Fast Multipole Technique Research Articles

Powerful numerical method for analysis of electromagnetic scattering from multiple 3D coated targets buried under rough surface

ABSTRACT In this work, an efficient numerical method is proposed for three-dimensional (3D) electromagnetic scattering from multiple coated targets buried under a two-dimensional dielectric rough surface. In the proposed method, the interior region of each coated target is regarded as an independent subregion. The field within each coated target is modeled by 3D vector finite element method (FEM). The multiple scattering interactions between the buried targets and between targets and rough surface are considered by the electric field integral equation and the Poggio–Miller–Chang–Harrington–Wu integral equations, respectively. The fields inside and outside the targets are coupled together via boundary conditions on the boundary of each target. To improve the calculation efficiency and reduce the memory requirement, the fast multipole technique is used in our method. The presented method is numerically validated by comparing it with method of moments and conventional FEM-boundary integral method (BIM). The results indicate that, compared with traditional FEM-BIM, our method has significant advantages in computational efficiency and memory requirements.

Broadband seismic isolation effect of three-dimensional seismic metamaterials in a semi-infinite foundation: modelled by fast multipole indirect boundary element method

In this study, a fast multipole indirect boundary element method (FM-IBEM) is developed for studying the scattering of seismic waves by 3-D seismic metamaterials (SMs), and a 3-D rock soil composite seismic metamaterial (RSCSM) is proposed. Based on the indirect boundary element method, the fast multipole expansion technique is adopted to reduce the calculation time and memory requirements, and effectively solve the broadband scattering problem of seismic waves by large-scale SMs. The bandgap characteristics of three-dimensional SM are studied in detail, and the effects of the geometric and material parameters on the isolation effect of SMs are investigated. The results show that RSCSM can attenuate the ultralow frequency of a seismic wave, the bandgap width is 0.5–19 Hz, and the isolation effect is 37%. The cell size and material parameters significantly affect the bandgap characteristics of the RSCSM. This type of seismic metamaterial provides a new technique for seismic wave isolation and could be applied in engineering structures.

Solving of multi-connected curvilinear boundary value problems by the fast PIES

This paper presents the fast parametric integral equations system (PIES) in solving two-dimensional (2D) multi-connected curvilinear potential boundary value problems. The fast PIES is a combination of the fast multipole technique with a modified binary tree and original PIES. It was successfully employed for exploring the efficiency of the method applied in modelling and solving single-connected 2D potential problems. The fast PIES allows reducing the time of numerical computations (CPU time), as well as RAM utilization. However, application to the multi-connected problems requires appropriate changes in the fast multipole computations due to the small distance between some segments in internal parts of the boundary. The method is demonstrated with the examples of curvilinear potential problems dealing with perforated plates (plates with many circular holes).

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 algorithm for transient current through open quantum systems

Transient current calculation is essential to study the response time and capture the peak transient current for preventing meltdown of nanochips in nanoelectronics. Its calculation is known to be extremely time consuming with the best scaling $T{N}^{3}$ where $N$ is the dimension of the device and $T$ is the number of time steps. The dynamical response of the system is usually probed by sending a steplike pulse and monitoring its transient behavior. Here, we provide a fast algorithm to study the transient behavior due to the steplike pulse. This algorithm consists of two parts: algorithm I reduces the computational complexity to ${T}^{0}{N}^{3}$ for large systems as long as $T<N$; algorithm II employs the fast multipole technique and achieves scaling ${T}^{0}{N}^{3}$ whenever $T<{N}^{2}$ beyond which it becomes $T{log}_{2}N$ for even longer time. Hence it is of order $O(1)$ if $T<{N}^{2}$. Benchmark calculation has been done on graphene nanoribbons with $N={10}^{4}$ and $T={10}^{8}$. This algorithm allows us to tackle many large scale transient problems including magnetic tunneling junctions and ferroelectric tunneling junctions.

THE FMM-IBEM SOLUTION OF THREE-DIMENSIONAL SCATTERING OF ELASTIC WAVES

Boundary element method has unique advantages to solve elastic wave scattering problems, while the asymmetric and dense matrix equation greatly restrict the application of traditional boundary element in practical engineering. Therefore, based on the single-layer potential theory, combined with the fast multipole expansion technique, a new rapid indirect boundary element method is developed by using the Taylor series expansion of compressible and shear wave potential function for solving large scale elastic wave scattering problems efficiently and accurately. It is verified that this method greatly improves the computing speed and reduces the calculation storage. Then the elastic wave scattering problem for millions of DOFs can be solved effectively and accurately on an ordinary personal workstation. Finally, the scattering problem of P wave by ellipsoidal cavities group in elastic full-space is solved, and some important characteristics for the displacement and stress fields in steady state are discussed based on the numerical results. This method is especially suitable for solving the multiple scattering problems at low dimensionless frequency (ka<5.0).

A Fast-Multipole Unified Technique For The Analysis Of Potential Problems With The Boundary Element Methods

The proposed developments are based on a consistent implementation of the conventional, collocation boundary element method (BEM). A scheme is used to expand a generic (not problem-dependent) fundamental solution about hierarchical levels of source and field poles, which is particularly advantageous to make the technique seamlessly applicable to 2D and 3D problems of elasticity or potential, in terms of different types of curved elements for generally complicated geometry and topology. The proposed compact algorithm is more straightforward to lay out and seems to be more efficient than the ones available in the technical literature – particularly because the outermost loop refers to field nodes and geometry, in what may be called a reverse implementation. Some numerical results are shown for the conventional BEM, with validation and assessment for a few simple, but very large-scale, 2D potential problems with complicated geometry and topology for constant, linear and quadratic elements. Since iterative solvers are not required in this first step of numerical simulations, an isolated assessment of accuracy, computational effort and storage allocation of the proposed fast multipole technique becomes possible.

