Approximate force method reanalysis techniques in structural optimization

Based on the size variable-independence of the equilibrium equations in matrix format, a rational reduced basis reanalysis and an approximate reanalysis method are presented. Because the redundancy of structures is often low, and the decomposed coefficient matrix is known, these techniques can be advantageous in the structural optimization process. Several truss structures are studied for the purpose of numerical illustration.     

Approximate reanalysis in topology optimization

In the nested approach to structural optimization, most of the computational effort is invested in the solution of finite element analysis equations. In this study, the integration of an approximate reanalysis procedure into the framework of topology optimization of continuum structures is investigated. The nested optimization problem is reformulated to accommodate the use of an approximate displacement vector and the design sensitivities are derived accordingly. It is shown that relatively rough approximations are acceptable since the errors are taken into account in the sensitivity analysis. The implementation is tested on several small and medium scale problems, including 2‐D and 3‐D minimum compliance problems and 2‐D compliant force inverter problems. Accurate results are obtained and the savings in computation time are promising. Copyright © 2008 John Wiley & Sons, Ltd.     

A Lanczos-based method for structural dynamic reanalysis problems

In this paper, a new solution method for the modified eigenvalue problem with specific application to structural dynamic reanalysis is presented. The method, which is based on the block Lanczos algorithm, is developed for multiple low rank modifications to a system and calculates a few selected eigenpairs. Given the solution to the original system Ax = λx, procedures are developed for the modified standard eigenvalue Problem (A + ΔA)x? = λ x?, where

• 1 ΔA = Σ_jBS_jB^T, where S_j = S ∈ ?^{p × p}, p ? n and B ∈ ?^{n × p} is constant for all the perturbations S_j.
• 2 ΔA = Σ_iΣ_j B_iS_jB_i^T, where B_i ∈ ?^{n × p} may vary with the pertubations S_j.

The procedures are then extended for the reciprocal and generalized eigenvalue problems so that they are directly applicable to the structural dynamic reanalysis problem. Numerical examples are given to demonstrate the applications of the method.     

A universal method for structural static reanalysis of topological modifications

This paper presents a universal method, iterative combined approximation (iterative CA) approach, for structural static reanalysis of all types of topological modifications. The proposed procedure is basically an approximate two-step method. First, the newly added degrees of freedom (DOFs) are assumed to be linked to the original DOFs of the modified structure by means of the Guyan reduction so as to obtain the condensed equation. Second, the displacements of the original DOFs of the modified structure are solved by using the iterative CA approach. And the displacements of the newly added DOFs resulting from topological modification can be recovered. Four numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural static reanalysis of all types of the topological modifications and it is easy to implement on a computer. Copyright © 2004 John Wiley & Sons, Ltd.     

Sparse matrix factor modification in structural reanalysis

Structural reanalysis problems, such as in nonlinear finite element analysis or optimum design, involve progressive changes in the global stiffness matrix and its matrix factors. Although many studies have been devoted to the subject of matrix factor modification, most investigations have dealt with the problem separately from sparse matrix methods. This paper introduces a graph-theoretic model for the forward solution procedure which is applicable for identifying the modified entries of the matrix factors due to changes in the original matrix. Applications of this graph-theoretic model to existing refactorization methods are presented. The relation between substructuring and sparse matrix ordering strategies, and their effects on reanalysis are discussed. Modification of a sparse matrix associated with an n × n finite element grid ordered by the nested dissection scheme is analysed.     

Approximate direct-search minimax circuit optimization

A direct-search optimization strategy, involving pattern-searches, pattern-moves and a simplex algorithm, has been applied to a number of frequency-domain circuit problems. We show that an approximate minimax result can be obtained with our simple approach, which includes using a modified minimax error function. In the examples attempted, the method comes quite close to the accuracy, if not the efficiency, of true minimax optimizers, and yields an accuracy as good as or better than a least-pth (p = 10) gradient optimizer. The method is broadly appropriate wherever it is difficult or inconvenient to calculate the gradients required for conventional minimax optimization.     

