Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

This paper describes formulation and implementation of the fast multipole boundary element method （FMBEM） for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems

This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/planesymmetric acoustic wave problems.The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only.Moreover,a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived,and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating,translating and saving the multipole/local expansion coefficients of the image domain.The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems.As for exterior acoustic problems,the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method.Details on the implementation of the present method are described,and numerical examples are given to demonstrate its accuracy and efficiency.

Acceleration of unsteady vortex lattice method via dipole panel fast multipole method

The Unsteady Vortex Lattice Method(UVLM) is a medium-fidelity aerodynamic tool that has been widely used in aeroelasticity and flight dynamics simulations. The most timeconsuming step is the evaluation of the induced velocity. Supposing that the number of bound and wake lattices is N and the computational cost is O (N2), we present an OeNT Dipole Panel Fast Multipole Method(DPFMM) for the rapid evaluation of the induced velocity in UVLM. The multipole expansion coefficients of a quadrilateral dipole panel have been derived in spherical coordinates, whose accuracy is the same as that of the Biot-Savart kernel at the same truncation degree P.Two methods(the loosening method and the shrinking method) are proposed and tested for space partitioning volumetric panels. Compared with FMM for vortex filaments(with three harmonics),DPFMM is approximately two times faster for N2 [103,106]. The simulation time of a multirotor(N~104) is reduced from 100 min(with unaccelerated direct solver) to 2 min(with DPFMM).

A New Directional Algebraic Fast Multipole Method Based Iterative Solver for the Lippmann-Schwinger Equation Accelerated with HODLR Preconditioner

We present a fast iterative solver for scattering problems in 2D,where a penetrable object with compact support is considered.By representing the scattered field as a volume potential in terms of the Green’s function,we arrive at the Lippmann-Schwinger equation in integral form,which is then discretized using an appropriate quadrature technique.The discretized linear system is then solved using an iterative solver accelerated by Directional Algebraic Fast Multipole Method(DAFMM).The DAFMM presented here relies on the directional admissibility condition of the 2D Helmholtz kernel[1],and the construction of low-rank factorizations of the appropriate low-rank matrix sub-blocks is based on our new Nested Cross Approximation(NCA)[2].The advantage of the NCA described in[2]is that the search space of so-called far-field pivots is smaller than that of the existing NCAs[3,4].Another significant contribution of this work is the use of HODLR based direct solver[5]as a preconditioner to further accelerate the iterative solver.In one of our numerical experiments,the iterative solver does not converge without a preconditioner.We show that the HODLR preconditioner is capable of solving problems that the iterative solver can not.Another noteworthy contribution of this article is that we perform a comparative study of the HODLR based fast direct solver,DAFMMbased fast iterative solver,and HODLR preconditioned DAFMM based fast iterative solver for the discretized Lippmann-Schwinger problem.To the best of our knowledge,this work is one of the first to provide a systematic study and comparison of these different solvers for various problem sizes and contrast functions.In the spirit of reproducible computational science,the implementation of the algorithms developed in this article is made available at https://github.com/vaishna77/Lippmann_Schwinger_Solver.

A parallel fast multipole BEM and its applications to large-scale analysis of 3-D fiber-reinforced composites

In this paper, an adaptive boundary element method （BEM） is presented for solving 3-D elasticity problems. The numerical scheme is accelerated by the new version of fast multipole method （FMM） and parallelized on distributed memory architectures. The resulting solver is applied to the study of representative volume element （RVE） for short fiberreinforced composites with complex inclusion geometry. Numerical examples performed on a 32-processor cluster show that the proposed method is both accurate and efficient, and can solve problems of large size that are challenging to existing state-of-the-art domain methods.

Application of A Fast Multipole BIEM for Flow Diffraction from A 3D Body

A Fast Multipole Method (FMM) is developed as a numerical approach to the reduction of the computational cost and requirement memory capacity for a large in solving large-scale problems. In this paper it is applied to the boundary integral equation method (BIEM) for current diffraction from arbitrary 3D bodies. The boundary integral equation is discretized by higher order elements, the FMM is applied to avoid the matrix/vector product, and the resulting algebraic equation is solved by the Generalized Conjugate Residual method (GCR). Numerical examination shows that the FMM is more efficient than the direct evaluation method in computational cost and storage of computers.

A FAST FREE MEMORY METHOD FOR AN EFFICIENT COMPUTATION OF CONVOLUTION KERNELS

We introduce the Fast Free Memory method(FFM),a new implementation of the Fast Multipole Method(FMM)for the evaluation of convolution products.The FFM aims at being easier to implement while maintaining a high level of performance,capable of handling industrially-sized problems.The FFM avoids the implementation of a recursive tree and is a kernel independent algorithm.We give the algorithm and the relevant complexity estimates.The quasi-linear complexity enables the evaluation of convolution products with up to one billion entries.We illustrate numerically the capacities of the FFM by solving Boundary Integral Equations problems featuring dozen of millions of unknowns.Our implementation is made freely available under the GPL 3.0 license within the Gypsilab framework.

