Applications of a fast multipole Galerkin in boundary element method in linear elastostatics

Abstact Boundary element methods provide a powerful tool for solving boundary value problems of linear elastostatics, especially in complicated three–dimensional structures. In contrast to the standard Galerkin approach leading to dense stiffness matrices, in fast boundary element methods such as the fast multipole method the application of matrix–vector products can be realized with almost linear complexity. Since all boundary integral operators of linear elastostatics can be reduced to those of the Laplacian, the discretization of the corresponding single and double layer potentials of the Laplace operator has to be employed only. This technique results in a fast multipole method which is an efficient tool for the simulation of elastic stress fields in engineering and industrial applications. This work has been supported by the German Research Foundation DFG under the Grant SFB 404 Multifield Problems in Continuum Mechanics. Dedicated to George C. Hsiao on the occasion of his 70th birthday.     

Graphics processing unit (GPU) accelerated fast multipole BEM with level-skip M2L for 3D elasticity problems

In order to accelerate fast multipole boundary element method (FMBEM), in terms of the intrinsic parallelism of boundary elements and the FMBEM tree structure, a series of CUDA based GPU parallel algorithms for different parts of FMBEM with level-skip M2L for 3D elasticity are presented. A rigid body motion method (RBMM) for the FMBEM is proposed based on special displacement boundary conditions to deal with strongly singular integration and free term coefficients. The numerical example results show that our parallel algorithms obviously accelerates the FMBEM and can be used in large scale engineering problems with wide applications in the future.     

基于快速多极边界元法的沥青混凝土弹性模量预测

由于对沥青混凝土材料的研究无论是试验法还是经验法均建立在宏观层面上,无法与其细观结构建立本质的联系,因此利用快速多极边界元法(Fast Multipole Boundary Element Method,FMBEM),结合数字图像处理技术,实现沥青混凝土二维几何建模及弹性模量预测.通过数字图像处理技术识别拍摄得到的原始...     

Petascale molecular dynamics simulation using the fast multipole method on K computer

In this paper, we report all-atom simulations of molecular crowding — a result from the full node simulation on the “K computer”, which is a 10-PFLOPS supercomputer in Japan. The capability of this machine enables us to perform simulation of crowded cellular environments, which are more realistic compared to conventional MD simulations where proteins are simulated in isolation. Living cells are “crowded” because macromolecules comprise ∼30% of their molecular weight. Recently, the effects of crowded cellular environments on protein stability have been revealed through in-cell NMR spectroscopy. To measure the performance of the “K computer”, we performed all-atom classical molecular dynamics simulations of two systems: target proteins in a solvent, and target proteins in an environment of molecular crowders that mimic the conditions of a living cell. Using the full system, we achieved 4.4 PFLOPS during a 520 million-atom simulation with cutoff of 28 Å. Furthermore, we discuss the performance and scaling of fast multipole methods for molecular dynamics simulations on the “K computer”, as well as comparisons with Ewald summation methods.     

Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

In this paper, we consider the identification of a corrosion boundary for the two-dimensional Laplace equation. A boundary collocation method is proposed for determining the unknown portion of the boundary from the Cauchy data on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter is provided by the generalized cross-validation criterion. Numerical examples show that the proposed method is reasonable and feasible.     

Biomolecular electrostatics using a fast multipole BEM on up to 512 gpus and a billion unknowns

We present teraflop-scale calculations of biomolecular electrostatics enabled by the combination of algorithmic and hardware acceleration. The algorithmic acceleration is achieved with the fast multipole method (fmm) in conjunction with a boundary element method (bem) formulation of the continuum electrostatic model, as well as the bibee approximation to bem. The hardware acceleration is achieved through graphics processors, gpus. We demonstrate the power of our algorithms and software for the calculation of the electrostatic interactions between biological molecules in solution. The applications demonstrated include the electrostatics of protein–drug binding and several multi-million atom systems consisting of hundreds to thousands of copies of lysozyme molecules. The parallel scalability of the software was studied in a cluster at the Nagasaki Advanced Computing Center, using 128 nodes, each with 4 gpus. Delicate tuning has resulted in strong scaling with parallel efficiency of 0.8 for 256 and 0.5 for 512 gpus. The largest application run, with over 20 million atoms and one billion unknowns, required only one minute on 512 gpus. We are currently adapting our bem software to solve the linearized Poisson–Boltzmann equation for dilute ionic solutions, and it is also designed to be flexible enough to be extended for a variety of integral equation problems, ranging from Poisson problems to Helmholtz problems in electromagnetics and acoustics to high Reynolds number flow.     

A fast quasi-multiple medium method for 3-D bem calculation of parasitic capacitance

A quasi-multiple medium (QMM) method is proposed to accelerate the boundary element method (BEM) for the 3-D parasitic capacitance calculation. In the QMM method, a homogeneous dielectric is decomposed into a number of fictitious medium blocks, each with the same permittivity of original medium. By the localization character of BEM, the QMM method makes great sparsity to the coefficient matrix of the overall discretized BEM equations. Then, using storing technique of sparse matrix and iterative equation solvers, the sparsity is explored to greatly reduce CPU time and memory usage of BEM computation. The computational complexity of the QMM accelerated BEM for a single-medium model problem is analyzed, and it is concluded as O(N), if the number of iterations is bounded. Numerical results verify the theoretical analysis and show the accelerating efficiency of the QMM method for calculation of 3-D parasitic capacitance.     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.     