A preconditioned conjugate gradient approach to structural reanalysis for general layout modifications

This paper presents a preconditioned conjugate gradient approach to structural static reanalysis for general layout modifications. It is suitable for all types of layout modifications, including the general case in which some original members and nodes are deleted and other new members and nodes are added concurrently. The approach is based on the preconditioned conjugate gradient technique. The preconditioner is constructed, and an efficient implementation for applying the preconditioner is presented, which requires the factorization of the stiffness matrix corresponding to the newly added degrees of freedom only. In particular, the approach can adaptively monitor the accuracy of approximate solutions. Numerical examples show that the condition number of the preconditioned matrix is remarkably reduced. Therefore, the fast convergence and accurate results can be achieved by the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

Updating the Craig–Bampton reduction basis for efficient structural reanalysis

A Combined Approximation (CA) based reanalysis technique is proposed for updating the static modes in a reduction basis which can be used for sizing optimization problems. Although the proposed technique is utilized under the framework of the Craig–Bampton (CB) method, it can be employed with any condensation procedure that involves the computation of the static modes. An automated update scheme is also presented that switches the proposed technique with the exact analysis when the computational efficiency is lost. Moreover, the Enriched Craig–Bampton (ECB) method is studied for the reanalysis of the normal modes. The ECB‐ and the CA‐based techniques are merged for efficient update of the CB reduction basis. An academic test problem is utilized for the demonstration of the introduced concepts. Copyright © 2010 John Wiley & Sons, Ltd.     

Iterative reanalysis approximation-assisted moving morphable component-based topology optimization method

An Iterative Reanalysis Approximation- (IRA) assisted Moving Morphable Components- (MMCs) based topology optimization is developed (IRA-MMC) in this study. Compared with other classical topology optimization methods, Finite Element-based solver is replaced with the suggested IRA. In this way, the expensive computational cost can be significantly saved by several nested iterations. In the suggested algorithm, a hybrid optimizer based on Method of Moving Asymptotes approach and Globally Convergent version of Method of Moving Asymptotes is suggested to improve convergence ratio and avoid local optimum. Finally, the proposed approach is evaluated by some classical benchmark problems in topology optimization. The results show significant time saving without compromising accuracy.     

移动荷载作用下结构损伤的近似熵分析

当移动荷载接近或远离结构损伤部位时,结构的振动响应的幅值和非线性特征会发生变化,可以从中提取有关结构损伤的信息;近似熵可以表达一个时间序列的复杂性和内在模式。用移动荷载作用于结构上,对其振动响应数据进行近似熵计算,提取其非线性特征值,进而用神经网络进行结构损伤模式识别。通过一个移动荷载作用下简支梁的计算实例考察了这一方法的有效性。移动荷载的作用,使得损伤造成的结构非线性特征更加显著;研究、计算与实测案例表明近似熵能够有效地表征信号的非线性程度,而且对噪声干扰的敏感度低,可以作为神经网络模式识别的特征向量。     

A multiscale strategy for structural optimization

This paper presents a multiscale strategy dedicated to structural optimization. The applications concern the study of geometric details (such as holes, surface profiles, etc) within the structures with frictional contacts. The first characteristic of the method is that it uses a micro–macro approach. This approach is based on a domain decomposition into substructures and interfaces, which involves the resolution of independent ‘micro’ problems in each substructure and transfers ‘macro’ information only through the interfaces. The second characteristic is the use of a multiresolution strategy in order to reduce the computation cost for problems with evolving design parameters. The last characteristic is the capability to model the geometry of details without remeshing thanks to two features: the use of a local enrichment method , and the use of level set functions to easily modify the boundary of the detail during the optimization process. Copyright © 2008 John Wiley & Sons, Ltd.     

