三维弹性快速多极边界元法

将静电场多极展开法和广义极小残值法结合于三维弹性问题的边界元法,使其求解的计算量及所需内存量同节点的自由度总数成正比,变革计算结构,加快求解速度以适应大规模数值计算。两者结合的关键点在于边界元法基本解的合理分解,并用广义极小残值法(GMRES)求解方程。轧机支承辊变形场大规模数值算例的总自由度数首次达N=34008并获得成功。清晰地描述了支承辊和工作辊接触区的辊型。     

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,...     

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。     

高阶边界元奇异积分的一种通用高效计算方法

针对三维边界元法中曲面单元上的(弱、强、超)奇异积分提出了一种通用高效的计算方法。经极坐标变换,将奇异积分转化为常规积分;采用数值方法计算Cauchy主值积分和Hadamard有限项积分系数;引入保角变换和反曲变换消除因单元畸形或因积分点靠近单元边界而引起的周向积分奇异性。该方法可以统一处理(弱、强、超)奇异积分,并且只需要知道核函数的奇异阶数和少数几个点上的被积函数值,不依赖于积分和函数的具体选取;所需的积分点少,精度高,并且受单元畸形程度影响较小,稳定性好。采用该方法计算了声学和弹性力学中的典型奇异积分,并结合二阶Nystrm方法求解了弹性力学的边界积分方程,验证了方法的高精度和高效性。本文数值积分程序可向作者索取。     

基于比例边界有限元法的高阶透射边界应用

基于比例边界有限元法和连分式展开推导了无限域弹性动力分析的求解方程,实现了一种局部的高阶透射边界. 采用改进的连分式法求解无限域的动力刚度矩阵,克服了原连分式算法可能会造成矩阵运算病态的问题. 该局部高阶透射边界在时域里表示为一阶常微分方程组,其稳定性取决于其系数矩阵的广义特征值问题. 如果出现虚假模态,采用移谱法来校正系数矩阵以消除虚假模态. 通过两个算例验证了该高阶透射边界的精确性、鲁棒性.     

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。     

边界元技术中的全特解场方法

采用相应基本解来构造全特解场，据而确定各该问题边界元方程系数矩阵的全部元素。解算中不涉及具体插值，不用数值积分，避免了奇异积分的数值处理，而任意点的物理量计算（如应力、声强等等）不依赖于待解的边界未知量，因而算效奇高，精度特好。适于求解各类数学物理和工程技术问题。     

基于预修正快速傅里叶变换高阶边界元方法的多体水动力分析

高阶边界元方法在求解波浪对海上建筑物作用问题中具有诸多优势,但由于它所形成的矩阵是一个满阵,计算量和存储量均为未知量的平方量级,很难满足如多体水动力分析等大尺度多未知量问题的计算需要。本文采用预修正快速傅里叶变换高阶边界元方法（pFFT-HOBEM）,将计算量和存储量均降低到未知量的线性量级。通过对不同未知量时该方法与传统边界元法的计算量与存储量的对比,以及该方法自身各步骤计算时间的对比,研究了不同pFFT网格方案对计算量和存储量的影响,并提出了基于计算时间最小化原则的pFFT网格优化方法。采用本文方法研究了四柱结构在不同频率波浪作用下的作用力及波面分布,并对近场干涉发生时的物理现象进行了分析和讨论。     

pFFT快速边界元方法模拟三维声散射

研究了用pFFT快速边界元方法模拟声散射问题的关键技术。采用Burton—Miller方程消除了声学边界元方法中外问题解的不唯一现象。为此,文中研究了采用常量元时该方程中超奇异积分的计算方法。最后,通过对平面声波的刚性圆球声散射的数值模拟,验证了建立的声学pFFT快速边界元方法。     

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.     

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     

利用高阶分区并行算法实现直接数值模拟

在基于MPI环境的分布式内存机群上,结合高阶WENO-RF格式的特点,实现了5阶WENO-RF格式的分区并行计算方法,计算精度不受分区和节点数量影响。使用该分区并行算法以三维可压缩时间发展混合层为例进行了直接数值模拟,验证了并行算法的准确性,表明机群并行运算可以显著扩展微机的计算能力,并行效率高,减少了计算的墙上时间,适合在小型高速局域网内进行大规模数值模拟计算。     

Kernel-independent fast multipole method within the framework of regularized Stokeslets

The method of regularized Stokeslets (MRS) uses a radially symmetric blob function of infinite support to smooth point forces and allows for evaluation of the resulting flow field. This is a common method to study swimmers at zero Reynolds number where the Stokeslet is the fundamental solution corresponding to the kernel of the single layer potential. Simulating the collective motion of N micro-swimmers using the MRS results in at least N² pair-wise interactions. Efficient simulation of a large number of swimmers in free space is observed with the implementation of the kernel-independent fast multipole method (FMM) for radial basis functions. We illustrate the complexity of the algorithm on a simple test case where we study regularized point forces, showing that the method is of order N. Additionally, we explore accuracy in time for the MRS where the swimmers are modeled as Kirchhoff rods and the kernel-independent FMM is compared to the direct calculation using the standard MRS. Optimal hydrodynamic efficiency is also explored for different configurations of swimmers.     

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.     

适合于求解边界元方程组的GMRES算法的实用化和并行化研究

为了将GMRES算法应用于大型边界元方程组的求解,采用预条件技术和重正交技术相结合的方法实现了该算法的实用化,然后在实用化的基础上针对迭代算法具有良好并行性的特点,研究了该算法在网络机群环境下的并行化技术。数值试验和分析表明所用的这些技术是行之有效的,对于提高求解速度和增大求解问题的规模是有意义的。     

Applications of the dual integral formulation in conjunction with fast multipole method to the oblique incident wave problem

In this paper, the dual integral formulation is derived for the modified Helmholtz equation in the propagation of oblique incident wave passing a thin barrier (zero thickness) by employing the concept of fast multipole method (FMM) to accelerate the construction of an influence matrix. By adopting the addition theorem, the four kernels in the dual formulation are expanded into degenerate kernels that separate the field point and the source point. The source point matrices decomposed in the four influence matrices are similar to each other or only to some combinations. There are many zeros or the same influence coefficients in the field point matrices decomposed in the four influence matrices, which can avoid calculating the same terms repeatedly. The separable technique reduces the number of floating‐point operations from O((N)²) to O(N log^a(N)), where N is the number of elements and a is a small constant independent of N. Finally, the FMM is shown to reduce the CPU time and memory requirement, thus enabling us to apply boundary element method (BEM) to solve water scattering problems efficiently. Two‐moment FMM formulation was found to be sufficient for convergence in the singular equation. The results are compared well with those of conventional BEM and analytical solutions and show the accuracy and efficiency of the FMM. Copyright © 2008 John Wiley & Sons, Ltd.     

An interpolation formula of the derivatives of higher order

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Nielsen's and Euler's operators of higher order in analytical mechanics

In this paper, the definitions of Nielsen’s and Euler’s operators of higher order are presented. These operators are concerned in analysis for systems with holonomic constraints and non-holonomic constraints of higher order. Some theorems that indicating relation between the two operators are established. Moreover, using the theorems, the new equations of mechanical systems with constraints of higher order are derived. Finally, an example is given.     

Numerical schemes with high order of accuracy for the computation of shock waves

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