声学Burton-Miller方程边界元法GPU并行计算

基于GPU,对声学Burton-Miller积分方程的边界元解法进行并行计算.提出并行计算格式和程序实现方法,以及Burton-Miller方程中各类奇异(包括强奇异、超奇异)积分的GPU计算和局部修正方法.典型算例结果表明,在特征频率处可获得正确的解,具有较高精度,可在普通个人计算机上快速完成自由度超过2×10⁵的声学边界元分析.为计算声学及相关工程领域的中、大规模声场分析问题提供一种快速、高效、简便的数值计算工具.     

二维Helmholtz边界超奇异积分方程解析研究

基于常规边界元法及超奇异边界积分方程复线性耦合的Burton-Miller方法应用于无限域声学问题的最大难点在于处理超奇异积分（二维问题）.目前,此类超奇异积分主要使用各种弱奇异/正则化方法求解,而这些弱奇异/正则化方法具有时间消耗大等弱点.基于围道积分定理,本文给出一种使用常值单元的二维Helmholtz边界超奇异积分的解析表达式.在有限部分积分意义下,所有的奇异和超奇异积分可以解析表达.数值算例表明该解析表达式是有效的.     

Helmholtz边界积分方程的多频计算

提出了利用无穷级数展开的方法,将波数从Helmholtz边界积分方程的特解中分离出来,使随波数变化的系统矩阵变为波数的矩阵级数形式,同时证明了级数截断时的收敛性。数值结果表明,结合CHIEF方法,用级数展开的方法不仅能有效地克服频域内非唯一现象,节省计算时间;而且当频率较高时,在单元粗剖分下也能得到满意的结果。     

热应力边界元法中几乎超奇异积分的计算

通过分部积分变换将热弹性力学应力边界积分方程中的超奇异积分转化为强奇异积分,然后与另一个强奇异积分求和,得到仅含几乎强奇异的热应力自然边界积分方程.再对其中的几乎强奇异积分施以正则化,消除了热弹性力学边界元法中的几乎奇异积分,可以准确计算出热弹性力学问题中近边界内点的热应力.算例证明了方法的有效性.     

边界元奇异与近奇异数值积分方法及其应用于大规模声学问题

提出了综合处理Burton-Miller方法所导致的奇异积分与近奇异积分问题的数值求积方法,以此改进了基于常量元素的常规边界元和低频快速多极边界元方法。对于奇异积分问题,利用Hadamard有限积分方法进行解决;对于近奇异积分问题,则采用极坐标变换法和PART方法(Projection and Angular&;Radial Transformation)进行克服。与解析解和LMS Virtual.Lab商业软件的结果比较验证了方法的正确性,并对比分析了奇异积分与近奇异积分对计算精度的影响。采用低频快速多极子方法以加速常规边界元法的计算效率,计算分析了计算复杂度,并成功实现了34万自由度大规模问题的计算。结果表明,近奇异积分问题主要由超奇异核函数引起,对计算精度的影响不容忽略;快速多极边界元法的精度与常规边界元法一致,但计算复杂度要远低于后者。      

动力边界元法中的强奇异积分问题

本文讨论了动力边界元法中的奇异积分问题,对其中的强奇异积分提出了一个有效的计算方法.该方法从合非零初始态的边界积分方程出发,利用动力方程的特解间接地确定了主系数(即所谓强奇异积分),从而避免了直接计算强奇异积分的困难.根据该方法编制了计算程序,并给出了一个简单算例。     

三维静电场线性插值边界元中的解析积分方法

提出求解三维静电场的三角形线性插值边界元解析积分方法.针对含1/R和1/R²的积分项,将单元形状函数分解为常数项、含x的线性项和含y的线性项,从而将边界单元积分简化为6个基本积分组合,并导出其解析计算公式,避免了因形状函数改变而导致的重复计算.该方法不仅可以准确计算远离奇异情况下的边界元积分,而且可以准确计算一阶和二阶接近奇异积分以及一阶奇异积分.计算结果表明,在接近奇异积分和奇异积分比较突出的问题中,当数值积分方法不能给出正确结果时,用同样的边界元网格,解析积分方法可以给出正确的结果,提高了三维静电场线性插值边界元法的计算精度.     

基于Burton-Miller方程的轴对称结构声学边界元方法

提出了一种求取轴对称结构任意边界条件下声辐射特性的边界元方法。采用Burton和Miller改进型公式将高阶奇异项转化为弱奇异项之和,保证声辐射参数的唯一性,且计算简单精确。将结构表面声压与振速按照旋转轴角度进行Fourier级数展开,利用级数的正交性建立各项待定系数的求解公式;然后转化格林函数的法向偏导为切向偏导,方便直接计算各项积分,并将面积分公式表示为沿结构边界的线积分和沿旋转角度的积分;进一步采用二次等参单元离散结构边界线,建立声压与振速的关系矩阵,从而确定结构声辐射参数。以脉动球源和横向振动球源为例计算,与解析解和传统边界元法结果作对比,说明该方法的有效精确性。     

解弹性力学第二类边界积分方程的求积法与分裂外推

利用Sidi奇异求积公式,提出了解曲边多角形域上线性弹性力学第二类边界积分方程的求积法,即离散矩阵的每个元素的生成只需赋值不需计算任何奇异积分.通过估计离散矩阵的特征值和利用Anselone聚紧收敛理论,证明了近似解的收敛性;同时得到了误差的多参数渐近展开式;通过并行地解粗网格上的离散方程,利用分裂外推获得了高精度近似解和后验误差.     

