三维弹性问题Taylor展开多极边界元法的误差分析

Taylor展开多极边界元法有效的提高了边界元法的求解效率,使之可用于大规模问题的计算。然而,由于计算中对基本解进行了Taylor级数展开,与传统边界元方法相比计算精度有所下降。本文主要针对三维弹性问题Taylor展开多极边界元法的计算精度和误差进行研究。文中对两种方法的计算精度进行了比较;研究了核函数的Taylor展开性质;推导了三维弹性问题基本解的误差估计公式;给出了Taylor展开多极边界元法中远近场的划分原则。通过具体的算例,证明了该方法的正确性和误差估计公式的有效性,说明了影响Taylor展开多极边界元法求解精度的因素。     

快速多极虚边界元法对含圆孔薄板有效弹性模量的模拟分析

针对虚边界元法,引入快速多极展开和广义极小残值法(GMRES)的思想,以形成快速多极虚边界元法的求解思想,并将此方法用于含圆孔薄板有效弹性模量的模拟分析.由于本文方法采用了"源点"多极展开和"场点"局部展开的组合处理方案,从而使得原问题方程组求解的计算耗时量和储存量降至与所求问题的计算自由度数成线性比例.本文工作的研究目的在于:提高虚边界元法在普通台式机上的运算能力和拓宽虚边界元法对大规模复杂问题的求解(或数值模拟).文中给出了均布圆孔的正方形薄板和之字形分布圆孔薄板二个算例,以验证该方法的可行性,计算精度和计算效率.     

二维弹性新型快速多极虚边界元的最小二乘法

在虚边界元最小二乘法的方程求解中采用新型的快速多极展开和广义极小残值法,提出了一种二维弹性新型快速多极虚边界元最小二乘法的求解思想。基于二维弹性问题原有的快速多极虚边界元最小二乘法的展开格式,通过引入对角化的概念,以更新展开传递格式;相对于原有快速多极算法,该方法可进一步提高计算效率且仍能保证具有较高的计算精度。数值算例说明了该方法的可行性、计算效率、计算精度均较高。     

弹性动力学高阶核无关快速多极边界元法

基于核无关的快速多极方法, 发展了一种弹性动力学问题的快速、高精度边界元分析方法. 采用基于二次曲面单元的Nystr?m 离散, 将边界积分方程转化为求和形式, 可以方便地进行加速计算;由于采用二次元, 边界元分析精度很高. 将一种新型快速多极方法用于Nystr?m 边界元法的加速计算, 该方法的数值实现简便、不依赖于积分方程基本解的表达式, 因此通用性很好;该方法还具有最优的计算量和存储量、精度高且可以控制. 结合Nystr?m 边界元系数矩阵和快速多极方法转换矩阵的特点, 提出一种大幅度降低边界元内存消耗的策略. 数值结果表明, 该方法无论在分析精度, 还是计算速度和内存消耗上, 都大大优于同类方法, 是一种快速、通用的工程弹性动力学问题大规模数值分析方法.      

二维正交各向异性位势问题的高阶单元快速多极边界元法

研制了一种适用于二维正交各向异性位势问题的高阶单元(线性单元和二次单元)快速多极边界元法.在快速多极边界元法中,源点对于远场区域的积分采用快速多极展开式计算,而对于近场区域的积分则直接进行计算.高阶单元的使用使得近场积分,尤其是奇异积分和几乎奇异积分的计算更加复杂.通过引入复数表达对其进行简化,若边界采用线性单元插值,...     

规划-迭代型弹塑性摩擦接触多极边界元法

提出基于多极展开法的规划-迭代型的高度非线性方程的IGMRES(m)高效求解法,并建立其收敛性理论。本法适用于三维弹塑性摩擦接触多极边界元法,有效处理弹塑性摩擦接触迭代的繁杂和费时问题。数值试验证明,本求解法在确保数值精度的前提下,可减少迭代次数,显著提高计算效率。     

快速多极子展开技术在高阶边界元方法中的实现

高阶边界元法以较常数元方法计算精度高存储低而在工程计算中得到了广泛的应用,但由于其平方存储和计算量的本质,无法应用于大型工程问题中。本文将快速多极子方法（FMM）应用于高阶边界元中从而使其计算量和存储量分别降为O（Nlog N）和O（N）。通过无限区域中水流绕射算例的数值计算,对FMM高阶边界元法与传统高阶边界元法的运算速度和内存消耗进行了分析对比,结果表明对于大型计算问题FMM高阶边界元算法更加有效。     

