Block diagonalization of adjacency and Laplacian matrices for graph product; applications in structural mechanics

Eigenvalues and eigenvectors of graphs have many applications in structural mechanics and combinatorial optimization. For a regular space structure, the visualization of its graph model as the product of two simple graphs results in a substantial simplification in the solution of the corresponding eigenproblems. In this paper, the adjacency and Laplacian matrices of four graph products, namely, Cartesian, strong Cartesian, direct and lexicographic products are diagonalized and efficient methods are obtained for calculating their eigenvalues and eigenvectors. An exceptionally efficient method is developed for the eigensolution of the Laplacian matrices of strong Cartesian and direct products. Special attention is paid to the lexicographic product, which is not studied in the past as extensively as the other three graph products. Examples are provided to illustrate some applications of the methods in structural mechanics. Copyright © 2006 John Wiley & Sons, Ltd.     

Geometrically nonlinear analysis of circulant structures using an efficient eigensolution method

In this paper, an efficient method is presented for the geometrically nonlinear analysis of circulant structures. A structure is called regular if its model can be formed as the product of two subgraphs, and if one of the subgraphs is a cycle then it is termed circulant. In the present method, we deal with the eigensolution of circulant structures by simply finding the eigenvalues and eigenvectors of the blocks of the stiffness matrix rather than those of the entire structural matrix, leading to a considerable reduction in the computational time. The developed method utilizes concepts from product graphs and linear algebra for the nonlinear analysis of structures. Graph products are used to perform the eigensolution of circulant structures through those of its constituting subgraphs. Here, using the existing numerical methods, nonlinear static and dynamic analyses can be performed. In the presented method instead of solving the characteristic equation in each iteration of each step, the eigensolution of the structure is obtained using the eigensolution of the primary structure (the structure of the first iteration of the first step) together with some simple mathematical operations. In the present method instead of heavy matrix operations such as inverting a matrix and solution of the corresponding equations, one needs only to perform simple matrix multiplications and additions. The advantage of the presented method becomes more apparent when it is applied to the nonlinear analysis of a structure, where analysis should be performed many times.     

An efficient method for decomposition of regular structures using graph products

In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. A structural model is called regular if they can be viewed as the direct or strong Cartesian product of some simple graphs known as their generators. The eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigenvalues of its generators. The second eigenvalue of the Laplacian of a graph is also obtained using a much faster and much simple approach than the existing methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Factorization for efficient solution of eigenproblems of adjacency and Laplacian matrices for graph products

Many structural models can be generated as the graph products of two or three subgraphs known as their generators. The main types of graph products consist of Cartesian, strong Cartesian, direct, and lexicographic products. In this paper, a general method is presented for the factorization of these graph products, such that the eigenvalues of the entire graph are obtained as the union of the eigenvalues of the weighted subgraphs defined here. The adjacency and Laplacian matrices for each graph product are studied separately. For graphs with missing elements (cut‐outs), the eigenvalues are calculated with the additional use of the Rayleigh quotient approach. The main idea stems from the rules recently developed by the authors for block diagonalization of matrices. These products have many applications in computational mechanics, such as ordering, graph partitioning, dynamic analysis, and stability analysis of structures. Some of these applications are addressed in this paper. Copyright © 2007 John Wiley & Sons, Ltd.     

Inverse Sub‐Structuring Method for Multi‐Coordinate Coupled Product Transport System

A new high‐accuracy transfer function is selected, and an inverse sub‐structuring method is developed for the analysis of the dynamic characteristics of a three‐sub‐structure coupled product transport system. The closed‐form analytical solution to inverse sub‐structuring analysis of multi‐coordinate coupled multi‐ sub‐structure product transport system is derived. The proposed method is validated by a lumped mass spring damper model; the predicted frequency response functions (FRFs) of sub‐structures and the coupling stiffness, in addition to the most concerned system‐level FRF, are compared with the direct computations, showing exact agreement. Then, FRF tests of a physical prototype of the multi‐coordinate coupled product transport system with three sub‐structures are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown sub‐structure‐level FRFs and coupling stiffness purely from system‐level FRFs. The suggested method may help obtain the main controlling factors and contributions from the various structure‐borne paths for the product transport system, which may certainly facilitate the cushioning packaging design. Copyright © 2013 John Wiley & Sons, Ltd.     

