix. In this paper, we propose a method for the approximation of the periodic corrector problem based on boundary integral equations. The fully populated matrices obtained by the discretization of the integral operators are successfully dealt with using the H-matrix format. R esum e.

Solving multizone and multicrack elastostatic problems: A fast multipole symmetric Galerkin boundary element method approach

Symmetric Galerkin boundary element methods (SGBEMs) for three-dimensional elastostatic problems give rise to fully populated (albeit symmetric) matrix equations, entailing high solution times for large models. This paper is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks. The subdomain coupling approach is based on a minimal set of interfacial unknowns (i.e. one displacement and one traction vector at any interfacial point) that are defined globally for the complete multizone configuration. Then, unknowns for each subdomain are defined in terms of the global unknowns, with appropriate sign conventions for tractions induced by subdomain numbering. This formulation (i) automatically enforces the perfect-bonding transmission conditions between subdomains, and (ii) is globally symmetric. The subsequent FM-SGBEM basically proceeds by assembling contributions from each subregion, which can be computed by means of an existing single-domain FM-SGBEM implementation such as that previously presented by the authors (Pham et al., Eng Anal Bound Elem 2012;36:1838–47 [36]). Along the way, the computational performance of the FM-SGBEM is enhanced through (a) suitable storage of the near-field contribution to the SGBEM matrix equation and (b) preconditioning by means of nested GMRES. The formulation is validated on numerical experiments for 3D configurations involving many cracks and inclusions, and of sizes up to N≈106.

Fast isogeometric boundary element method based on independent field approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.

A fast directional BEM for large-scale acoustic problems based on the Burton–Miller formulation

In this paper, a highly efficient fast boundary element method (BEM) for solving large-scale engineering acoustic problems in a broad frequency range is developed and implemented. The acoustic problems are modeled by the Burton–Miller boundary integral equation (BIE), thus the fictitious frequency issue is completely avoided. The BIE is discretized by using the Nyström method based on the curved quadratic elements, leading to simple numerical implementation (no edge or corner problems) and high accuracy in the BEM analysis. The linear systems are solved iteratively and accelerated by using a newly developed kernel-independent wideband fast directional algorithm (FDA) for fast summation of oscillatory kernels. In addition, the computational efficiency of the FDA is further promoted by exploiting the low-rank features of the translation matrices, resulting in two- to three-fold reduction in the computational time of the multipole-to-local translations. The high accuracy and nearly linear computational complexity of the present method are clearly demonstrated by typical examples. An acoustic scattering problem with dimensionless wave number kD (where k is the wave number and D is the typical length of the obstacle) up to 1000 and the degrees of freedom up to 4 million is successfully solved within 10h on a computer with one core and the memory usage is 24GB.

A fast approach coupling Boundary Element Method and plane wave approximation for wave interaction analysis in sparse arrays of wave energy converters

A computational approach is developed to investigate wave interaction effects in sparse arrays of floating bodies (such as wave energy converters) based on linear potential theory. In particular, the wave diffraction and radiation problems in a multiple body array are solved in reasonable time and accuracy. In contrast to previous approaches that have considered all bodies in the array as a single module, the present approach treats each body in the array as an isolated body. The interactions resulting from the scattered wave field among the bodies are then taken into account via plane wave approximation in an iterative manner. The boundary value problem corresponding to an isolated body is solved by the Boundary Element Method (BEM). The approach is useful for wave periods in the range 4–15s, provided that the bodies are separated by at least five times the characteristic dimension of a body. The main advantage of the approach is that the computational time and memory requirements are significantly less than that of conventional BEM. In this paper, first, the numerical results for hydrodynamic coefficients computed by the proposed approach are validated against conventional BEM. Next, the wave interaction effects on power production are investigated in arrays of 50 wave energy converters.

Simple sensitivity calculation for inverse design problems in electrical engineering

Purpose – The purpose of this paper is to present a simple approach for calculation of the sensitivities in the free-form inverse design problems. The approach is based on the analogy with the similar tasks used in the signal-processing analysis. In the proposed case it is not required to solve an adjoint problem as in the most of the similar optimization tasks. The simulation engine used in the background is a Fast Boundary Element Method. The approach is validated on some known benchmark problems. Design/methodology/approach – Inverse design is recognized nowadays as a crucial scientific grand challenge. Contrary to the conventional approach (“Given the structure, find the properties”) it purses a new paradigm (“Given the desired property, find the structure”). Inverse class of problems has a broad application area, from the material-, medical-, bio- to the engineering-class of problems. When dealing with the inverse design in free-form optimization of the engineering problems the typical approach is to calculate the adjoint problem. Calculation of the adjoint problem mostly requires the costly calculation of the gradients, which makes the whole optimization procedure rather expensive due to the high computational burden required for their solution. Findings – In this paper it is proposed a novel Simple Sensitivity Approach to get in a fast way the response (sensitivity) function of the analyzed structure. The simulation engine used in the background is the Fast Boundary Element Method. Originality/value – Novel approach for inverse design when performing the free-form optimization of engineering problems.