Parallel computational strategies for structural optimization

The objective of this paper is to investigate the efficiency of various computational algorithms implemented in the framework of structural optimization methods based on evolutionary algorithms. In particular, the efficiency of parallel computational strategies is examined with reference to evolution strategies (ES) and genetic algorithms (GA). Parallel strategies are implemented both at the level of the optimization algorithm, by exploiting the natural parallelization features of the evolutionary algorithms, as well as at the level of the repeated structural analysis problems that are required by ES and GA. In the latter case the finite element solutions are performed by the FETI domain decomposition method specially tailored to the particular type of problems at hand. The proposed methodology is generic and can be applied to all types of optimization problems as long as they involve large‐scale finite element simulations. The numerical tests of the present study are performed on sizing optimization of skeletal structures. The numerical tests demonstrate the computational advantages of the proposed parallel strategies, which become more pronounced in large‐scale optimization problems. Copyright © 2003 John Wiley & Sons, Ltd.     

结构减振设计的协同优化分析

建立一种适用于工程结构减振设计的协同优化模型。在学科级优化子模型中,分别使用了不同的目标函数;在系统级优化子模型中,以如何获得尽可能接近于子学科独立优化的最优解为目标函数。系统级目标函数消除了量纲和数量级的影响。该模型采用分布协同的双层结构,允许在每一级优化中使用不同的优化策略。以减速器和板梁柱混合结构为例,对其进行了单学科和多学科优化分析。实例结果表明,在结构减振设计时,采用协同优化模型进行优化设计,不仅可以有效的降低结构动力特性参数,而且优化后结构的综合性能更佳,说明该模型能应用于实际工程结构的减振设计。     

Bilateral filtering for structural topology optimization

Filtering has been a major approach used in the homogenization‐based methods for structural topology optimization to suppress the checkerboard pattern and relieve the numerical instabilities. In this paper a bilateral filtering technique originally developed in image processing is presented as an efficient approach to regularizing the topology optimization problem. A non‐linear bilateral filtering process leads to a suitable problem regularization to eliminate the checkerboard instability, pronounced edge preserving smoothing characteristics to favour the 0–1 convergence of the mass distribution, and computational efficiency due to its single pass and non‐iterative nature. Thus, we show that the application of the bilateral filtering brings more desirable effects of checkerboard‐free, mesh independence, crisp boundary, computational efficiency and conceptual simplicity. The proposed bilateral technique has a close relationship with the conventional domain filtering and range filtering. The proposed method is implemented in the framework of a power‐law approach based on the optimality criteria and illustrated with 2D examples of minimum compliance design that has been extensively studied in the recent literature of topology optimization and its efficiency and accuracy are highlighted. Copyright © 2005 John Wiley & Sons, Ltd.     

Response surface approximations for structural optimization

Response surface methodology can be used to construct global and midrange approximations to functions in structural optimization. Since structural optimization requires expensive function evaluations, it is important to construct accurate function approximations so that rapid convergence may be achieved. In this paper techniques to find the region of interest containing the optimal design, and techniques for finding more accurate approximations are reviewed and investigated. Aspects considered are experimental design techniques, the selection of the ‘best’ regression equation, intermediate response functions and the location and size of the region of interest. Standard examples in structural optimization are used to show that the accuracy is largely dependent on the choice of the approximating function with its associated subregion size, while the selection of a larger number of points is not necessarily cost-effective. In a further attempt to improve efficiency, different regression models were investigated. The results indicate that the use of the two methods investigated does not significantly improve the results. Finding an accurate global approximation is challenging, and sufficient accuracy could only be achieved in the example problems by considering a smaller region of the design space. © 1998 John Wiley & Sons, Ltd.     

Solving optimization problems by searching along a curve

Abstract

This paper proposes a curve for the one‐dimensional search method in optimization problems. The search is along a curve instead of a line. This curve is determined by a parameter [u] from the following three points: (1) the initial point X_o, (2) the Cauchy point X_c, (3) the Newton point X_N, and has the following four characteristics: (1) tangents to the steepest descent direction at X_o, (2) passes through X_N, (3) decreases monotonically from X_o to X_N for a quadratic function, (4) no complex computation on the parameter u. Therefore, the proposed method has the following advantages: (1) it is globally convergent, (2) it is locally q‐quadratically (or q‐superlinearly for quasi‐Newton point) convergent, (3) the search procedure is as simple as the line search method.     

