The Boundary Element Method with a Fast Multipole Accelerated Integration Technique for 3D Elastostatic Problems with Arbitrary Body Forces

A line integration boundary element method (LIBEM) is proposed for three-dimensional elastostatic problems with body forces. The method is a boundary-only discretization method like the traditional boundary element method (BEM), and the boundary elements created in BEM can be used directly in the proposed method for constructing the integral lines. Finally, the body forces are computed by summing one-dimensional integrals on straight lines. Background cells can be used to cut the lines into sub-lines to compute the integrals more easily and efficiently. To further reduce the computational time of LIBEM, the fast multipole method is applied to accelerate the method for large-scale computations and the details of the fast multipole line integration method for 3D elastostatic problems are given. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

Design of absorbing material distribution for sound barrier using topology optimization

A topology optimization approach based on the boundary element method (BEM) and the optimality criteria (OC) method is proposed for the optimal design of sound absorbing material distribution within sound barrier structures. The acoustical effect of the absorbing material is simplified as the acoustical impedance boundary condition. Based on the solid isotropic material with penalization (SIMP) method, a topology optimization model is established by selecting the densities of absorbing material elements as design variables, volumes of absorbing material as constraints, and the minimization of sound pressure at reference surface as design objective. A smoothed Heaviside-like function is proposed to help the SIMP method to obtain a clear 0–1 distribution. The BEM is applied for acoustic analysis and the sensitivities with respect to design variables are obtained by the direct differentiation method. The Burton–Miller formulation is used to overcome the fictitious eigen-frequency problem for exterior boundary-value problems. A relaxed form of OC is used for solving the optimization problem to find the optimal absorbing material distribution. Numerical tests are provided to illustrate the application of the optimization procedure for 2D sound barriers. Results show that the optimal distribution of the sound absorbing material is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.     

Fast multipole method based solution of electrostatic and magnetostatic field problems

Abstact Applications of boundary element methods (BEM) to the solution of static field problems in electrical engineering are considered in this paper. The choice of a suitable BEM formulation for electrostatics, steady current flow fields or magnetostatics is discussed from user's point of view. The dense BEM matrix is compressed with an enhanced fast multipole method (FMM) which combines well-known BEM techniques with the FMM approach. An adaptive grouping scheme for problem oriented meshes is presented along with a discussion on the influence of the mesh to the efficiency of the FMM. The computational costs of the FMM algorithm are analyzed for typical problems in practice. Finally, some electrostatic and magnetostatic numerical examples demonstrate the simple usability and the efficiency of the FMM. Communicated by: U. Langer     

求解二维随机多区域声波散射问题的快速多极边界元方法

快速多极算法(FMM)是求解大尺度边界元问题的一种很有效的快速算法.应用快速多极算法求解二维随机多区域声散射问题的边界积分方程.首先给出了求解该问题的边界积分方程,进而给出快速多极算法求解的算法实现过程以及积分算子的相应多极展开、局部展开和相应系数的转化关系式.最后通过对数值例子的计算表明快速多极算法在求解随机多区域声散射问题时的可行性及高效性,其求解存储量和计算量都是O(N).     

An investigation of boundary element methods for the exterior acoustic problem

This paper is concerned with the efficient determination of acoustic fields around arbitrary-shaped finite structures in an infinite three-dimensional acoustic medium. The boundary integral formulation due to Burton and Miller is used to overcome the non-existence and non-uniqueness problems associated with classical integral equation formulations of this problem. A class of numerical approximation schemes is developed and the results of applying these schemes to a number of test problems are discussed and compared. The choice of the parameters of the method is critically considered. In particular, the gains to be made, in terms of solution accuracy, by using higher-order approximations to the unknown boundary function, rather than the commonly employed piecewise constant representation, are examined in view of their additional computational costs.     

Boundary element methods for transient convective diffusion. Part I: General formulation and 1D implementation

A general formulation of higher-order boundary element methods (BEM) is presented for time-dependent convective diffusion problems in one- and multi-dimensions. Free-space time-dependent convective diffusion fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Linear, quadratic and quartic time interpolation functions are introduced in this paper for approximate representation of time-dependent boundary temperatures and normal fluxes. Closed form time integration of the kernels is mandatory to attain both accuracy and efficiency of the numerical approach. A complete set of time integrals for the one-dimensional formulation is presented here for the first time in the literature.     

Approximate inverse preconditioning for the fast multipole BEM in acoustics

Abstact Acoustic radiation from vibrating structures is simulated by a Galerkin boundary element method based on the Burton–Miller approach. The boundary element operators are evaluated by the fast multipole method that allows large-scale computations in the medium frequency range. Two iterative solvers are considered: the generalized minimal residual method and a multigrid solver. Both approaches can be accelerated greatly by the presented approximate inverse preconditioner. Communicated by: U. Langer Research of the author is supported by the Deutsche Forschungsgemeinschaft in the framework of the collaborative research centre SFB 404 “Multifield Problems in Solid and Fluid Mechanics”     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

Structural shape optimisation using boundary elements and the biological growth method

A numerical evolutionary procedure for the structural optimisation for stress reduction of two-dimensional structures is presented in this paper. The proposed procedure couples the biological growth method (BGM) with the boundary element method (BEM). The boundary-only intrinsic characteristic of BEM together with its accuracy in the boundary displacement and stress solutions make BEM especially attractive for solving shape-optimisation problems. Two formulations of BEM are used in this work: the standard for two-dimensional elastostatics for the stress analysis and the dual reciprocity method (DRM), which is used to model the swelling or shrinking of the material. Two examples are analysed to illustrate the proposed methodology and to demonstrate its versatility and robustness.     