子波谱方法在三维声辐射和声散射中的应用

将边界变量用二维子波展开,获得了三维任意边界条件声辐射和声散射的边界积分方程的子波谱方法.采用以子波为权函数的Gauss积分法计算子波谱方法的系数,获得了与传统边界元法相同的计算量,克服了普通积分法计算子波系数计算量大甚至难以收敛的缺点;采用Duffy的方法解决了子波谱方法中的奇异积分,使其能够用普通的Gauss积分法计算.算例表明:子波谱方法系数矩阵压缩率超过50%以后,计算精度仍然高于传统边界元方法.     

A method for multi-frequency calculation of boundary integral equation in acoustics based on series expansion

The paper presents a method to solve the problem of multi-frequency calculation of Helmholtz boundary integral equation in acoustics. Based on series expansion, system matrices are independent of wavenumber and become the matrix power series of wavenumber. As a result, all matrices in the matrix power series are only dependent on the structure geometry. In addition, an element transform method to calculate the singular integral and Cauchy singular integral is also discussed because the singular integral need to be solved using the method. The convergence of the series expansion method is also proved in this paper. The effectiveness of the method is confirmed by two numerical examples.     

Numerical quadrature for singular and near-singular integrals of boundary element method and its applications in large-scale acoustic problems

The numerical quadrature methods for dealing with the problems of singular and near-singular integrals caused by Burton-Miller method are proposed,by which the conventional and fast multipole BEMs(boundary element methods) for 3D acoustic problems based on constant elements are improved.To solve the problem of singular integrals,a Hadamard finite-part integral method is presented,which is a simplified combination of the methods proposed by Kirkup and Wolf.The problem of near-singular integrals is overcome by the simple method of polar transformation and the more complex method of PART(Projection and Angular Radial Transformation).The effectiveness of these methods for solving the singular and near-singular problems is validated through comparing with the results computed by the analytical method and/or the commercial software LMS Virtual.Lab.In addition,the influence of the near-singular integral problem on the computational precisions is analyzed by computing the errors relative to the exact solution.The computational complexities of the conventional and fast multipole BEM are analyzed and compared through numerical computations.A large-scale acoustic scattering problem,whose degree of freedoms is about 340,000,is implemented successfully.The results show that,the near singularity is primarily introduced by the hyper-singular kernel,and has great influences on the precision of the solution.The precision of fast multipole BEM is the same as conventional BEM,but the computational complexities are much lower.     

基于等几何分析的边界元法求解Helmholtz问题

将基于一类局部双变量B样条函数的等几何分析方法和Burton-Miller方法相结合,分析三维Helmholtz问题.对于某些从二维参数域映射到三维空间具有奇异点的参数曲面,该方法可以有效地避免奇异点处大量奇异与近奇异积分的计算.数值算例表明该方法具有较好的计算精度和计算效率.复杂问题的分析表明,该方法具有良好的工程应用前景.     

大规模声学边界元法的GPU并行计算

提出一种大规模声学边界元法的高效率、高精度GPU并行计算方法.基于Burton-Miller边界积分方程,推导适于GPU的并行计算格式并实现了传统边界元法的GPU加速算法.为提高原型算法的效率,研究GPU数据缓存优化方法.由于GPU的双精度浮点运算能力较低,为了降低数值误差,研究基于单精度浮点运算实现的doublesingle精度算法.数值算例表明,改进的算法实现了最高89.8%的GPU使用效率,且数值精度与直接使用双精度数相当,而计算时间仅为其1/28,显存消耗也仅为其一半.该方法可在普通PC机(8GB内存,NVIDIA Ge Force 660 Ti显卡)上快速完成自由度超过300万的大规模声学边界元分析,计算速度和内存消耗均优于快速边界元法.     

Use of nonsingular boundary integral formulation for reducing errors due to near-field measurements in the boundary element method based near-field acoustic holography

In the conformal near-field acoustic holography (NAH) using the boundary element method (BEM), the transfer matrix relating the vibro-acoustic properties of source and field depends solely on the geometrical condition of the problem. This kind of NAH is known to be very powerful in dealing with the sources having irregular shaped boundaries. When the vibro-acoustic source field is reconstructed by using this conformal NAH, one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components. The conventional acoustic BEM based on the Kirchhoff-Helmholtz integral equation has the singularity problem in the close near field of the source surface. This problem stems from the singular kernel of the Green function of the boundary integral equation (BIE) and the singularity can influence the reconstruction accuracy greatly. In this paper, the nonsingular BIE is introduced to the NAH calculation and the holographic BIE is reformulated. The effectiveness of nonsingular BEM has been investigated for the reduction of reconstruction error. Through interior and exterior examples, it is shown that the resolution of predicted field pressure could be improved in the close near field by employing the nonsingular BIE. Because the BEM-based NAH inevitably requires the field pressure measured in the close proximity to the source surface, the present approach is recommended for improving the resolution of the reconstructed source field.