Efficient structural optimization using reanalysis and sensitivity reanalysis

Optimization of large-scale structures using conventional formulations often involves much computational effort. Repeated solutions of the analysis and sensitivity analysis equations usually require most of this effort. The computational cost may become prohibitive in large-scale structures having complex analysis models. To alleviate this difficulty, various procedures are integrated in this study into a general optimization approach. The approach is suitable for different classes of response types and optimization methods, including linear and non-linear response; static and dynamic response; direct and gradient optimization methods. Combined approximations are used for reanalysis and repeated sensitivity analysis. The advantage is that the efficiency of local approximations and the improved quality of global approximations are combined to obtain effective solution procedures. Approximate reanalysis and finite-difference sensitivity reanalysis are considered for each intermediate design during the solution process. Reductions in the computational effort may reach several orders of magnitude. Typical numerical examples show that the results achieved by the approach presented are similar to those obtained by exact reanalysis and sensitivity analysis.     

A unified reanalysis approach for structural analysis,design, and optimization

This study presents a unified reanalysis approach for structural analysis, design, and optimization that is based on the Combined Approximations (CA) method. The method is suitable for various analysis models (linear, nonlinear, elastic, plastic, static, dynamic), different types of structures (trusses, frames, grillages, continuum structures), and all types of design variables (cross-sectional, material, geometrical, topological). The calculations are based on results of a single exact analysis. The computational effort is usually much smaller than that needed to carry out a complete analysis of modified designs. Accurate results are achieved by low-order approximations for significant changes in the design. It is possible to improve the accuracy by considering higher-order terms, and exact solutions can be achieved in certain cases. The solution steps are straightforward, and the computational procedures presented can readily be used with general finite element systems. Typical results are demonstrated by numerical examples.     

Efficient reanalysis for topological optimization

An efficient reanalysis method for the topological optimization of structures is presented. The method is based on combining the computed terms of a series expansion, used as high quality basis vectors, and coefficients of a reduced basis expression. The advantage is that the efficiency of local approximations and the improved quality of global approximations are combined to obtain an effective solution procedure.The method is based on results of a single exact analysis and can be used with a general finite element system. It is suitable for different types of structures, such as trusses, frames, grillages, etc. Calculations of derivatives is not required, and the errors involved in the approximations can readily be evaluated.Several numerical examples illustrate the effectiveness of the solution procedure. It is shown that excellent results can be achieved with small computational effort for very large changes in the cross-sections and in the topology of the structure.     

An improved strategy for GAs in structural optimization

An improved strategy for genetic algorithms in structural optimization is presented in this paper. In the improved genetic algorithms, the terms of the feasible, infeasible individual strings, and the related space for the individual strings are given. In initializing the population and generating the individual strings of the next generations, the feasible individual strings are only chosen. The approach to structural approximation analysis by artificial neural networks is also adopted, in order to reduce the expensive computation arising from the constraints evaluations. The effectiveness of the improved strategy for genetic algorithms is shown by the numerical examples of the optimum weights of five- and 10-bar structures.     

Matrix iterative methods for structural reanalysis

The theory of linear, stationary, norm-reducing type iterations for the solution of linear, simultaneous equations is briefly reviewed and the genesis of simple iterition, Jacobi iteration and Gauss-Seidel iteration is shown to be the consequence of ‘splitting’ the coefficient matrix in different ways. For positive definite, sparse matrices arising in structural applications, block Gauss-Seidel iteration is shown to be effective for both reanalysis and initial analysis, through its influence as a norm-reducing aid which results in more pronounced ‘diagonal dominance’ and a better initial choice of starting vector. A numerical example is used to show the effectiveness of the method.     

Non-numerical modeling techniques in structural optimization

Structural optimization encompasses much more than just solving numerical optimization problems. As computer capabilities increase, the entire process of modeling structural optimization problems must be considered. In particular, the creation, transformation and evaluation of the underlying concepts, rather than just brute numerical power, are becoming a more and more dominant factor in finding safe-guarded solutions. In this paper a layer-based model of the total structural optimization process is presented. Each layer contains individual components, a major number of which are of non-numerical nature.Computerization of the non-numerical components requires new programming paradigms. The selection of an appropriate optimization method, to be discussed as a typical non-numerical problem, is difficult because a wide variety of distinct methods exist. Therefore, automated assistance based upon experience and knowledge gained through current research is of prime interest.     

