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

Finite Element Method (FEM1) is pervasively used in most of 3D product design analysis, in which Computer Aided Design (CAD) models need to be converted in to mesh models first and then enriched with some material features and boundary conditions data, etc. The interaction between CAD models and FEM models is intensive. Boundary Element Method (BEM) has been expected to be advantageous in large-scale problems in recent years owing to its reduction of the dimensionality and its reduced complexity in mesh generation. However, the BEM application has so far been limited to relatively small problems due to the memory and computational complexity for matrix buildup are O(N2). The fast multipole BEM (FMBEM) combined with BEM and fast multipole method (FMM) can overcome the defect of the traditional BEM, and provides an effective method to solve the large-scale problem. Combining GPU parallel computing with FMBEM can further improve its efficiency significantly. Based on the three-dimensional elastic mechanics problems, the parallelisms of the multipole moment (ME), multipole moment to multipole moment (M2M) translation, multipole moment to local expansion (M2L) translation, local expansion to local expansion (L2L) translation and near-field direct calculation were analyzed respectively according to the characteristics of the FMM, and the parallel strategies under CUDA were presented in this paper. Three main major parts are included herein: (1) FMBEM theory in 3D elastostatics, (2) the parallel FMBEM algorithm using CUDA, and (3) comparison the GPU parallel FMBEM with BEM, FEM and FMBEM respectively by engineering examples. Numerical example results show the 3D elastostatics GPU FMBEM not only can speed up the boundary element calculation process, but also save memory which can be effective to solve the large-scale engineering problems.

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

It is suitable to solve the acoustic problems by using the fast multipole boundary element method (FMBEM), since the FMBEM can accelerate the matrix-vector multiplication dramatically by reducing the CPU time and memory of conventional boundary element method to O(N log2N) and O(N) respectively. We propose a 3D acoustic FMBEM based on Burton-Miller formulation in this paper. A new adaptive algorithm is applied to the diagonal form FMBEM, and a new proposed analytical moment formulation is used in the moment computation. Both of them further improve the efficiency of FMBEM. Acoustic scattering of soft sphere at resonant frequency is investigated to validate the accuracy of solution using Burton-Miller formulation. Comparisons of solution to the multi-radiating spheres problem with the one solved by Bapat's program demonstrate the accuracy and the efficiency of our algorithm in solving large-scale acoustic problems. In the end, we use our algorithm to analyze the inner sound filed of a car and dolphin acoustic scattering.

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

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

A fast multilevel multipole algorithm (FMM) is derived for the Helmholtz equation and adopted to the symmetric Galerkin boundary element method (BEM) for acoustics.The FMM allows to evaluate a matrix-vector product of the BEM with a computational cost of O(N log 2 N ), thus leading to a significant reduction of computation time and memory requirements compared to standard BEM formulations.This allows the simulation of large scale acoustic models.A coupling algorithm based on Lagrange multipliers is proposed for the simulation of structure-acoustic field interaction.Finite plate elements are coupled to a Galerkin boundary element formulation of the acoustic domain.The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non-matching grids.The resulting saddle point problem is solved by an approximate Uzawa-type scheme in which the matrix-vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method.The algorithm is demonstrated on the example of a cavity backed elastic panel.

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

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 ☆

This paper describes an application of the fast multipole boundary element method (FMBEM) in simulating the viscoelastic behavior of asphalt concrete (AC). FMBEM has some merits including easy meshing of complicated geometries, accuracy in solving singular fields, high calculation efficiency, and low data storage requirements. These merits lead to many applications in multiple time-step and multi-inclusion problems, such as prediction of AC creep behavior. In this paper, a fast multipole formulation and an algorithm for two-dimensional linear viscoelastic problems are introduced first. To consider the bonding between the asphalt matrix and aggregates, a viscoelastic interface model is used to simulate the interfacial imperfection. A numerical model of asphalt concrete is then constructed based on image processing techniques, and the model is capable of taking into account the effects of the size and content of the air voids. The creep compliance of AC is predicted by the developed FMBEM arithmetic and subsequently compared with the experimental results. It has been shown that the developed method can describe and simulate the correct creep evolutionary trend of AC. DOI: 10.1061/(ASCE)MT.1943- 5533.0000603. © 2013 American Society of Civil Engineers. CE Database subject headings: Asphalts; Concrete; Micromechanics; Viscoelasticity; Imaging techniques; Boundary element method. Author keywords: Asphalt concrete; Micromechanics; Fast multipole boundary element method; Viscoelastic properties; Image processing technique.

