Multipole Boundary Element Method Research Articles

Wideband fast multipole boundary element method for flow-induced noise analysis based on Lighthill's equation

In order to reduce the computational load for the analysis of three-dimensional flow-induced noise based on the Lighthill equation using the boundary element method (BEM), an efficient method is developed on the basis of the fast multipole BEM (FMBEM). The common hierarchical cell structure for grouping boundary element nodes, sound-receiving points and flow-induced noise sources is used; this enables coefficients of the FMBEM to be reused during the calculation of sound pressure at boundary element nodes and receiving points, thereby further accelerating the analysis. In addition, to deal with the increasing hierarchical level for encompassing flow-induced noise sources, which are often widely distributed around the object to be analyzed, a wideband version of the FMBEM is applied. Following the computational procedure, two application examples are presented for discussion. First, noise from a cylinder located in a flow is analyzed to discuss the accuracy of the method. Then, a problem with large number of degrees-of-freedom and flow-induced noise sources is analyzed to demonstrate its applicability to large-scale problems.

Fast Multipole Boundary Element Method for Aerodynamic Sound Field Analysis Based on Lighthill’s Equation

The primary disadvantage of the aerodynamic sound field analysis based on the Lighthill’s equation using the boundary element method (BEM) is the computational time; this is mainly because contributions from numerous aerodynamic sound sources are computed at all boundary element nodes and sound-receiving points. This study proposes a fast method for computing source contributions based on the fast multipole method (FMM). Along with the fast multipole BEM, which is already in practical use as a fast BEM, the analysis is substantially accelerated. The use of a common hierarchical cell structure for grouping boundary element nodes, sound-receiving points and aerodynamic sound sources, enables coefficients of the FMM to be reused, thereby further accelerating the analysis. To deal with the increasing hierarchical level, a wideband FMM is applied. In the sound radiation analysis of a quadrupole source located in a free field, the accuracy is validated. Sound radiation from a cylinder located in a flow is analyzed as a practical problem; the accuracy and numerical settings are discussed. Finally, the proposed method is applied to a problem with more than 0.4 million degrees-of-freedom and more than 3 million aerodynamic sound sources to demonstrate its applicability to large-scale problems.

Analysis of 2-D elastic solid with multiple V-notches by a fast multipole BEM with a novel singular element with multi-order asymptotic terms

A novel fast multipole boundary element method (BEM) is developed to analyze the stress fields of plane structures with multiple V-notches. To accurately evaluate the singular stresses in the vicinity of the V-notch tip, a novel singular element with multi-order asymptotic terms (SEMAT) is first proposed. The tractions and displacements on the SEMAT are expressed as the asymptotic expansions, where both the first singular term and other higher-order terms are considered. The SEMAT is then introduced into the fast multipole BEM, and several singularity problems of boundary integrals on the SEMAT are treated with caution. Finally, the fast multipole BEM with the SEMAT is used to obtain the whole stress and displacement fields for the plane V-notch/crack structures. Numerical examples show that the present method is accurate for analyzing the V-notch with opening angle of arbitrary size in both single and bonded materials. It is not necessary to use a dense mesh in modeling, and the computational efficiency of the present method is high for large-scale problems.

A new boundary element formulation for modeling and simulation of three‐temperature distributions in carbon nanotube fiber reinforced composites with inclusions

A new boundary element method (BEM) formulation is developed in this paper for modeling and simulation of nonlinear three‐temperature (3T) distributions in carbon nanotube (CNT) fiber‐reinforced composites embedded with many rigid‐line inclusions that can be treated as having constant temperature distributions. The fast multipole method (FMM) has been applied to accelerate the proposed BEM solution of the considered problem of a large number of nano‐inclusions, and therefore, the computational complexity and computational time are reduced. The CPU time and memory requirement are also reduced by implementing generalized modified shift‐splitting (GMSS) iteration method which does not require the entire matrix to be stored in the memory. Then, we introduced a new formula which can play an important role in solving thermal problems arising in CNT nanotechnology. Effective thermal conductivity of fiber‐reinforced composites can be evaluated using the computed temperature and heat flux fields. The fast multipole BEM results are depicted graphically to show the effects of CNT manufacturing methods and thermal conductivities of different CNT composites on the temperature and heat flux distributions. The computational performance of the proposed methodology has been demonstrated. The fast multipole BEM results are compared graphically with the finite element method (FEM) results and with the experimental results reported in the literature. Excellent agreements are found in both cases, which clearly demonstrate the validity, accuracy, and efficiency of our proposed algorithm and modeling and simulation technique. The findings and solutions of this study contribute to the further development of CNT nanotechnology applications and devices.

Application of fast multipole boundary element method for two-dimensional nonlinear interface debonding of particulate composites

A fast multipole boundary element method (BEM) is used herein to simulate the two-dimensional interfacial debonding of particulate composites. The behavior of the interface between the inclusion and the matrix is modeled using a nonlinear constitutive relationship. Interface debonding is implemented by decoupling the node pair. In the proposed method, the degree of freedom (DOF) of the interfacial traction changes to that of displacement of the free surface; thus, the number of DOFs of each node pair prior to and after decoupling is the same, which facilitates the simulation better than the finite element method. In simulating the softening stage of the nonlinear interfacial deformation, the stress in the node pair is assumed to be unloaded to zero and then reloaded to a new equilibrium state. A fast multipole expansion technique is applied within the developed BEM to solve the large-scale problem of debonding particulate composites whereby millions of DOFs can be calculated via a step-wise calculation method that provides high precision interface stress results. A comparison of these results with the analytical solution for a single inclusion case demonstrates the high stress accuracy and effectiveness of the developed fast multipole BEM to solve large-scale nonlinear interface problems.

Fast multipole method for poroelastodynamics

Wave propagation phenomena occur in reality often in semi-infinite two-phase (porous) regions. It is well known that such problems can be handled well with the poroelastodynamic Boundary Element Method (BEM). But, it is also well known that the BEM with its dense matrices becomes prohibitive with respect to storage and computing time. This is especially true for poroelastodynamics, where in the best case four degrees of freedom per node are required. As well, the fundamental solution of poroelastodynamics is computationally expensive.Here, a fast multipole BEM is proposed to circumvent those points. The Chebyshev interpolation-based FMM significantly reduces the memory consumption of the system matrix and thus allows for larger problem sizes to be treated. As well, it requires fewer evaluations of the fundamental solution. To employ an iterative solver, the use of a transformation of the material data is mandatory. Numerical tests show the expected almost linear complexity of the proposed method.

Modeling of multiple crack propagation in 2-D elastic solids by the fast multipole boundary element method

In this paper, a fast multipole boundary element method (BEM) is presented for modeling crack propagation in two-dimensional (2-D) linear elastic solids. A dual boundary integral equation (BIE) formulation using a linear combination of the displacement and traction BIEs is applied to model cracks in this BEM. Constant boundary elements are used to discretize the BIEs and the fast multipole method (FMM) is applied to accelerate the solution of the BEM system of equations. Numerical examples of multiple crack propagation in 2-D elastic domains and under cyclic loading, including perforated plates with multiple holes and cracks, are presented to show the effectiveness and efficiency of the developed fast multipole BEM.

A new simple multidomain fast multipole boundary element method

A simple multidomain fast multipole boundary element method (BEM) for solving potential problems is presented in this paper, which can be applied to solve a true multidomain problem or a large-scale single domain problem using the domain decomposition technique. In this multidomain BEM, the coefficient matrix is formed simply by assembling the coefficient matrices of each subdomain and the interface conditions between subdomains without eliminating any unknown variables on the interfaces. Compared with other conventional multidomain BEM approaches, this new approach is more efficient with the fast multipole method, regardless how the subdomains are connected. Instead of solving the linear system of equations directly, the entire coefficient matrix is partitioned and decomposed using Schur complement in this new approach. Numerical results show that the new multidomain fast multipole BEM uses fewer iterations in most cases with the iterative equation solver and less CPU time than the traditional fast multipole BEM in solving large-scale BEM models. A large-scale fuel cell model with more than 6 million elements was solved successfully on a cluster within 3 h using the new multidomain fast multipole BEM.

Transient elastic wave analysis of 3-D large-scale cavities by fast multipole BEM using implicit Runge–Kutta convolution quadrature

Existing boundary element methods (BEMs) have been proven to be efficient numerical techniques particularly in modeling wave propagation problems but still remain limitations in solving large-scale problems. The main disadvantages may be caused by several problems, including high cost and large amount of computer memory for the computation. A time-domain BEM in which a collocation method is used for the time discretization also shows numerical instability for small CFL (Courant–Friedrichs–Lewy) number. In this work, we propose a new three-dimensional (3-D) approach for solving transient elastic wave propagation problems of large-scale spherical and spheroidal cavities by the convolution quadrature BEM accelerated by fast multipole method (FMM) using an implicit Runge–Kutta scheme (in abbreviation, we name it as IRK-based CQ-FMBEM). Two special techniques pertaining to the FMM including the scaling of modified spherical Bessel functions and the truncation method are used to enhance the efficiency and stability of our proposed method. This approach is found to be particularly suitable for 3-D large-scale problems as its high accuracy, stability of time-marching process, and computational efficiency. These properties are subsequently illustrated through numerical examples dealing with large-scale wave problems of 3-D single spherical and spheroidal cavities and multiple equally and irregularly spherical cavities. The accuracy of the present formulation is verified by comparing the obtained results with available reference solutions in the literature. Some numerical aspects of time increments, sizes and topological shapes of cavities, and their influences on the deformations and the waveforms are investigated. Also, detailed investigation of the computational efficiency of our proposed method, such as CPU time, required memory, etc. is presented. All the implementation tasks are carried out using the Tokyo Tech. supercomputer which is called TSUBAME 2.5 [42].

Boundary Condition Related Mixed Boundary Element and its Application in FMBEM for 3D Elastostatic Problem

A boundary condition (BC) related mixed element method is presented to address the corner problem in boundary element method (BEM) for 3D elastostatic problems. In this method, noncontinuous elements (NCEs) are only used at the displacement-prescribed corners/edges and continuous elements (CEs) in other places, which can decrease the degrees of freedom (DOFs) compared to the approach using NCEs at all corners/edges. Moreover, an automatic generation algorithm of BC related mixed linear triangular elements is implemented with the help of 3D modeling engine ACIS, and the boundary element analysis (BEA) is integrated into CAD systems. In order to solve large scale problems, the fast multipole BEM (FMBEM) with mixed elements is proposed and utilized in the BEA. The examples show that the node shift scheme adopting 1/4 is optimal and the BEM/FMBEM using mixed elements can produce more accurate results by only increasing a small number of DOFs.

A fast multipole boundary element method for modeling 2-D multiple crack problems with constant elements

A fast multipole boundary element method (BEM) for solving 2-D multiple crack problems in linear elastic fracture mechanics is presented in this paper. For multiple crack problems, both the degrees of freedom (DOFs) and the size of system matrices increase quickly as the number of cracks increases, and the conventional BEM cannot support such large systems. Instead of using the singular quarter-point boundary elements at the crack tips, constant line elements are applied to symmetrically discretize the outer boundaries and crack surfaces in the present approach. In order to keep the accuracy within a limited acceptable range, a relatively large number of constant elements are required to discretize the crack surfaces. The crack opening displacement (COD) fields of the multiple crack problems are obtained by the fast multipole BEM. Stress intensity factors (SIFs) are extracted from the obtained displacement fields near the crack tip by using one point COD formula. Comparison of the CODs between the fast multipole BEM and a finite element method using ANSYS are illustrated to show the feasibility of the proposed approach. With the acceleration of fast multipole technique, multi-crack problems can be dealt with desktop PCs. Several numerical examples are presented for computing the SIFs of cracks to study the effectiveness and the efficiency of the proposed approach. The numerical results clearly demonstrate the potentials of the fast multipole BEM for solving 2-D large-scale multi-crack problems by using constant elements.

A fast multipole boundary element method for solving two-dimensional thermoelasticity problems

A fast multipole boundary element method (BEM) for solving general uncoupled steady-state thermoelasticity problems in two dimensions is presented in this paper. The fast multipole BEM is developed to handle the thermal term in the thermoelasticity boundary integral equation involving temperature and heat flux distributions on the boundary of the problem domain. Fast multipole expansions, local expansions and related translations for the thermal term are derived using complex variables. Several numerical examples are presented to show the accuracy and effectiveness of the developed fast multipole BEM in calculating the displacement and stress fields for 2-D elastic bodies under various thermal loads, including thin structure domains that are difficult to mesh using the finite element method (FEM). The BEM results using constant elements are found to be accurate compared with the analytical solutions, and the accuracy of the BEM results is found to be comparable to that of the FEM with linear elements. In addition, the BEM offers the ease of use in generating the mesh for a thin structure domain or a domain with complicated geometry, such as a perforated plate with randomly distributed holes for which the FEM fails to provide an adequate mesh. These results clearly demonstrate the potential of the developed fast multipole BEM for solving 2-D thermoelasticity problems.

Slow convergence of the BEM with constant elements in solving beam bending problems

Constant elements offer many advantages as compared with other higher-order elements in the boundary element method (BEM). With the use of constant elements, integrals in the BEM can be calculated accurately with analytical integrations and no corner problems need to be addressed. These features can make fast solution methods for the BEM (such as the fast multipole, adaptive cross approximation, and pre-corrected fast Fourier transform methods) especially efficient in computation. However, it is well known that the collocation BEM with constant elements is not adequate for solving beam bending problems due to the slow convergence or lack of convergence in the BEM solutions. In this study, we quantify this assertion using simple beam models and applying the fast multipole BEM code so that a large number of elements can be used. It is found that the BEM solutions do converge numerically to analytical solutions. However, the convergence rate is very slow, in the order of h to the power of 0.55–0.63, where h is the element size. Some possible reasons for the slow convergence are discussed in this paper.

Multilevel fast multipole method based on a potential formulation for 3D electromagnetic scattering problems

A combination of the multilevel fast multipole method (MLFMM) and boundary element method (BEM) can solve large scale photonics problems of arbitrary geometry. Here, MLFMM-BEM algorithm based on a scalar and vector potential formulation, instead of the more conventional electric and magnetic field formulations, is described. The method can deal with multiple lossy or lossless dielectric objects of arbitrary geometry, be they nested, in contact, or dispersed. Several examples are used to demonstrate that this method is able to efficiently handle 3D photonic scatterers involving large numbers of unknowns. Absorption, scattering, and extinction efficiencies of gold nanoparticle spheres, calculated by the MLFMM, are compared with Mie's theory. MLFMM calculations of the bistatic radar cross section (RCS) of a gold sphere near the plasmon resonance and of a silica coated gold sphere are also compared with Mie theory predictions. Finally, the bistatic RCS of a nanoparticle gold-silver heterodimer calculated with MLFMM is compared with unmodified BEM calculations.

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.

Analyzing Acoustic Radiation Modes of Baffled Plates With a Fast Multipole Boundary Element Method

In the analysis of an acoustic radiation mode of a baffled plate, Rayleigh integral with free space Green's function is involved. The boundary element method (BEM) is one of the approaches to compute its modes and radiation efficiencies. In this paper a fast multipole BEM in conjunction with an iterative solver based on the implicit restart Arnold method is proposed to efficiently and accurately evaluate acoustic radiation modes and efficiencies. Even though a 3D free space Green's function is used here, a quad tree is used for the hierarchical tree structure of the boundary mesh instead of an oct tree, which can speed up the fast multipole BEM. Similar to the analytical integration of moment evaluations, the analytical integration is also employed to compute the local expansion coefficients which further improves the efficiency of the fast multipole BEM for the analysis of an acoustic radiation mode of baffled plates. Comparison between numerical and theoretical radiation efficiencies of a baffled circular plate vibrating as a piston shows that the fast multipole BEM proposed here can give results with very good accuracy. The computation of the eigenvalues and eigenvectors of a baffled rectangular plate further reveals the efficiency in CPU time, smaller memory size, and accuracy of the fast multipole BEM in the analysis of an acoustic radiation mode.

