基于Woodbury非线性方法的迭代算法对比分析

在结构局部非线性求解过程中,刚度矩阵仅部分元素发生改变,此时切线刚度矩阵可写成初始刚度矩阵与其低秩修正矩阵和的形式,每个增量步的位移响应可用数学中快速求矩阵逆的Woodbury公式高效求解,但通常情况下迭代计算在结构非线性分析中是不可避免的,因此迭代算法的计算性能也对分析效率有重要影响。该文以基于Woodbury非线性方法为基础,分别采用Newton-Raphson （N-R）法、修正牛顿法、3阶两点法、4阶两点法及三点法求解其非线性平衡方程,并对比分析5种迭代算法的计算性能。利用算法时间复杂度理论,得到了5种迭代算法求解基于Woodbury非线性方法平衡方程的时间复杂度分析模型,定量对比了5种迭代算法的计算效率。通过2个数值算例,从收敛速度、时间复杂度和误差等方面对比了各迭代算法的计算性能,分析了各算法适用的非线性问题。最后,计算了5种算法求解基于Woodbury非线性方法平衡方程的综合性能指标。     

一种适用于求解大规模结构的分步接触算法及其工程应用

大规模结构接触非线性问题的求解是当前工程界研究的热点和难点。该文基于传统的Lagrange乘子法提出了一种新的分步接触算法。该算法的基本原理是将接触问题分两步求解,第一步求解由整体系统力系平衡方程构成的控制方程,第二步求解接触局部区域的约束方程。该算法利用Lagrange乘子来精确模拟接触约束条件,同时对传统的Lagrange乘子法进行了解耦降维处理,所需存储量小、易于实现并行化,且通过引入缩放因子进一步提高了其求解效率,故非常适合高效求解大规模结构的接触问题。经典Hertz接触算例和平面双缝坝算例的结果验证了该算法的正确性,考虑内外衬接触非线性的穿黄隧洞整体模型工程应用算例说明了该算法的有效性。     

复线性方程组的预处理MCG算法

复线性方程组在科学与工程计算的诸多领域中有着重要的应用价值,如何高效的求解复线性方程组,一直是人们所关心的问题．目前对于复线性方程组,常用的处理方式有以下两种：一种是直接对方程组迭代求解,另外一种是将其转化为实线性方程组后进行求解．本文主要从两种处理方式讨论了共轭梯度法（CG法）,并理论上证明了两种处理方式下的CG法具有相同的收敛性．之后基于变形共轭梯度法（MCG法）收敛速度的本质与CG法类似,只需将MCG法推广到复线性方程组进行研究,并且为了提高MCG法的收敛速度,提出了一种预处理MCG法．最后,通过数值算例验证了算法与理论分析的一致性,以及预处理算法的有效性．     

梁的强非线性超、次谐波共振

本采用改进的L—P法研究梁的强非线性振动，分析了超谐波共振和次谐波共振两种情况。首先建立了梁振动的Duffing方程，然后采用改进的L—P法分别求解超谐波响应和次谐波响应，最后给出两个典型算例，其计算结果与增量谐波平衡法和经典的L—P法进行比较。算例表明，改进的L—P法得出的计算值与增量谐波平衡法所得出的结果非常吻合，但是经典的L—P法就失效了。     

求解动态有限元法中二次矩阵特征值问题的迭代摄动法

基于矩阵摄动分析理论，本文建立了一种求解动态有限元法中二次矩阵特征值问题的迭代摄动法，算例结果表明，该方法较以往的求解方法（如二分法和Ｓｔｕｒｍ序列联合法）具有计算工作量少，求解精度高的优点。     

一种新型线性化迭代算法及其在结构优化准则方程组求解中的应用

求解一元非线性方程的埃特金算法是一种线性化自动迭代算法,其每次迭代需要计算两次函数值。将其推广到结构优化非线性准则方程组的迭代求解,可实现结构优化迭代求解的完全自动化。为克服其每次迭代需要两次结构分析的缺点,构造了一种新型线性化迭代解法,称为Atiken Chen算法,该算法利用前次结构分析信息,每次迭代只需一次结构分析,从而大大提高了结构优化迭代计算的效率与自动化程度。算例验证了该算法的可行性和优越性。     

