结构设计的统一优化准则法

基于函数二次台劳展开将原问题化为一系列显式的近似问题,基于Kuhn-Tucker条件直接给出一种新的设计迭代式,近似问题解的序列将收敛于原问题解。为克服Hessian矩阵求逆的困难,引入线性方程组SOR松驰型迭代解法,使该法具有很好的效率。对包括复合材料机翼结构的若干算例进行了计算,结果表明,同其它方法相比,本方法的收敛效率是令人满意的。 对于应力、位移约束,文中统一给出了其一阶、二阶导数计算公式。特别是二阶导数的计算,基于矩阵运算的性质得到了一种简便易行的表达式。与其它现有的二阶导势计算方法相比,本文的方法具有更高的效率。     

非线性动力学常微分方程组高精度数值积分方法

建立了一种求解非线性动力学常微分方程组初值问题的新方法．若非线性函数一阶导数存在，则给出解的积分方程表达式，计算得到按规定误差要求的高精度数值解．引入一般自治或非自治非线性系统的首次近似Jacobi矩阵，不作任何假设重构等价的非线性常微分方程组，简捷而有广泛的适应性，不改变方程的本质，但其主项构成线性化方程组，其它项则代表非线性函数高阶余项而不涉及Taylor级数展开计算，给出该方程组初值问题的Duhamel卷积分解析表达式，在时间步长内进行数值积分选代求解，在指定误差内快速收敛，逐步递推获得非线性常微分方程的瞬态响应和全时域高精度数值解．积分解连续满足微分方程组而不是在离散的步长端点上满足代数方程组，打破了传统用增量法在离散点上建立的代数方程组迭代求解，从而使传统Euler型逐步积分法的各种差分格式算法改变成真正的积分格式算法．数值计算中给出指数矩阵递增展开式，变矩阵乘法为乘积系数的加法，避免了大量矩阵自乘而大大提高计算效率．算法验证为无条件稳定，则保证对线性常微分方程而言，计算中舍入误差的传播不会扩散，不出现计算机字长有限而引起舍入误差导致计算不确定性问题．基于以上理论和数值方法，计算了线性非线性算例并进行了分析，验证了本方法简捷而有广泛的适应性，可以有足够的精确性．     

非线性问题的变分迭代方法及其应用

本文应用变分的概念，提出了求解非线性问题的加速迭代方法．这一方法的基本思想是：先给出问题的近似解，再引进一乘子校正其近似解；而乘子可用变分的概念最优确定，几个例子说明这种方法是有效的．     

线性广义特征值问题的摄动迭代法

本文给出了一种适用于迭代计算的矩阵摄动法,它是进行广义特征值问题Ax=λBx的摄动重分析的一种高精度算法,同时也可用于改进由其它矩阵摄动分析方法提供的近似解的精度。实际算例表明,当结构参数修改量不太大时,采用这种摄动迭代法进行特征值问题的精确重分析是十分有效的。     

非线性振动系统的多项式向量方法

对于常微分方程描述的非线性振动系统,当采用摄动方法求近似解时,先是给出满足各阶近似解的二阶常微分方程组,继而依次对每一个常微分方程进行求解,以致多自由度非线性振动系统的求解过程相当繁琐.文章针对常微分方程表示的非线性振动系统,提出了一种求解非线性振动系统近似解的多项式向量方法,该方法将二阶常微分方程组表示成一阶状态方程组,将非线性部分写成常数矩阵和多项式向量之积的形式.然后,采用直接摄动方法,获得每个幂次近似解所满足的一组状态方程,此时状态方程的非线性部分成为常数矩阵和前一幂次近似解作为元素组成的多项式向量的乘积.进一步,借助Toeplitz矩阵将多项式向量之乘法表示成矩阵形式,以解决多项式相乘带来的幂次方系数的确定问题,再根据一阶非齐次方程组的求解方法,获得状态方程组的全部近似解析解.多项式向量方法将二阶常微分描述的非线性振动求解过程转换为一阶非齐次状态方程组的求解问题,计算过程主要是矩阵和向量之间乘法运算,提高了计算效率和程序化水平.     