Seismic response analysis of nonlinear structures

A solution method for the response of a class of nonlinear viscoelastic shear building structures subjected to stochastic excitation has been developed by means of a stochastic equivalent linearlization technique. The nonlinear viscoelastic properties of the structures are modelled in terms of an equation of motion which linearly involves the auxiliary variables as part of the restoring force and in terms of the auxiliary equation which describes a nonlinear relationship among the story displacements and auxiliary variables and their time derivatives. This auxiliary equation is linearized using a stochastic linearization technique. The integration of the equation of motion together with the linearized auxiliary equation is carried out numerically using the state-vector formulation. In doing so, an iterative upgrading of the values of the linearization coefficients is performed simultaneously for all the stories in the first time interval until a convergence criterion is satisfied. The iterative process is then repeated in the time intervals that follow, until the entire time interval for which the dynamic analysis is performed is covered. The analysis is based on the modal method which, however, requires the use of complex eigenvalues and eigenvectors. The proposed analysis produces covariance functions of the story displacements and velocities among other response quantities. These covariance functions play an important role in estimating structural reliability. The covariance function thus developed agrees very well with that obtained by the Monte Carlo technique.     

一种结构静力重分析方法及其在层合板逐次失效分析中的应用

本文提出了一种结构静力重分析方法,通过引入结构刚体位移特征向量,可以消除刚度矩阵的奇异性,从而求出结构广义柔度矩阵,位移一般解可在边界条件尚未引入之前导出。对于给定载荷及边界条件的静力问题,可以通过求解一个阶数较小的线性系统来获得位移解。对于局部修改的结构,相对于新结构的广义柔度矩阵可以迅速加以修正。这种静力重分析方法可以用于载荷、边界条件及结构单元同时或分别改变时的静力重分析问题。 之后,这种重分析方法被应用于复合材料层合板逐次失效分析,用以进行受破坏结构的刚度重分析。用有限元法求得各铺层的应力之后,引入破坏准则来判断铺层失效及失效模式,在本构方程中引入三个参数来修改失效铺层的刚度。层合板的破坏扩展及极限强度通过逐渐增加载荷迭代求得。文中提供了多个算例,计算结果与实验数据及文献吻合良好。     

Lifetime cost optimization with time-dependent reliability

Product lifetime cost is largely determined by product lifetime reliability. In product design, the former is minimized while the latter is treated as a constraint and is usually estimated by statistical means. In this work, a new lifetime cost optimization model is developed where the product lifetime reliability is predicted with computational models derived from physical principles. With the physics-based reliability method, the state of a system is indicated by computational models, and the time-dependent system reliability is then predicted for a given set of distributions and stochastic processes in the model input. A sampling approach to extreme value distributions of input stochastic processes is employed to make the system reliability analysis efficient and accurate. The physics-based reliability analysis is integrated with the lifetime cost model. The integration enables the minimal lifetime costs including those of maintenance and warranty. Two design examples are used to demonstrate the proposed model.     

Inverse Sub‐structuring Method for Rigidly Coupled Product Transport System based on Frequency Response Function Testing Probe Technique

The inverse sub‐structuring method has been recently proposed and applied for inverse analysis of product transport system, to predict the component‐level frequency response functions (FRFs) and the coupling dynamic stiffness from only the system‐level FRFs. However, previous applications of this method were all developed based on the assumption that the components were coupled by flexible couplings. Actually, increasing more components are welded or bolted to construct a coupled system, which should be treated as rigidly coupled system. The aim of this paper is to derive a new FRF‐based inverse sub‐structuring method for the analysis of the dynamic characteristics of a two‐component coupled product transport system with rigid couplings. And then a so‐called FRF testing probe technique is proposed and applied to measure the difficult‐to‐monitor FRFs at the coupling interface. The developed method is verified by a lumped‐mass model, showing exact agreement. Finally, the experiment on a physical prototype of two‐substructure coupled product transport system is performed to further check the accuracy of the suggested method. The proposed method is an extension of previous inverse sub‐structuring method and may help to obtain the main controlling factors and contributions from the various structure‐borne paths for product transport system. Copyright © 2016 John Wiley & Sons, Ltd.     