基于改进模拟退火算法的复合材料层合板频率优化

针对复合材料层合板频率优化问题,结合可行规则法和直接搜索模拟退化算法,提出了一种自适应模拟退火(SA)改进算法。层合板优化目标是基频、频率带隙以及给定基频和带隙约束的层合板厚度。设计变量包括铺层角度和铺层数两种离散变量。改进算法的自适应新点产生模块采用依赖温度的动态调整搜索半径,改善了直接搜索模拟退化(DSA)算法易陷入局部极值的缺陷,而可行规则法的引入提高了SA算法求解约束问题的效率和简易性。采用Ritz法进行频率响应分析以考虑弯扭耦合影响。不同铺层数、角度增量和长宽比时的层合板3类算例结果显示:改进算法能有效求解层合板频率优化,可获得更多或更好的铺层顺序全局优化解。     

基于组合时间积分算法的碰撞问题振荡现象修正

该文提出了一种能够有效改善碰撞问题振荡现象的修正方法。基于有限元方法，采用Lagrange乘子法施加接触约束条件，并与一种无条件稳定的隐式组合时间积分算法相结合对无摩擦动接触问题进行求解。通过引入附加Lagrange乘子对组合时间积分算法得到的速度和加速度进行修正，使其能够满足速度和加速度形式的接触约束的一致性条件。数值算例结果表明：该文提出的修正方法能够有效地改善碰撞问题中初始接触时速度、接触力等的振荡现象，进一步提高了碰撞问题的求解精度。     

基于有限元与宽频快速多极边界元的二维流固耦合声场分析

采用有限元/快速多极边界元法进行水下弹性结构的辐射和散射声场分析。Burton-Miller法用于解决传统单Helmholtz边界积分方程在求解外边界值问题时出现的非唯一解的问题。该文采用GMRES和快速多极算法加速求解系统方程。针对传统快速算法在高频处效率低和对角式快速算法在低频处不稳定这一问题,该文通过结合这两种快速算法形成宽频快速算法来克服。同时该文通过观察不同参数条件设置下,宽频快速多极法得到的数值结果在计算精度和计算时间上的变化,得到最优的参数组合值。最后通过数值算例验证该文算法的正确性和有效性。     

弹塑性隐式阻尼迭代法

作者给出了联立求解平衡方程、屈服条件和塑性流动方程的弱解形式, 同时提出了两种位移增量迭代算法:一是以应变增量和塑性因子为独立基本未知量的联立求解迭代算法;二是以应变增量为独立基本未知量的数值求解方法。为了避免弹塑性刚度矩阵的奇异性, 改善迭代收敛速度, 针对上述两种迭代算法本文提出了隐式阻尼迭代法, 并编写了相应脚本文件和计算流程, 并基于有限元程序自动生成系统(FEPG)生成了求解弹塑性问题的有限元源程序。最后, 通过算例验证了所提出的迭代法和计算程序的可靠性和有效性。     

A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method

A new efficient convergence criterion, named the reducible design variable method (RDVM), is proposed to save computational expense in topology optimization. There are two types of computational costs: one is to calculate the governing equations, and the other is to update the design variables. In conventional topology optimization, the number of design variables is usually fixed during the optimization procedure. Thus, the computational expense linearly increases with respect to the iteration number. Some design variables, however, quickly converge and some other design variables slowly converge. The idea of the proposed method is to adaptively reduce the number of design variables on the basis of the history of each design variable during optimization. Using the RDVM, those design variables that quickly converge are not considered as design variables for the next iterations. This means that the number of design variables can be reduced to save the computational costs of updating design variables. Then, the iteration will repeat until the number of design variables becomes 0. In addition, the proposed method can lead to faster convergence of the optimization procedure, which indeed is a more significant time saving. It is also revealed that the RDVM gives identical optimal solutions as those by conventional methods. We confirmed the numerical efficiency and solution effectiveness of the RDVM with respect to two types of optimization: static linear elastic minimization, and linear vibration problems with the first eigenvalue as the objective function for maximization. Copyright © 2012 John Wiley & Sons, Ltd.     