基于边界元法求解三维弹性摩擦接触问题

边界元法求解三维摩擦接触问题,其中一个关键点在于如何确定滑移方向。即当出现相对滑移时,滑移方向如何确定。当前常采用的方法是,粘结点利用切向面力得到滑移方向,滑移点利用切向相对位移得到滑移方向。不过该方法难以保证收敛性。针对这一问题,本文采用滑移方向预测技术得到滑移方向。即以后出现相对滑移时,滑移方向采用预测技术中得到的滑移方向。由于摩擦接触问题和历史加载相关,本文采用增量法求解。不同摩擦系数下的数值结果都证明了本文算法的有效性和收敛性及滑移方向预测技术的有效性。     

预条件GMRES(m)法迭代求解大规模边界元弹性问题

提出了一种基于单元节点的块雅可比预条件方法,扩大了边界元法的计算规模,使之可用于大规模工程问题的求解.数值实验说明了这种预条件技术的有效性,表明预条件GMRES(m)算法具有较好的收敛特性,适合于求解大规模问题边界元弹性问题所形成的稠密非对称线性方程组.     

二维弹性接触问题中随机边界元法与可靠性分析

发展了二维弹性接触问题中的随机边界元法，推导并建立了相应的随机边界元基本方程，并将所发展的方法用于静强度的可靠性分析，讨论了其数值解技术。通过算例分析表明，本文发展的方法是可行的。     

Application of a new fast multipole BEM for simulation of 2D elastic solid with large number of inclusions

A fast multipole method (FMM) is applied for BEM to reduce both the operation and memory requirement in dealing with very large scale problems. In this paper, a new version of fast multipole BEM for 2D elastostatics is presented and used for simulation of 2D elastic solid with a large number of randomly distributed inclusions combined with a similar subregion approach. Generalized minimum residual method (GMRES) is used as an iterative solver to solve the equation system formed by BEM iteratively. The numerical results show that the scheme presented is applicable to certain large scale problems. The project supported by the National Nature Science Foundation of China (10172053) and the Ministry of Education     

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 fiber-reinforced 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.The project supported by the National Natural Science Foundation of China (10472051).The English text was polished by Yunming Chen.     

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green’s tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green’s tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process.     

Wideband fast multipole boundary element method: Application to acoustic scattering from aerodynamic bodies

A wideband adaptive multi‐level fast multipole method (MLFMM) is used to accelerate the matrix–vector products arising from a boundary element method (BEM) formulation which solves the Burton–Miller boundary integral equation (BIE). The wideband MLFMM presented here applies a plane wave expansion formulation with fast interpolation and filtering for calculations in the high‐frequency regime and a partial wave expansion formulation with rotation‐coaxial translation in the low‐frequency regime. The iterative solvers GMRES, Bi‐CGSTAB and CGS are tested and compared and a block diagonal preconditioner is used to improve the condition number of the BEM matrices and to accelerate the convergence of the iterative solvers. Details on the implementation of the formulations are described, including the treatment of singular integrals. Results for acoustic scattering from a wing plus engine nacelle configuration for a prescribed source in a subsonic uniform flow are presented for a broad range of frequencies in order to assess the implemented capability. Copyright © 2010 John Wiley & Sons, Ltd.     

Collision detection for complicated polyhedra using the fast multipole method or ray crossing

The purpose of this paper is a comparison of two different methods that can be used for collision detection. One method is called the ray-crossing method, a commonly used geometrical approach. The other method is the fast multipole method, usually used for boundary element methods, which is also applied for collision detection purposes here. Both methods are especially of interest when the collision for arbitrarily shaped polyhedra has to be detected. Here, both methods are described and compared for different examples of complex shaped polyhedra with up to 5 × 10⁵ faces and more than 5 × 10⁵ test points regarding efficiency and required calculation time.     

用边界元方法数值分析绕平壁上一薄平板的STOKES流动

本文用边界元方法研究了绕平壁上一薄平板的Ｓｔｏｋｅｓ流动，数值模拟了薄平板与平壁成不同倾角ξ＝π／２时，平板左右两侧形成大小相等的两个涡旋，其涡心到角点的距离为平板高的０．５４倍；当ζ＝５π／１２时，平板两侧仍然形成两个涡旋，不过平板右侧５π／１２角内的涡旋明显大于板左侧的涡旋，其涡心距角点的距离右侧是平板高的０．６８倍，左侧是０．２９倍；当ξ＝π／３时，平板左侧的涡旋已经消失，右侧π／３角内角然     

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.