矩阵分块加速迭代法

本文提出一种求解结构静力和动力问题的迭代方法,它将刚度矩阵和质量矩阵分为若干块,然后分块求解,再将得到的解在子空间中进行“综合”,求得原方程的近似解,经迭代收敛于精确解.这一方法所需内存少、收敛快。为小型、微型机求解大型结构提供了实用的方法。     

精细辛算法的高效格式和简化计算

对精细辛几何算法设计了高效的迭代过程，减少了精细积分的计算量，同时提出了精细辛算法的简化形式，避免了复杂的矩阵求逆运算，并给出了相应的误差估计，最后编制了程序进行验证，证明了所采取的方法能够使计算快捷，精度高，稳定性好．     

约束层阻尼板动力学问题的半解析解

利用条形传递函数方法(SDTFM)得到了约束层阻尼(CLD)板动力学问题的半解析解.首先对CLD板沿纵向离散成多个条形单元,基于Hamilton原理推导出条形单元的刚度矩阵和质量矩阵,仿照有限元法组集得到系统的总刚度矩阵和总质量矩阵.经Laplace变换后引入状态向量,采用分布参数传递函数方法在状态空间内建立CLD板的控制方程并进行求解.最后以对边固支和悬臂CLD板为例,得到了板的动力学特性和频响曲线,并与NASTRAN或相关文献结果进行了比较,吻合良好,验证了该方法的有效性.从推导过程和算例可以看出,该方法所需的单元数目少,获得的是半解析解,计算效率高且准确可靠.     

摄动法结合Pad逼近在结构拓扑重分析中的应用

讨论了矩阵摄动理论结合Pad啨逼近在结构拓扑修改重分析中的应用,利用分步迭代的方法来取得高精度的近似解。定义过渡方程并利用原始结构信息得到其精确解;利用正交化的摄动基作Pad啨逼近,并采用迭代的方法得出对过渡方程解的增量,从而得到修改结构的近似解。     

摄动法结合Padé逼近在结构拓扑重分析中的应用

讨论了矩阵摄动理论结合Padé逼近在结构拓扑修改重分析中的应用,利用分步迭代的方法来取得高精度的近似解.定义过渡方程并利用原始结构信息得到其精确解;利用正交化的摄动基作Padé逼近,并采用迭代的方法得出对过渡方程解的增量,从而得到修改结构的近似解.     

大型结构特征值问题的混合粒度并行算法

本文提出一种求解大形结构特征值问题的粗细粒度混合并行算法：在子结构模态综合粗粒度并行算基础上,综合系统的特性值问题采用细粒度并行方式求解。细粒度并行包括子空间迭代法的子结构并行算法、雅可比分块并行计算的方法和一种Ｎｅｗｔｏｎ－Ｒａｐｈｏｎ迭代法在多处理器上任力均衡分配的有效策略。子空间迭代法的子结构并行计算的实施是利用子结构的刚度阵和质量阵而不必完全组集系统刚度阵和国求综合系统的特征值问题。利用雅     

直接基于单元平衡的变截面梁反应分析方法

固定形状的单元位移插值函数不能合理地近似变截面梁内部的位移变化,从而影响了传统梁单元用于计算变截面梁的精度.采用直接基于单元平衡的思想给出了计算变截面梁反应的有限元方法,解决了单元位移插值函数局限性所带来的问题.导出了变截面梁单元的单元刚度矩阵、单元等效节点荷载和单元一致质量矩阵.在此基础上,利用编制的程序进行了算例验证与分析.算例验证了本文理论的正确性,表明本文方法具有很高的计算精度.     

AN APPROXIMATE PROJECTION SCHEME FOR INCOMPRESSIBLE FLOW USING SPECTRAL ELEMENTS

An approximate projection scheme based on the pressure correction method is proposed to solve the Navier–Stokes equations for incompressible flow. The algorithm is applied to the continuous equations; however, there are no problems concerning the choice of boundary conditions of the pressure step. The resulting velocity and pressure are consistent with the original system. For the spatial discretization a high-order spectral element method is chosen. The high-order accuracy allows the use of a diagonal mass matrix, resulting in a very efficient algorithm. The properties of the scheme are extensively tested by means of an analytical test example. The scheme is further validated by simulating the laminar flow over a backward-facing step.     

Detection of crack location using cracked beam element method for structural analysis

The local flexibility introduced by cracks changes the dynamic behavior of the structure and, by examining this change, crack position and magnitude can be identified. In order to model the structure for FEM analysis, a special finite element for a cracked Timoshenko beam is developed. Shape functions for rotational and translational displacements are used to obtain the consistent mass matrix for the cracked beam element. Effect of the crack on the stiffness matrix and consistent mass matrix is investigated. Proposed is a procedure for identifying cracks in structures using modal test data.     