Sequential algorithms for structural design optimization under tolerance conditions

A deterministic optimization usually ignores the effects of uncertainties in design variables or design parameters on the constraints. In practical applications, it is required that the optimum solution can endure some tolerance so that the constraints are still satisfied when the solution undergoes variations within the tolerance range. An optimization problem under tolerance conditions is formulated in this article. It is a kind of robust design and a special case of a generalized semi-infinite programming (GSIP) problem. To overcome the deficiency of directly solving the double loop optimization, two sequential algorithms are then proposed for obtaining the solution, i.e. the double loop optimization is solved by a sequence of cycles. In each cycle a deterministic optimization and a worst case analysis are performed in succession. In sequential algorithm 1 (SA1), a shifting factor is introduced to adjust the feasible region in the next cycle, while in sequential algorithm 2 (SA2), the shifting factor is replaced by a shifting vector. Several examples are presented to demonstrate the efficiency of the proposed methods. An optimal design result based on the presented method can endure certain variation of design variables without violating the constraints. For GSIP, it is shown that SA1 can obtain a solution with equivalent accuracy and efficiency to a local reduction method (LRM). Nevertheless, the LRM is not applicable to the tolerance design problem studied in this article.     

Physical decomposition of thin plate bending problems and their solution by mesh-free methods

We develop and compare a number of mesh-free formulations for solution of plate bending problems using either the Moving Least Square-approximation or Point Interpolation Method-approximation. For solution of the original biharmonic problem, we develop only the local weak formulation to avoid the 4th order derivatives of deflections. The decomposition of the biharmonic operator leads to lower order derivatives of field variables in both the developed strong and local weak formulations which are applicable to arbitrary boundary value problems for thin plate bending. The modified differentiation technique is proposed, in order to increase the accuracy of higher order derivatives of field variables. The accuracy, convergence of accuracy and computational efficiency are studied in simple boundary value problems for circular plate. The discussed methods give reasonable numerical results when applied to decomposed problem, while the methods applied to original biharmonic problem either fail or give unreliable results.     

Fast analytical method describing the postbuckling behavior of long,symmetric, balanced laminated composite plates under biaxial compression and shear

The present work deals with the postbuckling behavior of an infinitely long plate consisting of laminated composites with symmetrical, balanced lay-up. Along the longitudinal edges the plate is elastically clamped with torsional springs. As loading situation combinations of biaxial compression and shear are accounted for. Aiming at high computational efficiency the problem is solved by variational methods employing a shape function with only few variables. Inserting the shape function into the compatibility condition of in-plane strains, a closed-form solution for the Airy stress function can be obtained. The resulting equilibrium condition is then used to derive the load–deflection relationship by means of the Galerkin procedure. All other state variables such as displacements and stresses can then be calculated. Comparative finite element analyses reveal that apart from the cases involving transversal compression, where the application should be handled with care, the present solution procedure can serve to obtain good approximations of the stability behavior in the early postbuckling region with negligible computational effort.     

Selecting Bayesian Acceptance Plans for Quality Control by Pattern Search

This paper reports the results of an experimental evaluation of the effectiveness of pattern search as a solution algorithm for determining optimal Bayesian single sampling acceptance plans for quality control over a broad range of such problems. Since the effectiveness of pattern search is presumably dependent on the choice of the pattern search parameters, the following factors were systematically manipulated as independent variables: (1) starting vector, (2) pattern multiplier, (3) step size, and (4) sophistication of exploratory move (one versus two dimensional). The effectiveness measures (dependent variables) were: (1) % expected total cost obtained by pattern search above the optimal expected total cost, (2) central processing unit (CPU) time, and (3) number of search iterations. While the results showed that solution quality and computational efficiency were affected by some of the search parameters both in terms of main and interaction effects, pattern search as a solution routine in quality control acceptance sampling was shown to be seriously deficient when used in a single application. However, 20 multiple applications of pattern search using single dimensional exploration and varying search parameters overcame this deficiency by efficiently converging on an optimal solution. Such a solution procedure compares favorably to other existing solution methods.     