FAST MULTIPOLE SINGULAR BOUNDARY METHOD FOR LARGE-SCALE PLANE ELASTICITY PROBLEMS

The singular boundary method （SBM） is a recent meshless boundary collocation method that remedies the perplexing drawback of fictitious boundary in the method of fundamental solutions （MFS）. The basic idea is to use the origin intensity factor to eliminate singularity of the fundamental solution at source. The method has so far been applied successfully to the potential and elasticity problems. However, the SBM solution for large-scale problems has been hindered by the operation count of O（N^3） with direct solvers or O（N^2） with iterative solvers, as well as the memory requirement of O（N^2）. In this study, the first attempt was made to combine the fast multipole method （FMM） and the SBM to significantly reduce CPU time and memory requirement by one degree of magnitude, namely, O（N）. Based on the complex variable represen- tation of fundamental solutions, the FMM-SBM formulations for both displacement and traction were presented. Numerical examples with up to hundreds of thousands of unknowns have successfully been tested on a desktop computer. These results clearly illustrated that the proposed FMM-SBM was very efficient and promising in solving large-scale plane elasticity problems.

Fast Multipole Accelerated Boundary Integral Equation Method for Evaluating the Stress Field Associated with Dislocations in a Finite Medium

In this paper,we develop an efficient numericalmethod based on the boundary integral equation formulation and new version of fast multipole method to solve the boundary value problem for the stress field associated with dislocations in a finite medium.Numerical examples are presented to examine the influence from material boundaries on dislocations.

Fast multipole boundary element analysis of 2D viscoelastic composites with imperfect interfaces

A fast multipole boundary element method(FMBEM)is developed for the analysis of 2D linear viscoelastic composites with imperfect viscoelastic interfaces.The transformed fast multipole formulations are established using the time domain method. To simulate the viscoelastic behavior of imperfect interfaces that are frequently encountered in practice,the Kelvin type model is introduced.The FMBEM is further improved by incorporating naturally the interaction among inclusions as well as eliminating the phenomenon of material penetration.Since all the integrals are evaluated analytically,high accuracy and fast convergence of the numerical scheme are obtained.Several numerical examples,including planar viscoelastic composites with a single inclusion or randomly distributed multi-inclusions are presented.The numerical results are compared with the developed analytical solutions,which illustrates that the proposed FMBEM is very efficient in determining the macroscopic viscoelastic behavior of the particle-reinforced composites with the presence of imperfect interfaces.The laboratory measurements of the mixture creep compliance of asphalt concrete are also compared with the prediction by the developed model.

Fast Multipole BEM for Simulation of 2-D Solids Containing Large Numbers of Cracks

The fast multipole method was used to solve the traction boundary integral equation for 2-D crack analysis. The use of both multipole and local expansions reduces both the computational complexity and the memory requirement to O(N). The multipole expansion uses a complex Taylor series expansion to reduce the number of multipole moments. The generalized minimum residual method solver (GMRES) was selected as the iterative solver. An improved preconditioner for GMRES was developed which uses less CPU time and less memory. A new initial candidate vector for the iterative solver was developed to further improve the efficiency. The numerical examples apply the method to the analysis of cracks in infinite 2-D space with the largest model having 900 000 degrees of freedom.

A Preconditioned 3-DMulti-Region Fast Multipole Solver for Seismic Wave Propagation in Complex Geometries

The analysis of seismic wave propagation and amplification in complex geological structures requires efficient numerical methods.In this article,following up on recent studies devoted to the formulation,implementation and evaluation of 3-D single-and multi-region elastodynamic fast multipole boundary element methods(FM-BEMs),a simple preconditioning strategy is proposed.Its efficiency is demonstrated on both the single-andmulti-region versions using benchmark examples(scattering of plane waves by canyons and basins).Finally,the preconditioned FM-BEM is applied to the scattering of plane seismic waves in an actual configuration(alpine basin of Grenoble,France),for which the high velocity contrast is seen to significantly affect the overall efficiency of the multi-region FM-BEM.

The relation between a microscopic threshold-force model and macroscopic models of adhesion

This paper continues our recent work on the relationship between discrete contact interactions at the microscopic scale and continuum contact interactions at the macroscopic scale (Hulikal et al., J. Mech. Phys. Solids 76, 144-161, 2015). The focus of this work is on adhesion. We show that a collection of a large number of discrete elements governed by a threshold-force based model at the microscopic scale collectively gives rise to continuum fracture mechanics at the macroscopic scale. A key step is the introduction of an efficient numerical method that enables the computation of a large number of discrete contacts. Finally, while this work focuses on scaling laws, the methodology introduced in this paper can also be used to study rough-surface adhesion.

