Efficient methods for solving boundary integral equation in diffusive scalar problem and eddy current nondestructive evaluation

Abstract: In this paper, a half space adaptive fast multipole boundary facemethod (FMBFM) is presented for solving the three-dimensional half space ex-terior acoustic problems. In the presented method, the Burton-Miller equationbased on the conventional boundary integral equation (CBIE) and its hyper-singularboundary integral equation (HBIE) is used to deal with the fictitious eigenfrequen-cies problem. The half space Green’s function is employed, thus the tree structurein the fast multipole method can be used only for the real domain. The higher orderelements and an adaptive tree structure are used to improve the efficiency of theFMBFM. This half space adaptive FMBFM for half space acoustic problems is anextension of the adaptive FMBFM for full space acoustic problems developed bythe authors. Numerical examples for half space acoustic problems in this paperdemonstrate the efficiency and validity of this method.Keywords: fast multipole boundary face method, Burton-Miller equation, acous-tic problems, half space, modified hyper-singular boundary integral equation1 IntroductionAcoustic problems in half space are a major subject in engineering. Many researchworks have been done on this aspect (Seybert et al., 1988; Secgin et al., 2010; Caoet al., 2010). The half space Green’s function (Seybert et al., 1988) can be appliedby adding the source solution at the image point to the original full-space Green’sfunction for dealing with half space problems. Sound source localization and iden-tification of a refrigerator were accomplished by using the half space boundaryelement method (BEM) (Secgin et al., 2010). Using the half space Green’s func-tion, the discretization of the rigid infinite plane is removed and only the boundaries

Standard numerical models in acoustics rely on the isentropic Helmholtz equation. Its derivation assumes adiabatic and reversible, i.e., dissipation-free, wave propagation. Sound waves in fluids are, however, subject to viscous and thermal losses. These losses originate from viscous friction and heat conduction, leading to the formation of acoustic boundary layers. Considering these effects becomes significant in setups with acoustic cavities of similar dimension as the boundary layers. Recently, boundary element methods (BEM) accounting for the viscothermal dissipation have been proposed. These methods limit the discretization to the surface of the fluid domain and require significantly fewer degrees of freedom than comparable finite element models. However, the BEM coefficient matrices are fully populated, resulting in high computational costs and storage requirements. This study develops a new BEM formulation for acoustics in viscothermal fluids that uses the fast multipole method - a low-rank approximation technique based on a hierarchical subdivision of the computational domain - to alleviate this shortcoming. It is shown that the fast viscothermal BEM improves the algorithmic complexity over the conventional formulation, reducing both the memory requirements and solution time. The results indicate good scalability of the method making it feasible for larger applications such as acoustic metamaterials.

This paper discusses a formulation and its applications of the new Fast Multipole Method (FMM) to three-dimensional Boundary Integral Equation Method (BIEM) in elastostatic crack problems. It is shown, through numerical experiments, that the new FMM is more efficient than the original FMM.

A numerical solution for the diffraction of linear waves around a vertical cylinder is presented in this paper using the fast multipole boundary element method (FMBEM). The fast multipole method (FMM) is employed to accelerate the traditional boundary element method. The accuracy and efficiency of the FMBEM for 2D Helmholtz equation is demonstrated, and it’s proved that this approach is suitable for solving large-scaled problems. In addition, numerical results of water dynamics coefficients, which are related to the diffraction problems of linear waves around a cylinder, are supplied in this paper.

The fast multipole method (FMM) is very efficient in solving integral equations. This paper applies the method to solve large solid-solid boundary integral equations for elastic waves in two dimensions. The scattering problem is first formulated with the boundary element method. FMM is then introduced to expedite the solution process. By using the FMM technique, the number of floating-point operations of the matrix-vector multiplication in a standard conjugate gradient algorithm is reduced from O(N 2) to O(N 1.5), where N is the number of unknowns. The matrix-filling time and the memory requirement are also of the order N 1.5. The computational complexity of the algorithm is further reduced to O(N 4/3) by using a ray propagation technique. Numerical results are given to show the accuracy and efficiency of FMM compared to the boundary element method with dense matrix.

Fast multipole singular boundary method for Stokes flow problems

A modified dual-level fast multipole boundary element method based on the Burton–Miller formulation for large-scale three-dimensional sound field analysis