Multiple reciprocity hybrid boundary node method for potential problems

Meshless methods have some obvious advantages such as they do not require meshes in the domain and on the boundary, only some nodes are needed in the computation. Furthermore, for the boundary-type meshless methods, the nodes are even not needed in the domain and only distributed on the boundary. Practice shows that boundary-type meshless methods are effective for homogeneous problems. But for inhomogeneous problems, the application of these boundary-type meshless methods has some difficulties and need to be studied further.The hybrid boundary node method (HBNM) is a boundary-only meshless method, which is based on the moving least squares (MLS) approximation and the hybrid displacement variational principle. No cell is required either for the interpolation of solution variables or for numerical integration. It has a drawback of ‘boundary layer effect’, so a new regular hybrid boundary node method (RHBNM) has been proposed to avoid this pitfall, in which the source points of the fundamental solutions are located outside the domain. These two methods, however, can only be used for solving homogeneous problems. Combining the dual reciprocity method (DRM) and the HBNM, the dual reciprocity hybrid boundary node method (DRHBNM) has been proposed for the inhomogeneous terms. The DRHBNM requires a substantial number of internal points to interpolate the particular solution by the radial basis function, where approximation based only on boundary nodes may not guarantee sufficient accuracy.Now a further improvement to the RHBNM, i.e., a combination of the RHBNM and the multiple reciprocity method (MRM), is presented and called the multiple reciprocity hybrid boundary node method (MRHBNM). The solution comprises two parts, i.e., the complementary and particular solutions. The complementary solution is solved by the RHBNM. The particular solution is solved by the MRM, i.e., a sum of high-order homogeneous solutions, which can be approximated by the same-order fundamental solutions. Compared with the DRHBNM, the MRHBNM does not require internal points to obtain the particular solution for inhomogeneous problems. Therefore, the present method is a real boundary-only meshless method, and can be used to deal with inhomogeneous problems conveniently. The validity and efficiency of the present method are demonstrated by a series of numerical examples of inhomogeneous potential problems.     

Solution methods and initialization techniques in SFE analysis of structural stability

Computational efficiency is a significant issue in simulation-based stochastic stability analysis of structures with random material properties. The efficiency of the analysis is affected by the selection of eigenproblem solution method and, for a specific eigensolution, how the iterations are initialized. This study applies the inverse and shifted inverse iteration methods to solve the eigensystem for the lowest eigenpair (the smallest buckling load and corresponding buckled shape). The efficiency and accuracy of these two methods are investigated in combination with several initialization strategies to start the solution processes. Among these strategies is a novel tree-type ordering strategy. The impact of the use of the solution methods and initialization strategies on the efficiency of the stochastic stability analysis is investigated and is illustrated in analyses of three systems: a sway frame, a column with a spring support at the mid-span, and a two-bay frame structure.     

A 9-node co-rotational quadrilateral shell element

A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.     

基于改进遗传算法的海底地层参数反演计算

为了提高对海底地层参数变量的反演计算能力,设计了一种基于双种群协同进化策略的改进遗传算法。针对标准遗传算法局部搜索能力差且易于出现早熟现象的缺点,在标准遗传算法基础上引入双种群同时进行优化搜索,两个种群分别给予不同的控制参数,实现协同进化,最终给出一个综合的最优解。通过两个算例对遗传算法的寻优能力进行测试,实验结果表明,提出的改进算法不仅提高了搜索性能,并且对遗传控制参数的依靠度大大降低,特别是对大型复合参数反演问题的求解计算更为有效。     

Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems

In this study, we first discuss the moving least‐square approximation (MLS) method. In some cases, the MLS may form an ill‐conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved moving least‐square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill‐conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation method, a direct meshless BIE method, the boundary element‐free method (BEFM), for two‐dimensional elasticity is presented. Compared to other meshless BIE methods, BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied easily; hence, it has higher computational precision. For demonstration purpose, selected numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd.