双轴受压反对称角铺设复合材料层合板在固支边界下的后屈曲和模态跃迁

为有效分析双轴受压反对称角铺设复合材料层压板在固支边界下的后屈曲性能, 由渐近修正几何非线性理论推导其双耦合四阶偏微分方程(即应变协调方程和稳定性控制方程), 通过双Fourier级数将耦合非线性控制偏微分方程转换为系列非线性常微分方程, 从而获得相对简单的求解方法。使用广义Galerkin方法求解与角交铺设复合层合板相关的边界值问题, 研究了模态跃迁前后不同复杂程度的后屈曲模式。对四层固支边界复合层合板的数值模拟结果表明: 该解析法与有限元方法在主后屈曲区域的线性屈曲荷载计算结果吻合良好; 有限元方法在解靠近二次分岔点时失去收敛性, 而解析方法可深入后屈曲区域, 准确捕捉模态跃迁现象; 对于反对称角铺设层合板, 可仅用纯对称模态来定性预测主后屈曲分支、二次分岔荷载及远程跃迁路径。      

复合材料层合板振动的边界元法

本文用边界元法分析了复合材料特殊正交各向异性层合板的振动.为了克服在用边界元法求解正交各向异性层合板振动时寻求相应的基本解的困难,本文采用了傅立叶级数形式的近似基本解.算例说明了近似基本解方法的可行性和有效性.      

纤维增强复合材料层合板屈曲性态分析的边界元法

本文用边界元法分析了纤维增强复合材料正交各向异性层合板的屈曲性态。为了克服在用边界元法求解正交各向异性层合板屈曲时寻求相应的基本解的困维,本文采用了双重傅立叶级数和引用等效荷载的概念,建立了层合板屈曲临界荷载的特征方程。算例说明了本文方法的可行性和有效性。      

反对称角铺设复合材料层合板热后屈曲和模态跃迁分析

为有效分析反对称角铺设复合材料层压板热后屈曲性能,由渐近修正几何非线性理论推导双耦合四阶偏微分方程（即协调方程和动态控制方程）,通过双Fourier级数将耦合非线性控制偏微分方程转换为系列非线性常微分方程,从而获得相对简单的求解方法。使用广义Galerkin方法求解与角交铺设复合层合板相关的边界值问题,研究了模态跃迁前后不同复杂程度的后屈曲模式。通过四边简支、面内不可移边界下复合层合板的数值计算表明：该解析法与有限元方法在主后屈曲区域的计算结果有很好的吻合性;有限元方法在解靠近二次分岔点时失去收敛性,而解析法仍具有深入探索后屈曲区域和准确捕捉模态跃迁现象的能力。     

一种基于谱元法的层合板SEA参数计算方法

针对一般层合板统计能量分析(SEA)参数获取困难问题,建立了一种基于谱元法(SFEM)的SEA参数计算方法。采用三节点二次谱单元在层合板厚度方向进行网格划分,通过刚度矩阵与质量矩阵建立波数的特征方程;根据模态相似原则,利用皮尔逊相关系数对各阶模态的波数进行分类,进而获得层合板的模态密度、辐射效率等SEA参数。对单层薄板和三明治夹芯板进行了数值模拟研究,计算结果与其他理论值或实验值进行对比,验证了该方法的有效性,并以五层碳-碳正交各向异性板为例,考察了经典层合板理论和一阶剪切理论的分析偏差。最后计算了汽车玻璃层合板的内损耗因子,验证了基于SFEM计算层合板内损耗因子的有效性。     

复合材料层合板智能结构主动振动控制的边界元法

利用边界元法模拟智能结构的振动控制，推导出具有压电传感器及致动器的复合材料层合板的边界积分方程，应用负速度反馈控制律，研究了复合材料层合板智能结构主动振动控制问题，算例分析证明该方程的正确性。     

瞬态弹性动力学问题的半解析边界元法

本文用半解析有限元法对边界积分方程作离散化处理,通过引入基本解函数和半解析半离散试函数的二次半解析过程,使三维弹性动力学问题简化为一维数值计算。文中又采用移动边界元法来模拟波在半无限介质中传播的表面积分问题,分析计算了各种瞬态波在介质内传播,绕射及地面运动问题。计算结果表明,半解析边界元法不仅计算精度高,而且工作量大大降低,具有较高的经济效益与应用价值。     

