基于辛数学方法的一维声子晶体禁带计算

将一维声子晶体的原胞简化为有限多个自由度的弹簧振子结构,在辛对偶变量体系下探讨晶格振动,引入辛数学方法确定波矢与本证值的色散关系。通过本证值计数法计算特征频率,从而得到禁带区间。与传统集中质量法相比,该算法的计算结果与之吻合很好,且提高了计算精度和计算效率,更重要的是在低频处收敛性更好。     

杂质吸收对一维声子晶体滤波器设计的影响

为了研究杂质吸收对一维声子晶体滤波器设计的影响,引入复波数并推导出一维掺杂声子晶体的转移矩阵,计算了一维掺杂声子晶体的透射系数随衰减系数的变化特征。得出：滤波透射峰的峰值随杂质的衰减系数增加而迅速减小,滤波透射峰的半高宽随衰减系数增加而增大。滤波透射峰的峰值和半高宽都随吸收杂质的厚度的增加而减小。在设计声子晶体滤波器时,必须考虑杂质吸收这一重要因素,应选择衰减系数小于0.0005k的掺杂材料,并且杂质的厚度应小于一个波长。     

爆轰波斜冲击金属介质理论在聚能装药药型罩设计中的应用研究

将爆轰波斜冲击金属介质理论引入聚能药型罩罩高参数设计中,运用LS-DYNA显式动力学软件对爆轰波斜入射铜介质进行了仿真计算,得到了接触位置压力峰值P与入射角φ0之间的关系,数值仿真结果和理论计算值、试验值均吻合得较好。基于仿真计算和理论分析,得出了圆锥形和球缺形药型罩点起爆条件下罩高参数确定的工程算法,并将其应用于一种组合式战斗部的设计中。该方法为药型罩高度的确定提供了基于理论分析方面的设计依据,丰富了聚能装药战斗部的设计方法。     

基于快速多极子基本解方法(FMM-MFS)的弹性波二维散射模拟研究

针对弹性波二维散射问题,发展一种新的快速多极子基本解方法（FMM-MFS）。方法基于单层位势理论,通过在虚边界上设置膨胀波线源和剪切波线源以构造散射波场,从而避免了奇异性的处理和边界单元离散;结合快速多极子展开技术（FMM）,大幅度降低了计算量和存储量,突破了传统方法难以处理大规模散射问题的瓶颈。以全空间孔洞对P、SV波的二维散射为例,给出了具体求解步骤,并在个人计算机上实现了上百万自由度问题的快速精确计算。在方法效率和精度检验基础上,分别以单孔洞和随机孔洞群对平面波（P、SV波）的散射为例进行计算模拟,揭示了孔洞（群）周围弹性波散射的若干重要规律。     

二维新型快速多极虚边界元配点法

以二维弹性问题为研究背景, 提出了一种二维新型快速多极虚边界元配点法的求解思想, 即采用新型的快速多极展开和运用广义极小残值法来求解传统的虚边界元配点法方程。相对常规快速多极展开技术, 该文针对二维弹性问题在原有的快速多极虚边界元法展开格式的基础上, 通过引入对角化的概念, 以更新展开传递格式, 欲达到进一步提高计算效率的目的。数值算例说明了该方法的可行性, 计算效率和计算精度。此外, 该文方法的思想具有一般性, 应用上具有扩展性。     

实复转换式衰减信号参数估计算法

为抑制衰减实信号中负频率成分对参数估计的影响,提出一种实复转换式参数估计算法。预估计采样信号频谱能量最大值点的索引值;构造只含有负频率成分的参考信号,并将采样信号和参考信号相减实现实复转换,以抑制负频率频谱泄漏的影响;利用频谱两点插值算法得到频率偏差、衰减因子和复幅值的粗估计值,并重新生成参考信号和复信号;通过迭代计算得到精确的频率、衰减因子、初幅值和初相位估计值。以频率估计为例的仿真实验结果表明:所提算法可有效地抑制负频率频谱泄漏的影响,提高中高信噪比条件下的频率估计精度,特别是信号频率较低时的频率估计精度,提升了频率估计的综合性能。此外,在科氏流量计中进行了实测实验,检验了所提算法的有效性。     