A high‐order mass‐lumping procedure for B‐spline collocation method with application to incompressible flow simulations

This paper presents new developments of the staggered spline collocation method for cost‐effective solution to the incompressible Navier–Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high‐accuracy of the B‐spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high‐order generalization of the mass‐lumping technique of the finite‐element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi‐consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier–Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd.     

结构模态分析的有限元并行模拟

以子结构模态综合分析为基础,提出一种求解大型结构特征值问题的并行解法．采用子结构模态综合算法,结构特征模态采用子空间迭代方式并行求解．这种子空间迭代法的子结构并行计算的实施是利用子结构的刚度阵和质量阵而不必完全组集系统刚度阵和质量阵求解综合系统的特征值问题．数值结果表明这种求解大型结构特征值问题的并行算法是可行有效的．     

Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides

We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in the unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with an overlap between the domains. Specific transmission conditions are used, so that at each step of the algorithm only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using a bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized. An original choice of transmission conditions is proposed which enhances the effect of the overlap and allows us to handle arbitrary anisotropic materials. As a by-product, we derive transparent boundary conditions for an arbitrary anisotropic waveguide. The transparency of these new boundary conditions is checked for two- and three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for two- and three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.     

A subdomain boundary element method for high‐Reynolds laminar flow using stream function‐vorticity formulation

The paper presents a new formulation of the integral boundary element method (BEM) using subdomain technique. A continuous approximation of the function and the function derivative in the direction normal to the boundary element (further ‘normal flux’) is introduced for solving the general form of a parabolic diffusion‐convective equation. Double nodes for normal flux approximation are used. The gradient continuity is required at the interior subdomain corners where compatibility and equilibrium interface conditions are prescribed. The obtained system matrix with more equations than unknowns is solved using the fast iterative linear least squares based solver. The robustness and stability of the developed formulation is shown on the cases of a backward‐facing step flow and a square‐driven cavity flow up to the Reynolds number value 50 000. Copyright © 2004 John Wiley & Sons, Ltd.     

Diagonalized consistent mass matrix and the dynamical finite element analysis of elastic-plastic impact in axisymmetrical problems

In this paper, the diagonalized consistent mass matrix is found for the triangular ring element in axisymmetrical problems. The results of this work eliminate the feeling of uncertainty and arbitrariness of lumped mass method on the one hand and the difficulty of computation due to non-diagonalized character of consistent mass method on the other. This paper gives also the foundations of the finite element analysis of elastic-plastic axisymmtrical impact problems.     

Dynamic finite element with diagonalized consistent mass matrix and elastic-plastic impact calculation

There are some common difficulties encountered in elastic-plastic impact codes such as EPIC^[1,2], NONSAP^[3] etc. Most of these codes use the simple linear functions usually taken from static problems to represent the displacement components. In such finite element formulation, the strain and stress components are constants in every element. In the equations of motion, the stress components in general appear in the form of their space derivatives. Thus, if we use such form functions to represent the displacement components, the effect of internal stresses to the equations of motion vanishes identically. The usual practice to overcome such difficulties is to establish as self-equilibrium system of internal forces acting on various nodal points by means of transforming equations of motion into variational form of energy relation through the application of virtual displacement principle. The nodal acceleration is then calculated from the total force acting on this node from all the neighbouring elements. The transformation of virtual displacement principle into the variational energy form is performed on the bases of continuity conditions of stress and displacement throughout the integrated space. That is to say, on the interface boundary of finite element, the assumed displacement and stress functions should be conformed. However, it is easily seen that, for linear form function of finite element calculation, the displacement continues everywhere, but not the stress components. Thus, the convergence of such kind of finite element computation is open to question. This kind of treatment has never been justified even in approximation sense. Furthermore, the calculation of nodal points needs a rule to calculate the mass matrix. There are two ways to establish mass matrix, namely lumped mass method and consistent mass method [4]. The consistent mass matrix can be obtained naturally through finite element formulation, which is consistent to the assumed form functions. However, the resulting consistent mass matrix is not in diagonalized form, which is inconvenient for numerical computation. For most codes, the lumped mass matrix is used, and in this case, the element mass is distributed in certain assumed proportions to all the nodal points of this element. The lumped mass matrix is diagonalized with diagonal terms composed of the nodal mass. However, the lumped mass assumption has never been justified. All these difficulties are originated from the simple linear form functions usually used in static problems.In this paper, we introduce a new quadratic form function for elastic-plastic impact problems. This quadratic form function possesses diagonalized consistent mass matrix, and non-vanishing effect of internal stress to the equations of motion. Thus with this kind of dynamic finite element, all above-said difficulties can be eliminated.