Mathematical and Numerical Aspects of the Adaptive Fast Multipole Poisson-Boltzmann Solver

This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole Poisson-Boltzmann(AFMPB)solver.We introduce and discuss the following components in order:the Poisson-Boltzmann model,boundary integral equation reformulation,surface mesh generation,the nodepatch discretization approach,Krylov iterative methods,the new version of fast multipole methods(FMMs),and a dynamic prioritization technique for scheduling parallel operations.For each component,we also remark on feasible approaches for further improvements in efficiency,accuracy and applicability of the AFMPB solver to largescale long-time molecular dynamics simulations.The potential of the solver is demonstrated with preliminary numerical results.

A Fast Accurate Boundary IntegralMethod for Potentials on Closely Packed Cells

Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties.However,accuracy deteriorates when the cell boundaries are close to each other.We present a boundary integralmethod in two dimensions which is specially designed tomaintain second order accuracy even if boundaries are arbitrarily close.Themethod uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy.For boundaries with many components we use the fast multipolemethod for efficient summation.We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium.We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals.Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region.A number of examples are presented.We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.

Exponential Convergence Theory of the Multipole and Local Expansions for the 3-D Laplace Equation in Layered Media

In this paper,we establish the exponential convergence theory for the multipole and local expansions,shifting and translation operators for the Green's function of 3-dimensional Laplace equation in layered media.An immediate application of the theory is to ensure the exponential convergence of the FMM which has been shown by the numerical results reported in[27].As the Green's function in layered media consists of free space and reaction field components and the theory for the free space components is well known,this paper will focus on the analysis for the reaction components.We first prove that the density functions in the integral representations of the reaction components are analytic and bounded in the right half complex wave number plane.Then,by using the Cagniard-de Hoop transform and contour deformations,estimates for the remainder terms of the truncated expansions are given,and,as a result,the exponential convergence for the expansions and translation operators is proven.

A High-Accurate Fast Poisson Solver Based on Harmonic SurfaceMapping Algorithm

Poisson’s equations in a cuboid are frequently solved in many scientific and engineering applications such as electric structure calculations,molecular dynamics simulations and computational astrophysics.In this paper,a fast and highly accurate algorithm is presented for the solution of the Poisson’s equation in a cuboidal domain with boundary conditions of mixed type.This so-called harmonic surface mapping algorithm is a meshless algorithm which can achieve a desired order of accuracy by evaluating a body convolution of the source and the free-spaceGreen’s function within a sphere containing the cuboid,and another surface integration over the spherical surface.Numerical quadratures are introduced to approximate the integrals,resulting in the solution represented by a summation of point sources in free space,which can be accelerated by means of the fast multipole algorithm.The complexity of the algorithm is linear to the number of quadrature points,and the convergence rate can be arbitrarily high even when the source term is a piecewise continuous function.

A fast algorithm for computing electromagnetic fields of a thin wire current source inside lossy ground

In this paper,a fast algorithm is presented to compute the electromagnetic fields of a thin wire current source inside lossy ground.The modified image method is used to evaluate the Sommerfeld integrals,and the fast multipole method(FMM)is utilized for solving the governing electric field integral equation(EFIE).The validation of the proposed algorithm is performed by comparing the results with that of using method of moment(MoM).The numerical example shows the flexibility,efficiency and accuracy of this algorithm.

A BIE-BASED DtN-FEM FOR FLUID-SOLID INTERACTION PROBLEMS

In this paper, we are concerned with the coupling of finite element methods and bound- ary integral equation methods solving the classical fluid-solid interaction problem in two dimensions. The original transmission problem is reduced to an equivalent nonlocal bound- ary value problem via introducing a Dirichlet-to-Neumann mapping by the direct boundary integral equation method. We show the existence and uniqueness of the solution for the corresponding variational equation. Numerical results based on the finite element method coupled with the standard Galerkin boundary element method, the fast multipole method and the NystrSm method for approximating the DtN mapping are provided to illustrate the efficiency and accuracy of the numerical schemes.

NUMERICAL STUDY OF THE INFLUENCE OF SURFACE ROUGHNESS OF CYLINDER ON FLOW STRUCTURE

In this paper, the influence of surface roughness on flow structure was numerically studied. An adaptive numerical method, the fast vortex method was employed. A mathematical roughness, which comes from the no slip condition of vortex method, was introduced. The numerical results indicate that the roughness has appreciable influence on the flow structure. The vortex shedding could be controlled if the forward multi layer boundary condition is exerted.