Faster energy convergence rate and more accurate calculation of the stress intensity factor KI have been reported, using the mixed finite element method, by Mirza and Olson (1978). However, the three node linear displacement and linear stress mixed element has five degrees of freedom (u, v, τxx τyy and τxy) per node and leads to a large half-bandwidth and consequently a very large computer storage requirement. Because of the superior performance of the mixed finite elements in linear elastic, stress singular problems it is desirable to use such elements near the crack tip. At the same time, away from the crack tip where stresses are distributed in a more regular manner, the conventional displacement finite elements may be employed. This leads to a transitional zone where the application of the Hellinger-Reissner mixed variational principle near the crack tip switches over to application of the potential energy theorem away from the crack tip. A transitional mixed finite element for connecting the linear displacement-linear stress (three node) mixed elements with the eight-node, displacement isoparametric elements have been developed by Leung (1982). This three element scheme leads to a much more efficient method computationally and yet maintains the energy convergence rate of 0(N−4) for both, the mixed and the eight node, isoparametric displacement elements. The energy release rate, using the equivalent of stiffness derivative technique by Park (1974), is used to calculate the stress intensity factor K•. Numerical calculations were found to be in very good agreement with the available results in the literature.

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.

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.

The surface electric field analysis of the converter valve shield system is a large-scale electrostatic field problem, which is difficult to analyse. The fast multipole boundary element method (FMBEM), which is suitable for solving large-scale problems, can accelerate the computation speed and conserve memory. However, the coefficient matrix implicitly formed by using the FMBEM is sometimes ill-conditioned, especially for large-scale problems; thus, the convergence of iteration is poor. In this paper, a fast solver is proposed to improve efficiency. First, an adaptive GMRES( m ) with variant restart parameter is adjusted for the Galerkin FMBEM. In addition, the sparse approximate inverse preconditioner is improved, and a new sparsity pattern is proposed for the multiscale problem derived from the converter valve shield system. The numerical results show that the accuracy can meet the engineering requirements compared with the finite element method. Compared with other solvers and preconditioners, the algorithm can achieve a satisfactory convergence rate and reduce the computation time. In addition, a single bridge shield system of ±160 kV converter valve is successfully analysed using the proposed method.

We propose a fast implementation of the boundary element method for solving the Poisson equation, which approximately determines the electrostatic field around solvated molecules of arbitrary shape. The method presented uses computational resources of order O(N) only, where N is the number of elements representing the dielectric boundary at the molecular surface. The method is based on the Fast Multipole Algorithm by Rokhlin and Greengard, which is used to calculate the Coulombic interaction between surface elements in linear time. We calculate the solvation energies of a sphere, a small polar molecule, and a moderately sized protein. The values obtained by the boundary element method agree well with results from finite difference calculations and show a higher degree of consistency due to the absence of grid dependencies. The boundary element method can be taken to a much higher accuracy than is possible with finite difference methods and can therefore be used to verify their validity. © 1995 by John Wiley & Sons, Inc.

An efficient fluid-structure interaction (FSI) method is established to quickly and accurately calculate the fluid load of three-dimensional (3D) elastic structure in the process of water slamming. In this method, the boundary element method (BEM) and computational structural dynamics (CSD) are coupled by thin plate spline (TPS) interpolation method and flow field reconstruction. In order to improve the calculation efficiency, the fast multipole algorithm is introduced into the BEM, the finite element mesh of the wetted part of the structure is directly used as the flow field boundary, and the structural dynamic response is analyzed by the direct integration method of state equation based on modal superposition. The numerical results are compared with the results of Nisewanger’s sphere water entry test and the existing literature, which proves the accuracy of the numerical calculation method established in this paper. The effects of time-domain iteration step size, mesh size, slamming velocity, fluid density and structural elastic effect on water slamming load are studied. The results show that the slamming velocity and fluid density have a great influence on the simulation results; the iteration step size has little influence on the accuracy of simulation calculation, and the mesh size has a certain influence on the accuracy of simulation calculation; elastic effect reduces the peak value of slamming load, considering elastic effect is necessary for accurate calculation of slamming load.