一种基于GPU 加速细粒度并行遗传算法的实现方法

为改善遗传算法对大规模多变量求解的性能,提出一种基于图形处理器(GPU)加速细粒度并行遗传算法的实现方法.将并行遗传算法求解过程转化为GPU纹理渲染过程,使得遗传算法在GPU中加速执行.实验结果表明,该算法抑制了早熟现象,增大了并行遗传算法的种群规模,提高了算法的运算速度,并为普通用户研究并行遗传算法提供了一种可行的方法.     

Acceleration of boundary element method by explicit vectorization

Although parallelization of computationally intensive algorithms has become a standard with the scientific community, the possibility of in-core vectorization is often overlooked. With the development of modern HPC architectures, however, neglecting such programming techniques may lead to inefficient code hardly utilizing the theoretical performance of nowadays CPUs. The presented paper reports on explicit vectorization for quadratures stemming from the Galerkin formulation of boundary integral equations in 3D. To deal with the singular integral kernels, two common approaches including the semi-analytic and fully numerical schemes are used. We exploit modern SIMD (Single Instruction Multiple Data) instruction sets to speed up the assembly of system matrices based on both of these regularization techniques. The efficiency of the code is further increased by standard shared-memory parallelization techniques and is demonstrated on a set of numerical experiments.     

Subdomain solutions of complex variable boundary element method

In this paper, a new subdomain solution of the boundary element method based on complex variable fundamental solutions for non-homogeneous materials is developed. Being different from the conventional BEM, subdomains in the method presented can be produced by considering not only the properties of materials, but also the geometry and correspondent boundary conditions of the problem. The formulation may be combined with other complex variable fundamental solutions to provide higher accuracy and better efficiency. The coupling formulations are given in matrix form, and the numerical procedure is described. the advantages and high efficiency of the present method are demonstrated by two numerical examples.     

Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems

In this paper the dual reciprocity boundary element method in the Laplace domain for anisotropic dynamic fracture mechanic problems is presented. Crack problems are analyzed using the subregion technique. The dynamic stress intensity factors are computed using traction singular quarter-point elements positioned at the tip of the crack. Numerical inversion from the Laplace domain to the time domain is achieved by the Durbin method. Numerical examples of dynamic stress intensity factor evaluation are considered for symmetric and non-symmetric problems. The influence of the number of Laplace parameters and internal points in the solution is investigated.     

An efficient method for solving elliptic boundary element problems with application to the tokamak vacuum problem

A method for regularizing ill-posed Neumann Poisson-type problems based on applying operator transformations is presented. This method can be implemented in the context of the finite element method to compute the solution to inhomogeneous Neumann boundary conditions; it allows to overcome cases where the Neumann problem otherwise admits an infinite number of solutions. As a test application, we solve the Grad–Shafranov boundary problem in a toroidally symmetric geometry. Solving the regularized Neumann response problem is found to be several orders of magnitudes more efficient than solving the Dirichlet problem, which makes the approach competitive with the boundary element method without the need to derive a Green function. In the context of the boundary element method, the operator transformation technique can also be applied to obtain the response of the Grad–Shafranov equation from the toroidal Laplace n=1 response matrix using a simple matrix transformation.     

Worst case bounds on the point-wise discretization error in boundary element method for the elasticity problem

In this work, point-wise discretization error is bounded via interval approach for the elasticity problem using interval boundary element formulation. The formulation allows for computation of the worst case bounds on the boundary values for the elasticity problem. From these bounds the worst case bounds on the true solution at any point in the domain of the system can be computed. Examples are presented to demonstrate the effectiveness of the treatment of local discretization error in elasticity problem via interval methods.     

A fast singular boundary method for 3D Helmholtz equation

This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of $O\left(N^3\right)$ with direct solvers or $O\left(N^2\right)$ with iterative solvers, as well as the memory requirement of $O\left(N^2\right)$. To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from $O\left(N^2\right)$ to $O\left(NlogN\right)$ or $O\left(N\right)$. Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy.     

Implementation of a boundary element method on distributed memory computers

In this paper, we analyse and compare different parallel implementations of the Boundary Element Method on distributed memory computers. We deal with the computation of two-dimensional magnetostatic problems. The resulting linear system will be solved using Householder transformation and Gaussian elimination. Experimental results are obtained on a Meiko Computing Surface with 32 T800 transputers.     

A complex variable boundary element method for elliptic partial differential equations in a multiple-connected region

A simple boundary element method based on the Cauchy integral formulae is proposed for the numerical solution of a class of boundary value problems involving a system of elliptic partial differential equations in a multiple-connected region of infinite extent. It can be easily and efficiently implemented on the computer.     

Application of boundary-domain element method to the free vibration problem of plate structures

The boundary-domain element method is applied to the free vibration problem of thin-walled plate structures. The static fundamental solutions are used for the derivation of the integral equations for both in-plane and out-of-plane motions. All the integral equations to be implemented are regularized up to an integrable order and then discretized by means of the boundary-domain element method. The entire system of equations for the plate structures composed of thin elastic plates is obtained by assembling the equations for each plate component satisfying the equilibrium and compatibility conditions on the connected edge as well as the boundary conditions. The algebraic eigenvalue equation is derived from this system of equations and is able to be solved by using the standard solver to obtain eigenfrequencies and eigenmodes. Numerical analysis is carried out for a few example problems and the computational aspects are discussed.