计算气动声学中的高阶Nodal-DG方法研究

气动噪声的直接模拟对数值格式的色散、耗散特性提出了严格的要求。基于描述声波的线性双曲方程,运用本征值方法分析了高阶Nodal-DG方法的色散、耗散特性。结果发现,对于任意给定的m阶多项式基函数,数值波解有m+1个值,但仅有一个能够表示对应微分方程的物理波传播方式,其余的都是寄生波,且两种波型的传播方向相反。通过与Tam的DRP格式和Lele的六阶紧致格式进行比较,发现在相同的计算精度下,Nodal-DG方法的有效求解波数范围介于DRP格式和六阶紧致格式之间。通过对初始扰动为高斯波形的计算比较发现,在较少的网格数下,Nodal-DG方法的计算结果可以与紧致格式的计算结果相比,但优于DRP格式的计算结果,非常适合于气动声学的数值模拟,为气动声场的直接计算提供了一种新的方法     

数据驱动与协方差驱动随机子空间法差异化分析

针对能有效从环境激励结构振动响应中获取模态参数的随机子空间法,传统观点认为无论在理论上或应用中数据驱动随机子空间法与协方差驱动随机子空间法在模态参数识别过程中表现一致,实际应用中表现不一致问题,理论上探讨两种方法出现差异的原因,并进行相应的数值模拟。研究结果表明：基于QR分解的数据驱动随机子空间法无论计算精度或对较弱势模态的识别能力均明显优于协方差驱动随机子空间法。     

数据驱动随机子空间法矩阵维数选择与噪声问题研究

数据驱动随机子空间法作为一种线性系统辩识方法,可以有效地从环境激励的结构振动响应中获取模态参数。其中,Hankel矩阵维数的选择直接影响到数据驱动随机子空间法消噪能力。本文理论上分析了噪声与数据驱动随机子空间法Hankel矩阵维数之间的关系,并基于归一化奇异值（SVD）、稳定图以及有限元模态识别结果（FE）,提出了一种评估数据驱动随机子空间法矩阵维数选择优劣的方法,并通过数值算例和导管架平台振动台试验系统地验证了该方法的有效性,结果表明：非方阵的Hankel矩阵使数据驱动随机子空间法具备更强的消噪能力和更高的模态识别精度。     

大型浮体水弹性作用的频域分析

对大型浮体水弹性响应的频域计算方法做了综述和介绍。分别介绍了干模态法、湿模态法和直接计算方法,对于干模态法的五种结构弹性模态函数做了介绍,研究了五种模态函数下计算结果的收敛速度。对于水动力分析方法的计算量和存储量做了分析,介绍了降低计算量和存储量的一些新计算方法,实现了一种柱坐标下的多极子展开高阶边界元方法,积分方程中的固角系数和柯西主值积分均采用直接方法计算,应用该方法计算了波浪与大型弹性浮体的相互作用问题,计算结果与实验值得到很好的吻合。     

Fast multipole DBEM analysis of fatigue crack growth

A fast multipole method (FMM) based on complex Taylor series expansions is applied to the dual boundary element method (DBEM) for large-scale crack analysis in linear elastic fracture mechanics. Combining multipole expansions with local expansions, both the computational complexity and memory requirement are reduced to O(N), where N is the number of DOF. An incremental crack-extension analysis based on the maximum principal stress criterion and the Paris law is used to simulate the fatigue growth of numerous cracks in a 2D solid. Some examples are presented to validate the numerical scheme.     