Automated structural optimization system for integrated topology and shape optimization

Abstract

This paper combines previously developed techniques for image‐preprocessing and characteristic image‐interpreting together with a newly proposed automated shape‐optimization modeling technique into an integrated topology‐optimization and shape‐optimization system. As a result, structure designers are provided with an efficient and reliable automated structural optimization system (ASOS). The automated shape‐optimization modeling technique, the key technique in ASOS, uses hole‐expanding strategy, interference analysis, and hole shape‐adjusting strategy to automatically define the design variables and side constraints needed for shape optimization. This technique not only eliminates the need to manually define design variables and side constraints for shape optimization, but during the process of shape optimization also prevents interference between the interior holes and the exterior boundary. The ASOS is tested in three different structural configuration design examples.     

Reverse adaptivity—a new evolutionary tool for structural optimization

This paper presents a new method, Reverse Adaptivity, for automatically generating solutions to initial design and redesign problems. The method is based on a combination of existing adaptive finite element methods and evolutionary structural optimization methods. The usual difficulties inherent in structural optimization problems, and the shortcomings of the evolutionary methods in tackling these difficulties, are reviewed as a prelude to discussing Reverse Adaptivity. Once the initial finite element problem is defined, the method proceeds with reverse adaptive analysis, which refines low stress regions of the finite element mesh by element subdivision. Following this, any low stress subdivided elements are removed and the process is repeated. With successive decrements of adapted element size, the process satisfies many of the shortcomings of existing evolutionary optimization methods, yet is simple to understand and can be readily implemented. The results produced by the method are superior to those produced by existing methods, yet can be obtained with highly practicable computational resources. As a demonstration, solutions to a number of well‐ known classical problems are presented, and highlight the method's ability to distinguish new classes of solutions for some problems. Full implementation and parameter details are also presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Sequential Approximate Optimization using Radial Basis Function network for engineering optimization

This paper presents a Sequential Approximate Optimization (SAO) procedure that uses the Radial Basis Function (RBF) network. If the objective and constraints are not known explicitly but can be evaluated through a computationally intensive numerical simulation, the response surface, which is often called meta-modeling, is an attractive method for finding an approximate global minimum with a small number of function evaluations. An RBF network is used to construct the response surface. The Gaussian function is employed as the basis function in this paper. In order to obtain the response surface with good approximation, the width of this Gaussian function should be adjusted. Therefore, we first examine the width. Through this examination, some sufficient conditions are introduced. Then, a simple method to determine the width of the Gaussian function is proposed. In addition, a new technique called the adaptive scaling technique is also proposed. The sufficient conditions for the width are satisfied by introducing this scaling technique. Second, the SAO algorithm is developed. The optimum of the response surface is taken as a new sampling point for local approximation. In addition, it is necessary to add new sampling points in the sparse region for global approximation. Thus, an important issue for SAO is to determine the sparse region among the sampling points. To achieve this, a new function called the density function is constructed using the RBF network. The global minimum of the density function is taken as the new sampling point. Through the sampling strategy proposed in this paper, the approximate global minimum can be found with a small number of function evaluations. Through numerical examples, the validities of the width and sampling strategy are examined in this paper.     

Game formulations for structural optimization under uncertainty

We consider structural optimization (SO) under uncertainty formulated as a mathematical game between two players –– a “designer” and “nature”. The first player wants to design a structure that performs optimally, whereas the second player tries to find the worst possible conditions to impose on the structure. Several solution concepts exist for such games, including Stackelberg and Nash equilibria and Pareto optima. Pareto optimality is shown not to be a useful solution concept. Stackelberg and Nash games are, however, both of potential interest, but these concepts are hardly ever discussed in the literature on SO under uncertainty. Based on concrete examples of topology optimization of trusses and finite element-discretized continua under worst-case load uncertainty, we therefore analyze and compare the two solution concepts. In all examples, Stackelberg equilibria exist and can be found numerically, but for some cases we demonstrate nonexistence of Nash equilibria. This motivates a view of the Stackelberg solution concept as the correct one. However, we also demonstrate that existing Nash equilibria can be found using a simple so-called decomposition algorithm, which could be of interest for other instances of SO under uncertainty, where it is difficult to find a numerically efficient Stackelberg formulation.