A refined global-local finite element analysis method

A refined global-local method was proposed to improve the efficiency of finite element analysis. The proposed method was based on the regular finite element method in conjunction with three basic step, i.e. the global analysis, the local analysis and the refined global analysis. In the first two steps, a coarse finite element mesh was used to analyse the entire structure to obtain the nodal displacements which were subsequently used as displacement boundary conditions for local regions of interest. These local regions with the prescribed boundary conditions were then analysed with refined meshes to obtain more accurate stresses. In the third step, a new global displacement distribution based on the results of the previous two steps was assumed for the analysis, from which much improved solutions for both stresses and displacements were produced. Numerical examples showed that the proposed method yielded accurate solutions with significant savings in computing time compared with the regular finite element method. Further, this method is suitable for parallel computation.     

A method for modal reanalysis of topological modifications of structures

A method for structural modal reanalysis for three cases of topological modifications, the number of degrees of freedom (DOFs) is unchanged, decreased, and increased, is presented. In this method, the newly added DOFs are linked to the original DOFs of the modified structure by means of the dynamic reduction so as to obtain the condensed equation. The methods for forming the stiffness and mass increments, Δ K and Δ M , resulting from the three cases of topological modifications of structures are discussed. The extended Kirsch method is used to improve the accuracy of the starting solutions of the initial structure. And then, the eigenvectors of newly added DOFs resulting from topological modification can be recovered. At last, the Rayleigh–Ritz analysis is used to evaluate the eigenvalues and eigenvectors for the modified structure. Three numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural modal reanalysis of three cases of the topological modifications and it is easy to implement on a computer. By comparing with previous method, it can be seen that the present method can give good approximate eigenvalues and eigenvectors, even if the topological modifications are very large. Copyright © 2005 John Wiley & Sons, Ltd.     

基于眼动追踪的产品族设计基因识别强度研究

目的 对小米品牌产品族中多品类的产品设计开展案例研究,探索其设计基因的构成特性,研究分析产品设计基因的识别强度差异性,进而初步形成产品设计基因的提取及构建方法,以及识别强度分析方法。方法 首先构建产品设计基因识别强度对比方法,通过卡片分类法、聚类分析法和多维尺度分析法筛选出典型样本;其次由问卷分析和设计形态分析提取典型基因;最后结合眼动追踪和熵权法综合评价基因识别强度差异。结果 形成小米品牌产品（家电与智能类）因子库,包含通用型基因5项,专用型基因18项,且通用型基因的识别强度高于专用型基因。结论 初步构建了基于眼动追踪的产品族设计基因识别强度研究方法,可以此作为基础规划及管理产品设计方案,使产品设计兼顾品牌基因的延续性及单品设计的合理性;同时还可为生态链企业研发同品牌、不同品类产品提供设计策略与参考。     

约束对象的产品族模型知识表示及配置方法

针对目前产品族模型存在冗余、配置效率低、不易更新的问题,研究了基于约束对象的产品族模型的知识表示方法,有效地降低了产品结构冗余以及配置的复杂度,实现模型知识的有效封装.在此基础上研究了基于约束和实例的配置推理方法,给出了具体的配置推理过程.最后,给出了一个摩托车产品族建模的案例.该模型及其推理方法能够方便地利用计算机实现,可用于建立一个高效的产品族配置系统.     

柔性组合结构的大挠度非线性有限元分析

本文应用预应力柔性索的非线性有限元模型,用折算弹性模量考虑索初始垂度的影响,引入主一从控制原理来克服柔性元素和刚性元素共同作用时易产生的方程病态,在结构变位后的基础上建立平衡方程,采用带有动坐标的刚度矩阵迭代法,分析了斜拉桥、悬索桥等柔性组合结构的大挠度问题。应用本文方法,已经成功地分析了多座斜拉桥实际结构。对于一般柔性组合结构,经过二至六次迭代即可达到相当的精度。     

STRIP DISTRIBUTED TRANSFER FUNCTION METHOD FOR ANALYSIS OF PLATES

A semi-analytical method, called the strip distributed transfer function method, is developed for analysis of plate structures that are composed of rectangular plates. In the method, a rectangular plate (substructure) is divided into a number of strips; the response of each strip is interpolated in the unknown nodal line displacements, which are functions of the strip longitudinal co-ordinate and time. The nodal line displacements are determined in an exact and closed form by the distributed transfer functions that are defined along the strips. Synthesis of the substructures using the strip distributed transfer functions yields accurate prediction of the static and dynamic response, natural frequencies and buckling loads of the structure. The proposed method is compared with some existing techniques in numerical examples.     