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.     

CUDA架构下的三维弹性静力学边界元并行计算

针对传统边界元法计算量大、计算效率低的问题,以三维弹性静力学的边界元法为对象,将基于CUDA的GPU并行计算应用到其边界元计算中,提出了基于CUDA架构的GPU并行算法.该算法首先对不同类型的边界元系数积分进行并行性分析,描述了相关的GPU并行算法,然后阐述了边界元方程组的求解方法及其并行策略.实验结果表明,文中算法较传统算法具有显著的加速效果.     

The BEM for numerical solution of partial fractional differential equations

A numerical method is presented for the solution of partial fractional differential equations (FDEs) arising in engineering applications and in general in mathematical physics. The solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape. The method is based on the concept of the analog equation, which in conjunction with the boundary element method (BEM) enables the spatial discretization and converts a partial FDE into a system of coupled ordinary multi-term FDEs. Then this system is solved using the numerical method for the solution of such equations developed recently by Katsikadelis. The method is illustrated by solving second order partial FDEs and its efficiency and accuracy is validated.     

Postbuckling analysis of plates on an elastic foundation by the boundary element method

The postbuckling behavior of plates on an elastic foundation is studied by using the boundary element method (BEM). A new fundamental solution of lateral deflection is derived through the resolution theory of a differential operator, and a set of boundary element formulae in incremental form is presented. By using these formulae, the BEM solution procedure becomes relatively simple. The results of a number of numerical examples are compared with existing solutions and good agreement is observed. It shows that the proposed method is effective for solving the postbuckling problems of plates with arbitrary shape and various boundary conditions.     

The use of continuous boundary elements in the boundary elements method for domains with non-smooth boundaries via finite difference approach

A numerical method is presented in this article to deal with the drawback of boundary elements method (BEM) at corner points. The use of continuous elements instead of the discontinuous ones has been recommended in the BEM literature widely because of the simplicity and accuracy. However the continuous elements lead to certain difficulties for problems where their domains contain corners. In this paper the finite difference method (FDM) has been applied to obtain some constraints for boundary points near the corners to deal with this drawback. Because of its simplicity and capability, the new scheme is applicable on BEM problems for all geometries, all governing equations and general boundary conditions, easily. Since the Dirichlet boundary condition is more critical than the other ones, we will focus on it in the numerical implementation. The numerical results show that the new treatment improves the accuracy of BEM significantly.     

舰船水下声场边界元法建模与计算机模拟

根据舰船声场的特性,将舰船水下声场简化为一特殊的半自由空间声辐射问题,利用边界元法(BEM)建立了舰船水下声场的数值计算模型。通过对声场边界条件的计算机仿真和计算,获得了部分计算结果。结果表明:由所建模型计算的舰船辐射声场与真实海域中的实测规律基本吻合,该文提出的方法是有效可行的。最后对存在的问题进行了讨论。     

多维多极值函数优化的和声退火算法

针对多极值实函数优化问题,本文结合和声搜索与模拟退火算法,提出了一种新的搜索算法,即和声退火算法。新算法保留了和声搜索的搜索机理,但对和声搜索中于和声记忆库外的搜索方法用超快速模拟退火算法作了改进,对和声记忆库内新解产生方法也作了相应的调整,从而提高了对多维问题的搜索效率。数值实验结果表明算法对和声搜索有明显的改进,收敛速度更快,跳出局部极值点的能力较强。新算法在解决多维多极值优化问题方面比遗传算法更具效率,值得进一步研究与推广应用。     

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty

The difficulty of solving the min–max optimal control problems (M-MOCPs) with uncertainty using generalised Euler–Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler–Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.     

Modelling and effective modification of smooth boundary geometry in boundary problems using B-spline curves

To create curves in computer graphics, we use, among others, B-splines since they make it possible to effectively produce curves in a continuous way using a small number of de Boor’s control points. The properties of these curves have also been used to define and create boundary geometry in boundary problems solving using parametric integral equations system (PIES). PIES was applied for resolution 2D boundary-value problems described by Laplace’s equation. In this PIES, boundary geometry is theoretically defined in its mathematical formalism, hence the numerical solution of the PIES requires no boundary discretization (such as in BEM) and is simply reduced to the approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained were compared with both exact and numerical solutions.     

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.     

针对声屏障的轨道交通的降噪研究

该文针对目前轨道交通中普遍使用的声屏障的降噪方案,提出了在轻轨轨道间增加双面吸声声屏障的新方案,以此来达到进一步降噪的目的。运用边界元方法,建立轨道声屏障降噪的边界元分析模型,利用边界元仿真软件SYSNOISE得到仿真结果,在此基础上预测声屏障的降噪效果。并研究分析了声屏障高度对降噪性能的影响,通过对比不同高度的降噪性能,得到最佳声屏障高度。最终通过对两种方案的降噪结果的对比,说明新的方案具有更好的降噪效果。