对称层合板复合材料的屈曲变形解析

提出了如何利用2 元多项式位移函数来分析层合板复合材料的屈曲问题。在解析中同时 考虑到层合板的面外剪切变形。解析表明屈曲载荷在多项式项数大于20 以上时趋于稳定, 可得到 高精度的解。本文并对对称角铺设层合板进行了系统地数值解析, 明确了在各种边界拘束条件下 材料的工程弹性系数, 层合板长宽比, 铺层数, 层合角对屈曲载荷的影响, 从而以达到优化设计。同 时, 验证了本解析法对任意边界条件下的层合板的屈曲解析是非常有效的。      

含孔复合材料层合板孔边的应力集中

针对含孔有限宽复合材料层合板的应力集中问题,提出一种计算孔边应力分布及应力集中因子的方法：先利用经典层板理论,将复合材料层合板化归为各向异性板;再将各向异性板等效转换为一偏轴拉伸的单向纤维层板;最后利用含孔偏轴单向板的孔边应力计算公式来分析一般铺层层合板孔边应力集中情况。根据所推导的计算公式,分析讨论了板宽/孔径比、铺层比例、铺层方式、材料性能参数等因素对孔边应力集中的影响。     

激光干涉仪反射镜三维温度场的快速多极边界元分析

温差造成光刻机激光干涉仪反射镜热变形,从而影响光刻机的精度。该文将基于二次单元的快速边界元法用于激光干涉仪反射镜的大规模温度场模拟。不连续单元的引入可以有效处理角点问题;新型快速多极算法用于边界元法的加速求解。建立统一的二次单元多极展开格式以处理混合边界。数值算例分析了快速多极边界元法的计算精度和效率,并和常规算法比较;使用该算法对激光干涉仪反射镜进行了大规模温度场计算,并和有限元法比较。结果表明:基于二次单元的快速多极边界元法可以高精度求解大规模三维传热问题。     

Boundary element analysis for axisymmetric heat conduction and thermal stress in steady state

In this paper, axisymmetric heat conduction and thermal stress problems with three types of boundary conditions are analysed by the boundary element method. The temperature and thermal stress fields for the piston of a diesel engine are calculated using triangular finite elements and constant boundary elements, respectively, and the two results agree. However, BEM needs fewer data, less computer time (about one-sixth that of FEM) and storage volume. The advantages of BEM are sufficiently demonstrated.     

Analytic formulations for calculating nearly singular integrals in two-dimensional BEM

There exist the nearly singular integrals in the boundary integral equations when a source point is close to an integration element but not on the element, such as the field problems with thin domains. In this paper, the analytic formulations are achieved to calculate the nearly weakly singular, strongly singular and hyper-singular integrals on the straight elements for the two-dimensional (2D) boundary element methods (BEM). The algorithm is performed after the BIE are discretized by a set of boundary elements. The singular factor, which is expressed by the minimum relative distance from the source point to the closer element, is separated from the nearly singular integrands by the use of integration by parts. Thus, it results in exact integrations of the nearly singular integrals for the straight elements, instead of the numerical integration. The analytic algorithm is also used to calculate nearly singular integrals on the curved element by subdividing it into several linear or sub-parametric elements only when the nearly singular integrals need to be determined. The approach can achieve high accuracy in cases of the curved elements without increasing other computational efforts. As an application, the technique is employed to analyze the 2D elasticity problems, including the thin-walled structures. Some numerical results demonstrate the accuracy and effectiveness of the algorithm.     

A fast multipole boundary element method for solving two-dimensional thermoelasticity problems

A fast multipole boundary element method (BEM) for solving general uncoupled steady-state thermoelasticity problems in two dimensions is presented in this paper. The fast multipole BEM is developed to handle the thermal term in the thermoelasticity boundary integral equation involving temperature and heat flux distributions on the boundary of the problem domain. Fast multipole expansions, local expansions and related translations for the thermal term are derived using complex variables. Several numerical examples are presented to show the accuracy and effectiveness of the developed fast multipole BEM in calculating the displacement and stress fields for 2-D elastic bodies under various thermal loads, including thin structure domains that are difficult to mesh using the finite element method (FEM). The BEM results using constant elements are found to be accurate compared with the analytical solutions, and the accuracy of the BEM results is found to be comparable to that of the FEM with linear elements. In addition, the BEM offers the ease of use in generating the mesh for a thin structure domain or a domain with complicated geometry, such as a perforated plate with randomly distributed holes for which the FEM fails to provide an adequate mesh. These results clearly demonstrate the potential of the developed fast multipole BEM for solving 2-D thermoelasticity problems.     