Approximate techniques of strctural reanalysis

A study is made of two approximate techniques for structural reanalysis. These include Taylor series expansions for response variables in terms of design variables and the reduced basis method. In addition, modifications to these techniques are proposed to overcome some of their major drawbacks. The modifications include a rational approach for the selection of the reduced basis vectors and the use of Taylor series approximation in an iterative process. For the reduced basis a normalized set of vectors is chosen which consist of the original analyzed design and the first-order sensitivity analysis vectors.The use of the Taylor series approximation as a first (initial) estimate in an iterative process, can lead to significant improvements in accuracy, even with one iteration cycle. Therefore, the range of applicability of the reanalysis technique can be extended.Numerical examples are presented of space truss structures. These examples demonstrate the gain in accuracy obtained by using the proposed modification techniques, for a wide range of variations in the design variables.     

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.     

An improved critical constraint method for structural optimization of product families

This paper discusses important improvements in the efficient Critical Constraint Method (CCM) for the optimization of structural product families subjected to multiple crash load cases. The method was first presented by Öman and Nilsson (Struct Multidisc Optim 41(5):797–815, 2010). However, the algorithm often converged towards an infeasible solution, which considerably limited the applicability of the method. Therefore, improvements are presented here to make the method more robust regarding feasible solutions, resulting in only a minor decrease in efficiency compared to the original method. The improvements include; a penalty approach to control the feasibility of the method by continuously pushing the solution out of the infeasible region, a dynamic contraction algorithm to increase the accuracy and robustness of the method by considering the optimization progress and variable history in the reduction of the step size, and the implementation of a parallel approach to further increase the efficiency of the method by enabling the full potential of large-scale computer clusters. Finally, the potential of the improved CCM algorithm is demonstrated on a large-scale industrial family optimization problem and it is concluded that the high efficiency of the method enables the usage of large product family optimization in the design process.     

Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods

Structural optimization with frequency constraints is highly nonlinear dynamic optimization problems. Genetic algorithm (GA) has greater advantage in global optimization for nonlinear problem than optimality criteria and mathematical programming methods, but it needs more computational time and numerous eigenvalue reanalysis. To speed up the design process, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented. This reanalysis technique is derived primarily on the Kirsch’s combined approximations method, which is also highly accurate for case of repeated eigenvalues problem. The required number of basis vectors at every generation is adaptively determined and the rules for selecting initial number of basis vectors are given. Numerical examples of truss design are presented to validate the reanalysis-based frequency optimization. The results demonstrate that the adaptive eigenvalue reanalysis affects very slightly the accuracy of the optimal solutions and significantly reduces the computational time involved in the design process of large-scale structures.     

A method for structural optimization which combines secondorder approximations and dual techniques

A structural optimization problem is usually solved iteratively as a sequence of approximate design problems. Traditionally, a variety of approximation concepts are used, but lately second-order approximation strategies have received most interest since high quality approximations can be obtained in this way. Furthermore, difficulties in choosing tuning parameters such as step-size restrictions may be avoided in these approaches. Methods that utilize second-order approximations can be divided into two groups; in the first, a Quadratic Programming (QP) subproblem including all available second-order information is stated, after which it is solved with a standard QP method, whereas the second approach uses only an approximate QP subproblem whose underlying structure can be efficiently exploited. In the latter case, only the diagonal terms of the second-order information are used, which makes it possible to adopt dual methods that require separability. An advantage of the first group of methods is that all available second-order information is used when stating the approximate problem, but a disadvantage is that a rather difficult QP subproblem must be solved in each iteration. The second group of methods benefits from the possibility of using efficient dual methods, but lacks in not using all available information. In this paper, we propose an efficient approach to solve the QP problems, based on the creation of a sequence of fully separable subproblems, each of which is efficiently solvable by dual methods. This procedure makes it possible to combine the advantages of each of the two former approaches. The numerical results show that the proposed solution procedure is a valid approach to solve the QP subproblems arising in second-order approximation schemes.Presented at NATO ASI Optimization of Large Structural Systems, Berchtesgaden, Germany, Sept. 23 – Oct. 4, 1991     