基于频响函数的结构健康监测主成分分析法

摘 要：基于测试频响函数,提出一种简单而有效的结构健康监测主成分分析（PCA）新方法。以结构的频响函数作为基本数据,首先将结构健康状态下的频响函数数据作为基本训练样本,通过PCA技术提取结构健康状态特征,并获得结构健康特征变换矩阵,即协方差的特征向量矩阵;然后再对损伤结构的测试频响函数数据进行转换以提取结构相应损伤状态特征;最后在二维PCA空间比较两次提取的结构状态特征分布图即可判断结构是否发生损伤并评估其损伤程度。两个数值算例表明基于频响函数的结构健康监测主成分分析新方法正确有效。该方法基于结构振动响应,与模型无关且诊断前无需大量的训练样本、计算量小、抗噪性能好,具有良好的应用前景。     

Design oriented analysis of imperfect truss structures—part II—approximate analysis

An approximate analysis method for calculating local and system geometric imperfection influences on the displacements, stresses and buckling load of imperfect truss structures, is developed. The structural response, as a function of the design variables, is expressed in this approximate analysis which is much faster and cheaper than the ‘accurate’ analysis. The accurate analysis for a particular design is used as a base for each approximate analysis which then gives the response information needed for other designs in the neighbourhood about the base point design. Comparison of the approximate behaviour predictions with accurate results based on the more refined analysis methods shows that the quality of the proposed approximate analysis procedure is adequate for guiding a structural design procedure. It is evident that the approximate analysis is capable of describing the nonlinear behaviour of the structure, but when nonlinearity increases, the quality of the approximate analysis tends to deteriorate.     

多点输入下大跨空间索结构的虚拟激励法

采用虚拟激励法分析大跨空间索结构的随机地震响应，从结构节点位移响应方差和加速度功率谱两个方面考查了一致输入与非一致输入下结构响应的差异，得出了一些重要结论。同时，计算还表明虚拟激励法是一种高效且精确的方法，对于分析振型密集的大跨空间索结构在多点输入下的地震响应是很适合的。     

Eigenvalue reanalysis by improved perturbation

A new efficient method of eigenvalue reanalysis has been developed based on the perturbation method such that first the differential equations which describe the change of the eigenvalues and eigenvectors of a system being modified are derived from the perturbation method and then the solutions of these equations are obtained by following them up with the solutions of more simple equations. The special features of this method are that it can give a clear correspondence between the eigenpairs before and after system modification, and that the computer run time spent by this method is far shorter than that of revised analysis by FEM when only a few selected modes are used in the form of an approximate computation. In addition, this approximate computation can give more accurate results than the perturbation method. The efficiency of the proposed method has been verified by applying it to a simple spring-mass model as well as a 600 degrees-of-freedom model which receive structural modification.     

多阶段预应力空间网架结构一次张拉计算法

该文提出多阶段预应力空间网架结构一次张拉计算法,将某一阶段第i次对i预应力杆增加的预应力值作为未知数,通过结构力学方法建立该阶段预应力准则方程式并求解之。然后可计算该阶段i次预应力各预应力杆的控制值(包括各预应力杆原有的内力值),后张预应力对先张预应力的影响值,以及网架结构杆件内力和节点变位,直至该阶段最后一次预应力时得到该阶段各预应力杆的设计内力值及网架结构内力和变位。重复这样的分析计算可得到多阶段预应力网架结构预应力杆的最终内力控制值及网架杆件内力和节点变位。采用该方法,预应力杆的内力值在预应力张拉过程中无需调整,可一次性完成预应力施加操作过程。该文中附有算例说明,计算结果表明:提出的多阶段预应力空间网架结构一次张拉计算法是正确、可靠和有效的。