An adaptive dual-information FMBEM for 3D elasticity and its GPU implementation

Combined the boundary element method (BEM) with the fast multipole method (FMM), the fast multipole BEM (FMBEM) is proposed to solve large scale problems. A key issue the FMBEM has to address is the element integrals, which usually consumes much time when the FMM for N -body problems is directly used. In order to accelerate element integrals, we present an adaptive FMBEM with a particular dual-information tree structure which contains both node and element information, and use it for 3D elasticity in this paper. In our adaptive FMBEM, the Multipole Expansions (ME), Moment-to-Local (M2L) translation, Local Expansions (LE), and the Near Field Direct Computation (NFDC) are level independent so that they are suitable for parallel computing. The examples show that the time of ME and NFDC in our FMBEM is almost 1/3 and 1/2 compared with that in a node-based FMBEM which deals with FMBEM in a particle interaction mode. We develop two GPU parallel strategies to accelerate the processes of ME, M2L and NFDC and implement them on a NVIDIA GTX 285 GPU, and the speedups to an Intel Core2 Q9550 CPU using 4 cores can reach 10.7 for ME, 16.2 for M2L, and 3.6 for NFDC.

Spherical dynamic models of top‐down tectonics

We use the Multipole–Boundary Element Method (MP‐BEM) to simulate regional and global geodynamics in a spherical 3‐D setting. We first simulate an isolated subducting rectangular plate with length (Llitho) and width (Wlitho) varying between 0.5 and 2 times the radius of the Earth (REarth) and with viscosity ηlitho varying between 100 and 500 times the upper mantle (ηUM), sinking in a layered mantle characterized by lower‐upper mantle viscosity ratioλ = ηLM/ηUM varying between 1 and 80. In a mantle with small upper/lower viscosity contrast (λ ≅ 1), trench and plate motions are weakly dependent on Wlitho; plate motion is controlled by slab pull if Llitho ≤ REarth, while for longer plates plate speed strongly decreases because of the plate basal friction and flow reorganization. An increasing viscosity ratio λ gradually breaks this pattern, and for λ ≅ 10 combined with Wlitho ≈ REarth(and greater) trench advance and retreat are simultaneously observed. These results offer a first‐order explanation of the origin of the size (Llitho ≈ Wlitho ≈ REarth) of the largest plates observed over the past 150 Myr. Finally, two global plate tectonic simulations are performed from reconstructed plates and slabs at 25 Ma before present and before 100 Ma, respectively. It is shown that MP‐BEM predicts present plate kinematics if plate‐mantle decoupling is adopted for the longest plates (Llitho > REarth). Models for 100 Ma show that the slab‐slab interaction between India and Izanagi plates at 100 Ma can explain the propagation of the plate reorganization from the Indian to the Pacific plate.

Theory of Fast Multipole Virtual Boundary Element Method and Discussion on Key Issues

The basic theory of fast multipole virtual boundary element method (VBEM) is discussed through expanding the fundamental solution, and the algorithm can make the complexities of operation and memory about solution of the equations to be of linear proportion to the freedoms of the problem. Numerical examples are presented to demonstrate the feasibility, accuracy and efficiency of the method. At the same time, the relationships between the order for expansion and the storage capacity, computing time, precision are analyzed, and the influence of boundary points in the leaf to the calculation efficiency is discussed. The corresponding reference value is put forward for the convenience of engineering application.

Analyzing Rolling Pressure of Rolling Strip by Multi-Object Contact Multi-Pole Boundary Element Method

For guiding production effectively and revealing rolling mechanisms, the analysis of rolling-pressure distribution regulation is very important. Because rolling strip belongs to multi-object contact engineering problem which involves coupling deformation, obviously the accuracy of disaggregate mathematical method is low. Therefore, in this paper,it is developed that multi-object, elasto-plastic and frictional contact multipole Boundary Element Method (BEM) by Fortran program. It can simulate five object synchronous contact accurately without artificial assumption.The three dimensional rolling-pressure distribution regulation of rolling strip has been obtained through this program. Through testing and calculating the load properties of bearing, the result shows that it is accordant for test and calculation. Therefore, multi-pole BEM is one of the effective numerical analytic tools of rolling contact problem, and the obtained result of rolling-pressure distribution can provide for reference value for analyzing rolling strip.