3-D boundary element method for cracks in a plate

New fundamental solutions which automatically satisfy boundary conditions at the interfaces of an elastic plate perfectly bonded to two elastic halfspaces are implemented in a 3-D boundary element method (BEM) for crack problems. The BEM features a new integration scheme for highly singular kernels. The capability is achieved through a part analytic and part numerical integration procedure, such that the analytic part of the integration is similar for all slip/opening variations, ‘Part-through’ elliptic cracks in an elastic plate with traction-free surfaces are analysed and the stress intensity factor (SIF) values along the crack front are found to compare favourably with widely accepted numerically obtained SIF results by Raju and Newman.¹     

An immersed boundary element method for level‐set based topology optimization

In this paper, we propose a new BEM for level‐set based topology optimization. In the proposed BEM, the nodal coordinates of the boundary element are replaced with the nodal level‐set function and the nodal coordinates of the Eulerian mesh that maintains the level‐set function. Because this replacement causes the nodal coordinates of the boundary element to disappear, the boundary element mesh appears to be immersed in the Eulerian mesh. Therefore, we call the proposed BEM an immersed BEM. The relationship between the nodal coordinates of the boundary element and the nodal level‐set function of the Eulerian mesh is clearly represented, and therefore, the sensitivities with respect to the nodal level‐set function are strictly derived in the immersed BEM. Furthermore, the immersed BEM completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization. By using the immersed BEM, we construct a concrete topology optimization method for solving the minimum compliance problem. We provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems

A new adaptive fast multipole boundary element method (BEM) for solving 3-D half-space acoustic wave problems is presented in this paper. The half-space Green's function is employed explicitly in the boundary integral equation (BIE) formulation so that a tree structure of the boundary elements only for the boundaries of the real domain need to be applied, instead of using a tree structure that contains both the real domain and its mirror image. This procedure simplifies the implementation of the adaptive fast multipole BEM and reduces the CPU time and memory storage by about a half for large-scale half-space problems. An improved adaptive fast multipole BEM is presented for the half-space acoustic wave problems, based on the one developed recently for the full-space problems. This new fast multipole BEM is validated using several simple half-space models first, and then applied to model 3-D sound barriers and a large-scale windmill model with five turbines. The largest BEM model with 557470 elements was solved in about an hour on a desktop PC. The accuracy and efficiency of the BEM results clearly show the potential of the adaptive fast multipole BEM for solving large-scale half-space acoustic wave problems that are of practical significance.     

On the validity of conforming BEM algorithms for hypersingular boundary integral equations

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying field is C ^1,α continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justification for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a finite approximation to the standard two-sided Hadamard finite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided finite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the field which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C ^1,α interpolations.     

A comparison of boundary element method formulations for steady state anisotropic heat conduction problems

In this paper the boundary element method (BEM) is numerically implemented in order to solve steady state anisotropic heat conduction problems. Various types of elements, namely, constant elements, continuous and discontinuous linear elements and continuous and discontinuous quadratic elements are used. The performances of these various BEM formulations are compared for both the direct well-posed Dirichlet problem and the inverse ill-posed Cauchy problem, revealing several features of the BEM. Furthermore, previously undetermined analytical solutions for the integrals associated with linear and quadratic elements are presented.     

Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton-Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numerical integration. Numerical examples show that using the analytical moments can reduce the CPU time by a lot as compared with that using the direct numerical integration. The percentage of CPU time reduction largely depends on the proportion of the time used for moments calculation to the overall solution time. Several examples are studied to investigate the effectiveness and efficiency of the developed diagonal fast multipole BEM as compared with earlier p³ fast multipole method BEM, including a scattering problem of a dolphin modeled with 404,422 boundary elements and a radiation problem of a train wheel track modeled with 257,972 elements. These realistic, large-scale BEM models clearly demonstrate the effectiveness, efficiency and potential of the developed diagonal form fast multipole BEM for solving large-scale acoustic wave problems.     

Parallel processing of quadratic boundary elements for axisymmetric elasto-static problems

The concern of this paper is on improving the computational efficiency of boundary element methods (BEM) through the development of parallel algorithms for use on massively parallel machines. The application is on the axisymmetric elasto-static problems with quadratic boundary elements. Different ways of parallel approaches are discussed and a parallel approach suited to the BEM numerical process is developed. Numerical results from both the parallel algorithm and a serial algorithm are given in the paper to illustrate the efficiency of the parallel approach.