Approximate structural reanalysis based on series expansion

In most optimal design procedures the analysis of the structure must be repeated many times. This operation, which involves much computational effort, is one of the main difficulties in applying optimization methods to large systems. This study deals with approximate reanalysis methods based on series expansion. Both design variables and inverse variables formulations are presented. It is shown that a Taylor series expansion of the nodal displacements or the redundant forces is equivalent to a series obtained from a simple iteration procedure. The series coefficients can readily be computed, providing efficient and high-degree polynomial approximations.To further improve the quality of the approximations, a modified nonpolynomial series is proposed. To reduce the amount of calculations, the possibility of reanalysis along a given line in the variables space is demonstrated. All the proposed procedures require a single exact analysis to obtain an explicit behaviour model along a line.The relationship between the various methods is discussed and numerical examples demonstrate applications. The results obtained are encouraging and indicate that the proposed methods provide efficient and high quality approximations for the structural behavior. This may lead to a wider use of optimization methods in the design of large structural systems.     

Computational efficiency of improved move limit method of sequential linear programming for structural optimization

The imporved move limit method of sequential linear programming is briefly explained. Comparison of computing efficiencies is made between the improved method and the conventional move limit method with six test problems. The usefulness of the method in the context of structural optimization is shown with the help of four examples.     

Benchmarking optimization solvers for structural topology optimization

The purpose of this article is to benchmark different optimization solvers when applied to various finite element based structural topology optimization problems. An extensive and representative library of minimum compliance, minimum volume, and mechanism design problem instances for different sizes is developed for this benchmarking. The problems are based on a material interpolation scheme combined with a density filter. Different optimization solvers including Optimality Criteria (OC), the Method of Moving Asymptotes (MMA) and its globally convergent version GCMMA, the interior point solvers in IPOPT and FMINCON, and the sequential quadratic programming method in SNOPT, are benchmarked on the library using performance profiles. Whenever possible the methods are applied to both the nested and the Simultaneous Analysis and Design (SAND) formulations of the problem. The performance profiles conclude that general solvers are as efficient and reliable as classical structural topology optimization solvers. Moreover, the use of the exact Hessians in SAND formulations, generally produce designs with better objective function values. However, with the benchmarked implementations solving SAND formulations consumes more computational time than solving the corresponding nested formulations.     

Post-optimal procedures for structural optimization

Structural optimization is a very well established design tool in several engineering fields when the problem is formulated with a single objective function and the feasible design region turns out to be convex. Nevertheless, many real problems lead to more complex formulations, sometimes because more than one local minima exist, or because more than one objective function must be included in the formulation. For such cases two procedures intended to enhance the capabilities of design optimization, namely, one approach to global optimization and a recent procedure to obtain sensitivity analysis in multiobjective optimization, are presented in the paper.     

Nonsmooth and nonconvex structural analysis algorithms based on difference convex optimization techniques

The impact of difference convex optimization techniques on structural analysis algorithms for nonsmooth and non-convex problems is investigated in this paper. Algorithms for the numerical solutions are proposed and studied. The relation to more general optimization techniques and to computational mechanics algorithms is also discussed. The theory is illustrated by a composite beam delamination example.     

Curvilinear search for structural optimization

This paper proposes an idea of curvilinear search for nonlinear programming. For the unconstrained and constrained problem, ordinary differential equations are derived to describe descending curves in which values of the objective function decrease. The strategy of the curvilinear search is to repeatedly perform a one-dimensional search at approximate descending curves. The curves are constructed by means of approximate analytical and numerical expressions that are investigated in the paper. The method is applied to solve the optimization problem of the truss structure. Computational results of examples show the efficiency of the method.     

Exact structural reanalysis by a first-order reduced basis approach

It is shown in this article that in truss structures, the first-order (two-term) reduced basis expressions provide exact displacements and stresses in terms of the cross-sectional variables. For changes inm members theexact solution is achieved by considering them first-order basis vectors. Each of the latter vectors is shown to be the constant sensitivity vector multiplied by a scalar variable. The multipliers of the reduced basis equations can readily be determined by solving anm × m set of equations. The main advantages of the method presented are as follows: (a) It is more efficient than the common exact analysis in cases where a limited number of members is changed. (b) Unlike the common approximations, the exact solution is achieved. This is particularly important in cases where the accuracy of the results obtained by approximate methods is not adequate. (c) The method can be used also in cases of changes in the structural layout (topological and geometrical optimization), where approximate methods might provide poor or meaningless results.     

An improved Henry gas solubility optimization for optimization tasks

Applied Intelligence - The henry gas solubility optimization (HGSO) is a new nature-inspired algorithm that mimics Henry Gas Solubility to solve global optimization problems. The main changes of...