Coupled finite element - hierarchical boundary element methods for dynamic soil-structure interaction in the frequency domain

SUMMARY This paper discusses the coupling of finite element and fast boundary element methods for the solution of dynamic soil–structure interaction problems in the frequency domain. The application of hierarchical matrices in the boundary element formulation allows considering much larger problems compared to classical methods. Three coupling methodologies are presented and their computational performance is assessed through numerical examples. It is demonstrated that the use of hierarchical matrices renders a direct coupling approach the least efficient, as it requires the assembly of a dynamic soil stiffness matrix. Iterative solution procedures are presented as well, and it is shown that the application of such schemes to dynamic soil–structure interaction problems in the frequency domain is not trivial, as convergence can hardly be achieved if no relaxation procedure is incorporated. Aitken's Δ2-method is therefore employed in sequential iterative schemes for the calculation of an optimized interface relaxation parameter, while a novel relaxation technique is proposed for parallel iterative algorithms. It is demonstrated that the efficiency of these algorithms strongly depends on the boundary conditions applied to each subdomain; the fastest convergence is observed if Neumann boundary conditions are imposed on the stiffest subdomain. The use of a dedicated solver for each subdomain hence results in a reduced computational effort. A monolithic coupling strategy, often used for the solution of fluid–structure interaction problems, is also introduced. The governing equations are simultaneously solved in this approach, while the assembly of a dynamic soil stiffness matrix is avoided. Copyright © 2013 John Wiley & Sons, Ltd.

A very fast and accurate boundary element method for options with moving barrier and time-dependent rebate

A numerical method to price options with moving barrier and time-dependent rebate is proposed. In particular, using the so-called Boundary Element Method, an integral representation of the barrier option price is derived in which one of the integrand functions is not given explicitly but must be obtained solving a Volterra integral equation of the first kind. This equation is affected by several kinds of singularities, some of which are removed using a suitable change of variables. Then the transformed equation is solved using a low-order finite element method based on product integration. Numerical experiments are carried out showing that the proposed method is extraordinarily fast and accurate. In particular a high level of accuracy is achieved also when the initial price of the underlying asset is close to the barrier, when the barrier and the rebate are not differentiable functions, or when the optionʼs maturity is particularly long.

Isogeometric boundary element method for the simulation of underground excavations

Isogeometric methods have become the latest trend in numerical simulation. Their attractiveness stems from the fact that the description of the geometry can be taken directly from CAD programs, avoiding the need for mesh generation. Since NURBS (non-uniform rational B-splines) functions that exhibit desirable properties such as high continuity and efficient refinement algorithms are also used for the description of the unknowns, a higher quality of results can be expected. This paper presents some results from the research project ‘Fast isogeometric boundary element methods for tunneling’. The aim of the project is to implement a fast and accurate method for the simulation of underground excavations without the need for the generation and subsequent refinement of a mesh.

Comparison of fast boundary element methods on parametric surfaces

We compare fast black-box boundary element methods on parametric surfaces in R3. 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.

Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses.

RWCap: A Floating Random Walk Solver for 3-D Capacitance Extraction of Very-Large-Scale Integration Interconnects

A floating random walk (FRW) solver, called RWCap, is presented for the capacitance extraction of very-large-scale integration (VLSI) interconnects. An approach, including the numerical characterization of the cross-interface transition probability and weight value, is proposed to accelerate the extraction of structures with multiple dielectric layers. A comprehensive variance reduction scheme based on the importance sampling and stratified sampling is proposed to improve the convergence rate of the FRW algorithm. Finally, the space management technique using an octree data structure and the parallel computing technique are presented to further improve the efficiency. Numerical experiments are carried out with the test cases generated under the 180 and 45-nm process technologies. They demonstrate that the proposed multidielectric FRW algorithm achieves up to 160× speedup over the FRW algorithm using spherical transition domains to cross dielectric interface, with very small memory overhead. The variance reduction techniques further bring 3× or more speedup without memory overhead and the loss of accuracy. The RWCap also outperforms other existing FRW algorithm and fast boundary element method solvers in terms of computational time or scalability. The experiments on an 8-core CPU machine show that the parallel RWCap is over 6× faster than its serial-computing version.

2010 非一様B-Splineに基づくアイソジオメトリック要素を用いた2次元Helmholtz方程式に対する高速多重極境界要素法

2010 非一様B-Splineに基づくアイソジオメトリック要素を用いた2次元Helmholtz方程式に対する高速多重極境界要素法

Large-scale thermal analysis of fiber composites using a line-inclusion model by the fast boundary element method

Abstract The fast boundary element method is applied for the three-dimensional large-scale thermal analysis of fiber-reinforced composites based on a line inclusion model. In this approach, fibers are treated as inclusions with temperature assumed constant over the circular cross-section and varying along the length direction. Therefore, fibers can be meshed with line elements, making both the modeling complexity and the number of unknowns significantly reduced. An interface integral boundary element method introduced by Gao in 2009 (Engineering Analysis with Boundary Elements 2009; 33: 539–546) is extended to generate a single-domain boundary integral equation for governing this line-inclusion problem. Thus in principle, fibers with arbitrary length can be modeled. The fast multipole method is employed for the fast analysis of such problems with large-scales. The largest composite model in a personal desktop computer has the number of fibers reaching 20,000. Numerical results clearly demonstrate validity of the proposed model and its potential for large-scale analysis of fiber-reinforced composites.

An application of fast multipole method to isogeometric boundary element method for Laplace equation in two dimensions

According to the concept of isogeometric analysis, we have developed a boundary element method (BEM) using B-spline basis functions for the two-dimensional Laplace equation, focusing on external Neumann problems. Further, we have applied the fast multipole method (FMM) to the present isogeometric BEM to reduce the computational complexity from O(n2) to O(n), where n is the number of control points to define the closed boundary of the computational domain. In a benchmark test, we confirmed that the FMM can accelerate the isogeometric BEM successfully. In addition, the proposed fast BEM can be an alternative of the standard fast BEM using the piecewise-constant elements. Finally, the feasibility of the proposed method for solving large-scale problems was demonstrated through numerical examples.