Fast Boundary Integral Equation Method Research Articles

Circular Slit Maps of Multiply Connected Regions with Application to Brain Image Processing

In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is O((M + 1)n), where M+1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require O((M+1)^3 n^3) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented.

Numerical Conformal Mapping onto the Exterior Unit Disk with a Straight Slit and Logarithmic Spiral Slits

This paper presents a fast boundary integral equation method for numerical conformal mapping of unbounded multiply connected regions onto a disk with an infinite straight slit and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method.

Numerical Conformal Mapping onto the Entire Complex Plane Bounded with Finite Straight Slit and Logarithmic Spiral Slits

This paper presents a fast boundary integral equation method with for computing conformal mappings of multiply connected regions. We consider the canonical region consists of the entire complex plane bounded by a finite straight slit on the line Im ω = 0 and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method.

A fast boundary integral equation method for point location problem

A numerical method based on the boundary integral equation is proposed for the point location problem. For a bounded domain, the integral value is close to 1.0 if a point is inside the domain, and is close to 0.0 when the point is outside the domain. For convenience of integration, the boundary of the domain can be discretized into boundary integral cells. The idea of isogeometric analysis can be easily coupled with the proposed method, i.e., using the parametric functions in geometric modeling to create the integral cells, which results in a mesh-free procedure for which the geometry can be exactly produced at all stages. Thus, the method can be applied to arbitrary shapes and easily embedded in computer-aided design (CAD) packages. The method is time-consuming if implemented directly; a fast multipole method is coupled with the proposed method to accelerate the integral procedure. Some examples of 2D and 3D cases are tested to show the accuracy and efficiency of the proposed method.

Boundary Integral Equations for Calculating Complex Eigenvalues of Transmission Problems

Resonance frequencies are complex eigenvalues at which the homogeneous transmission problems have non-trivial solutions. These frequencies are of interest because they affect the behavior of the solutions even when the frequency is real. The resonance frequencies are related to problems for infinite domains which can be solved efficiently with the Boundary Integral Equation Method (BIEM). We thus consider a numerical method of determining resonance frequencies with fast BIEM and the Sakurai-Sugiura projection Method (SSM). However, BIEM may have fictitious eigenvalues even when one uses M\"uller or PMCHWT formulations which are known to be resonance free when the frequency is real valued. In this paper, we propose new BIEs for transmission problems with which one can distinguish true and fictitious eigenvalues easily. Specifically, we consider waveguide problems for the Helmholtz equation in 2D and standard scattering problems for Maxwell's equations in 3D. We verify numerically that the proposed BIEs can separate the fictitious eigenvalues from the true ones in these problems. We show that the obtained true complex eigenvalues are related to the behavior of the solution significantly. We also show that the fictitious eigenvalues may affect the accuracy of BIE solutions in standard boundary value problems even when the frequency is real.

A Fast Boundary Integral Equation Method for Conformal Mapping of Multiply Connected Regions

This paper presents a fast boundary integral equation method for approximating the conformal mapping from bounded and unbounded multiply connected regions of connectivity $m+1$ onto the canonical region obtained by removing $m$ rectilinear slits from a strip. The method is based on a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the fast multipole method. The presented method requires $O((m+1)n\ln n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component. The numerical results of some test calculations illustrate that our method has the ability to handle regions with complex geometry and very high connectivity. Besides the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's 39 canonical slit regions.

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

SUMMARYThis article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two‐dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low‐frequency FMM and the high‐frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low‐frequency FMM and the quadrature order for the high‐frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite‐size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.

共有メモリー計算機における3次元時間域動弾性高速境界積分方程式法の並列化について

This paper discusses parallelisation strategies of a time domain fast boundary integral equation method for three dimensional elastodynamics for a large shared memory parallel computer. The performance of the OpenMP parallelised code is examined. Also in the case of MPI-OpenMP hybrid parallelisation, a numerical solution of the scattering probrem with more than one million DOFs is shown. It is concluded that the parallelisation with OpenMP is very effective, and that larger problem can be analysed with MPI-OpenMP hybrid parallelisation.

A fast BIEM for three-dimensional elastodynamics in time domain

A fast boundary integral equation method for three-dimensional (3D) elastodynamics in time domain is proposed. We discuss in detail the plane wave time domain (PWTD) algorithm in 3D elastodynamics, whose complexity is O(N s log 2N s N t ) or O( N s 3/2 N t), depending on the choice of the algorithms for the Legendre transform, where N s and N t are the spatial and temporal degrees of freedom, respectively. An O( N s 3/2 N t) implementation is made, whose efficiency is demonstrated in problems of the size of N s=O(10 4).

K-0311 3次元時間域動弾性問題に対する高速境界積分方程式法(S02-1 境界要素法とその周辺技術(1))(S02 境界要素法とその周辺技術)

In this article, we propose a fast boundary integral equation method (BIEM) for elastodynamic problems in 3D in time domain by extending the formulation for 2D problems developed by present authors recently. In order to obtain a fast solution method, we need to construct a fast algorithm for evaluating the layer potentials in BIE and combine it with an iterative solver for linear algebraic equations. As in 2D problems, we construct the fast algorithm using the plane wave expansions of the single and double layer kernels for elastodynamic problems and the hierarchical system of time-space domain. The fast method obtained here is expected to reduce the computational costs from O(N^2_S N^2_t) required for conventional BIEM to O(N_Slog^2N-S N^2_t) and to solve very large scale initial-boundary value problems where N_S and N_t are the spatial and temporal degrees of freedom, respectively.

Fast Boundary Integral Equation Method for Elastodynamic Problems in 2D in Time Domain.

Fast Boundary Integral Equation Method for Elastodynamic Problems in 2D in Time Domain.

147 2 次元時間域動弾性問題に対する高速境界積分方程式法

147 2 次元時間域動弾性問題に対する高速境界積分方程式法