Though computational molecular dynamics is an effective tool for nano-scale phenomenon analysis, computational costs associated with its computer simulation are extremely high. There are two major computational steps associated with computer simulation of dynamics of molecular structures. They are calculation of interatomic forces and formation and solution of the equations of motion. Currently, these two computational steps are treated separately in most commonly-used methods. For example, Fast Multipole Method (FMM) and Cell Multipole Method (CMM) have been used for calculation of interatomic forces, and Cartesian Coordinate Method (CCM) and Internal Coordinate Molecular Dynamics Method (ICMD) have been created for the formation and solution of equations of motion of an atomistic molecular system. In this paper, a new procedure is presented through a proper integration between multibody molecular algorithms (MMA) and fast multipole methods to improve computational efficiency for computer simulation of the dynamical behaviors of multibody molecular structures in polymers and biopolymers. For the computational costs associated with interatomic forces, a fast multipole method is used to calculate the interatomic forces due to the potentials. For the computational costs associated with formation and solution of equations of motion, a multibody molecular algorithm developed by the author in his previous work will be utilized to integrate with fast multipole methods. The algorithm significantly improves computational efficiency when comparing with its counterpart procedures. The fast multipole method begins by scaling all atoms into a box with coordinate ranges to ensure numerical stability of subsequent operations. The parent box is then divided into half in the direction of each Cartesian axis and each child box is then subdivided to form a computational family tree. The flow of calculations is carried out along the tree structure with five passes. The fast multipole method has been improved and modified to achieve better effectiveness and higher efficiency since it was created. The multibody molecular algorithm starts with numbering subsets, forming bond graph, and developing three computing passes along the tree structure of an atomistic molecular system. Computing data flows in the fast multipole method and the multibody molecular algorithm will properly line up with the parent-child recursive relationship along the configuration of the tree structure due to linear recursive natures of both fast multipole method and multibody molecular algorithm. Then the time spent on the recursive simulation passes in the fast multipole method for computing forces may overlap with the time spent on the three recursive computational passes in the multibody molecular algorithm for forming and solving equations of motion.

Fast multipole boundary element method using the binary tree structure with tight bounds: application to a calculation of an electrostatic force for the manipulation of a metal micro particle ☆

The complex variable fast multipole boundary element method for the analysis of strongly inhomogeneous media

In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this force calculation step can be significantly accelerated by the fast multipole method (FMM). We implemented and compared four different methods in isotropic and anisotropic elastic media: one based on the Taylor series expansion (Taylor FMM), one based on the spherical harmonics expansion (Spherical FMM), one kernel-independent method based on the Chebyshev interpolation (Chebyshev FMM), and a new kernel-independent method that we call the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of the most popular DD simulation softwares. The Spherical FMM employs a more compact multipole representation than the Taylor FMM does and is thus more efficient. However, both the Taylor FMM and the Spherical FMM are difficult to derive in anisotropic elastic media because the interaction force is complex and has no closed analytical formula. The Chebyshev FMM requires only being able to evaluate the interaction between dislocations and thus can be applied easily in anisotropic elastic media. But it has a relatively large memory footprint, which limits its usage. The Lagrange FMM was designed to be a memory-efficient black-box method. Various numerical experiments are presented to demonstrate the convergence and the scalability of the four methods.

The Method of Moments (MoM) is one of the most popular techniques for solving electromagnetic scattering problems. Its main advantage is that, when used to discretise a Boundary Integral Equation (BIE), only the surface of the objects needs to be discretised. Many other techniques require discretisation of the volume as well, leading to an increase of unknowns and requiring an artificial boundary to emulate infinity. The downside of the MoM is that the resulting linear system is dense, due to the Green's function, and often ill-conditioned. The density of the impedance matrix would lead to a prohibitive O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) complexity for both the CPU time and the memory. A number of techniques have been developed in the past, reducing this complexity. Among the most succesful are the Fast Multipole Methods (FMM), which reduce the complexity for a matrix-vector product to O (N log N). This matrix-vector product is the bottleneck of an iterative solution of the linear system. However, an iterative solution can only be efficient if the problem is sufficiently well-conditioned. The choice of BIE influences the conditioning number of the impedance matrix. The Electric Field Integral Equation (EFIE) leads to a highly accurate solution but is notoriously ill-conditioned. In comparison, the Magnetic Field Integral Equation (MFIE) is less accurate (for the same level of discretisation) but is well-posed. Also, on open surfaces the MFIE is not applicable. In order to alleviate the problems with the EFIE, a powerful preconditioner is required to stabilise the breakdown, which occurs in particular at low frequencies (LF). Recently, an approach has been proposed, allowing multiplication of the EFIE impedance matrix with itself, leading to a well-conditioned problem. In the next section, this approach will be briefly revisited. In order to maintain the O (N log N) complexity for the global solution, we employ FMM to accelerate the Calderon Preconditioner as well. A number of difficulties occur, but these can all be overcome. This combination of FMM and Calderon Preconditioning will be dealt with in the remaining sections. At the time of the conference, examples will be shown to illustrate the novel techniques.

The fast multipole method (FMM) is developed to analyse electromagnetic scattering using higher-order hierarchical Legendre basis functions, which have been proposed recently and make the impedance matrix, from method of moments, a low condition number. In this method, the point-to-point based-on-sample quadrature is used because the basis functions are defined on large surface patches of scatterers. The numerical implementations of matrix–vector multiply by FMM are particularly presented. These numerical results show that the proposed second- or third-order FMM (high-order FMM) is accurate and efficient.

Hi-BoX is a generic library implementing state-of-the-art fast direct and iterative solvers for existing codes based on Boundary Element Method (BEM) or Method of Moments (MoM). It benefits from recent advances in numerical methods, linear algebra and High Performance Computing (HPC). This includes new advances in H-matrix and Fast Multipole Method (FMM) and their hybridization, new approaches for handling a large number of right-hand sides, and finally the use of new parallelization techniques based on a runtime system simplifying the use of modern massively parallel and heterogeneous architectures.

An interpolation-based fast-multipole accelerated boundary integral equation method for the three-dimensional wave equation