The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems

This paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

Analysis of Solids with Numerous Microcracks Using the Fast Multipole DBEM

The fast multipole method (FMM) is applied to the dual boundary element method (DBEM) for the analysis of finite solids with large numbers of microcracks. The application of FMM significantly enhances the run-time and memory storage efficiency. Combining multipole expansions with local expansions, computational complexity and memory requirement are both reduced to O(N), where N is the number of DOFs (degrees of freedom). This numerical scheme is used to compute the effective in-plane bulk modulus of 2D solids with thousands of randomly distributed microcracks. The results prove that the IDD method, the differential method, and the method proposed by Feng and Yu can give proper estimates. The effect of microcrack non-uniform distribution is evaluated, and the numerical results show that non-uniform distribution of microcracks increases the effective in-plane bulk modulus of the whole microcracked solid.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast algorithm for three-dimensional electrostatics analysis: fast Fourier transform on multipoles (FFTM)

In this paper, we propose a new fast algorithm for solving large problems using the boundary element method (BEM). Like the fast multipole method (FMM), the speed-up in the solution of the BEM arises from the rapid evaluations of the dense matrix–vector products required in iterative solution methods. This fast algorithm, which we refer to as fast Fourier transform on multipoles (FFTM), uses the fast Fourier transform (FFT) to rapidly evaluate the discrete convolutions in potential calculations via multipole expansions. It is demonstrated that FFTM is an accurate method, and is generally more accurate than FMM for a given order of multipole expansion (up to the second order). It is also shown that the algorithm has approximately linear growth in the computational complexity, implying that FFTM is as efficient as FMM. Copyright © 2004 John Wiley & Sons, Ltd.     

A fast and accurate algorithm for a Galerkin boundary integral method

A fast and accurate algorithm is presented to increase the computational efficiency of a Galerkin boundary integral method for solving two-dimensional elastostatics problems involving numerous straight cracks and circular inhomogeneities. The efficiency is improved by computing the combined influences of groups, or blocks, of elements—with each element being an inclusion, a hole, or a crack—using asymptotic expansions, multiple shifts, and Taylor series expansions. The coefficients in the asymptotic and Taylor series expansions are computed analytically. Implementation of this algorithm involves a single- or multi-level grid, a clustering technique, and a tree data structure. An iterative procedure is adopted to solve the coefficients in the series expansions of boundary unknowns block by block. The elastic fields in each block are calculated by superposition of the direct influences from the nearby elements and the grouped far-field influences from all the other elements. This fast multipole algorithm is considerably more efficient for large-scale practical problems than the conventional approach.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.     

An <i>O(N)</i> Fast Multipole Hybrid Boundary Node Method for 3D Elasticity

The Hybrid boundary node method (Hybrid BNM) is a boundary type meshless method which based on the modified variational principle and the Moving Least Squares (MLS) approximation. Like the boundary element method (BEM), it has a dense and unsymmetrical system matrix and needs to be speeded up while solving large scale problems. This paper combines the fast multipole method (FMM) with Hybrid BNM for solving 3D elasticity problems. The formulations of the fast multipole Hybrid boundary node method (FM-HBNM) which based on spherical harmonic series are given. The computational cost is estimated and an O(N) algorithm is obtained. The algorithm is implemented on a computer code written in C++. Numerical results demonstrate the accuracy and efficiency of the proposed technique.     

An Adaptive Fast Multipole Boundary Element Method for Three-dimensional Potential Problems

An adaptive fast multipole boundary element method (FMBEM) for general three-dimensional (3-D) potential problems is presented in this paper. This adaptive FMBEM uses an adaptive tree structure that can balance the multipole to local translations (M2L) and the direct evaluations of the near-field integrals, and thus can reduce the number of the more costly direct evaluations. Furthermore, the coefficients used in the preconditioner for the iterative solver (GMRES) are stored and used repeatedly in the direct evaluations of the near-field contributions. In this way, the computational efficiency of the adaptive FMBEM is improved significantly. The adaptive FMBEM can be applied to both the original FMBEM formulation and the new FMBEM with diagonal translations. Several numerical examples are presented to demonstrate the efficiency and accuracy of the adaptive FMBEM for studying large-scale 3-D potential problems. The adaptive FMBEM is found to be about 50% faster than the non-adaptive version of the new FMBEM in solving the model (with 558,000 elements) for porous materials studied in this paper. The computational efficiencies and accuracies of the FMBEM as compared with the finite element method (FEM) are also studied using a heat-sink model. It is found that the adaptive FMBEM is especially advantageous in modeling problems with complicated domains for which free meshes with much more finite elements would be needed with the FEM.