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.

FAST MULTIPOLE BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL ELECTROMAGNETIC SCATTERING PROBLEM

We developed a fast numerical algorithm for solving the three-dimensional vectorial Helmholtz equation that arises in electromagnetic scattering problems. The algorithm is based on electric field integral equations and is essentially a boundary element method. Nyström's quadrature rule with a triangular grid is employed to linearize the integral equations, which are then solved by using a right-preconditioned iterative method. We apply the fast multipole technique to accelerate the matrix-vector multiplications in the iterations. We demonstrate the broad applications and accuracy of this method with practical examples including dielectric, plasmonic and metallic objects. We then apply the method to investigate the plasmonic properties of a silver torus and a silver split-ring resonator under the incidence of an electromagnetic plane wave. We show the silver torus can be used as a trapping tool to bind small dielectric or metallic particles.

Scattering by Conducting Bodies Coated With Bi-Isotropic Materials

Electromagnetic scattering by arbitrarily shaped conducting bodies coated with general bi-isotropic materials is formulated in terms of the surface integral equation method. In order to facilitate the implementation of the surface equivalence principle, a field decomposition scheme is utilized to split a bi-isotropic media into two equivalent isotropic media. By enforcing the boundary condition on the interfaces of the body, a set of coupled integral equations is finally obtained for the unknown surface currents and then numerically solved using the moment methods combined with the vector triangular basis function. The fast multipole technique has been embedded into the algorithm to accelerate the solution process. The validity of theoretical formulations is verified by numerical results and their comparisons. The calculated results for bi-isotropically coated conducting spheres and oblate spheroids are compared with the exact solution and the existing data, and excellent agreements are observed.

Numerical evaluation of the effective elastic moduli of rocks

In this paper we describe a two-dimensional numerical algorithm to calculate overall elastic properties of rocks. We assume that rocks can be viewed as an assemblage of blocks divided by discontinuity surfaces such as large-scale fractures. For each block we use the concept of representative volume element. We model a representative volume element as a finite body containing multiple defects in the form of inhomogeneities, pores, and microcracks and use simplified assumptions regarding the shapes of the defects. Effective moduli for all blocks can be used as input data for a boundary element (or finite element) computer code to model behavior of rock masses at a different length scale as an assemblage of homogeneous blocks of different shapes and elastic properties. The approach is based on a numerical solution of a hypersingular integral equation for a representative volume element. The use of global approximations and a specially tailored fast multipole technique allows one to efficiently solve large-scale problems and accurately calculate the elastic properties of an equivalent homogeneous material. An additional benefit of the approach is that it allows for an accurate representation of the displacement and stress fields anywhere within the material. A series of computer experiments is conducted to demonstrate the efficiency and accuracy of the algorithm. The possible extension of the approach for three-dimensional problems is discussed.

The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems

AbstractThis paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley &amp; Sons, Ltd.

An adaptive multigrid technique for evaluating long-range forces in biomolecular simulations

We propose an adaptive multigrid method for evaluating the nonlocal, long-range interactions in biomolecular models. The method provides O( h 3) accuracy for the forces, where h is the mesh size of the finest cubic grid, even though the stencil size is minimal. Two new algorithmic components are developed: an adaptive technique for spreading a three-dimensional charge distribution onto a space mesh, and a procedure for performing coarse-to-fine grid interpolation of the potential, force components, and their derivatives (grid-to-particle interpolation at the finest level) in a consistent fashion. The adaptive mesh used in the spreading procedure contains two types of nodes: regular, positioned at nodes cubic grid, and flexible, one per cubic cell. Thus, the charges located in each grid cell are spread to these nine nodes. The method conserves charge, dipole, and quadrupole moments of cells. The spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils. Implementation is also straightforward. Our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique.

A sparse multiresolution technique for fast capacitance computation

A technique for rapidly computing the capacitances of two-dimensional conducting structures is presented in this article. The method relies on a sparse multiresolution representation of the interactions between subsections of the conductors. This decomposition is used to substantially reduce the computational complexity of matrix-vector products required in the iterative solution of the method-of-movements system. It is shown that this technique leads to large CPU-time savings over those required by a standard iterative solver, and preserves high accuracy. Moreover, in contrast to the popular fast-multipole technique, the proposed method is not Green's-function specific; it can be used in conjunction with different static Green's functions representing varied configurations, including perfect electric conductors and stratified dielectric layers. © 1996 John Wiley & Sons, Inc.

A sparse multiresolution technique for fast capacitance computation

A technique for rapidly computing the capacitances of two-dimensional conducting structures is presented in this article. The method relies on a sparse multiresolution representation of the interactions between subsections of the conductors. This decomposition is used to substantially reduce the computational complexity of matrix-vector products required in the iterative solution of the method-of-movements system. It is shown that this technique leads to large CPU-time savings over those required by a standard iterative solver, and preserves high accuracy. Moreover, in contrast to the popular fast-multipole technique, the proposed method is not Green's-function specific; it can be used in conjunction with different static Green's functions representing varied configurations, including perfect electric conductors and stratified dielectric layers. © 1996 John Wiley & Sons, Inc.