Large scale structural optimization: Computational methods and optimization algorithms

Summary The objective of this paper is to investigate the efficiency of various optimization methods based on mathematical programming and evolutionary algorithms for solving structural optimization problems under static and seismic loading conditions. Particular emphasis is given on modified versions of the basic evolutionary algorithms aiming at improving the performance of the optimization procedure. Modified versions of both genetic algorithms and evolution strategies combined with mathematical programming methods to form hybrid methodologies are also tested and compared and proved particularly promising. Furthermore, the structural analysis phase is replaced by a neural network prediction for the computation of the necessary data required by the evolutionary algorithms. Advanced domain decomposition techniques particularly tailored for parallel solution of large-scale sensitivity analysis problems are also implemented. The efficiency of a rigorous approach for treating seismic loading is investigated and compared with a simplified dynamic analysis adopted by seismic codes in the framework of finding the optimum design of structures with minimum weight. In this context a number of accelerograms are produced from the elastic design response spectrum of the region. These accelerograms constitute the multiple loading conditions under which the structures are optimally designed. The numerical tests presented demonstrate the computational advantages of the discussed methods, which become more pronounced in large-scale optimization problems.     

Level-set methods for structural topology optimization: a review

This review paper provides an overview of different level-set methods for structural topology optimization. Level-set methods can be categorized with respect to the level-set-function parameterization, the geometry mapping, the physical/mechanical model, the information and the procedure to update the design and the applied regularization. Different approaches for each of these interlinked components are outlined and compared. Based on this categorization, the convergence behavior of the optimization process is discussed, as well as control over the slope and smoothness of the level-set function, hole nucleation and the relation of level-set methods to other topology optimization methods. The importance of numerical consistency for understanding and studying the behavior of proposed methods is highlighted. This review concludes with recommendations for future research.     

Structure topology optimization: fully coupled level set method via FEMLAB

This paper presents a procedure which can easily implement the 2D compliance minimization structure topology optimization by the level set method using the FEMLAB package. Instead of a finite difference solver for the level set equation, as is usually the case, a finite element solver for the reaction–diffusion equation is used to evolve the material boundaries. All of the optimization procedures are implemented in a user-friendly manner. A FEMLAB code can be downloaded from the homepage www.imtek.de/simulation and is free for educational purposes.     

Performance comparison of SAM and SQP methods for structural shape optimization

This paper presents a numerical performance comparison of a modern version of the well-established sequential quadratic programming (SQP) method and the more recent spherical approximation method (SAM). The comparison is based on the application of these algorithms to examples with nonlinear objective and constraint functions, among others: weight minimization problems in structural shape optimization. The comparison shows that both the SQP and SAM-algorithms are able to converge to accurate minimum weight values. However, because of the lack of a guaranteed convergence property of the SAM method, it exhibits an inability to consistently converge to a fine tolerance. This deficiency is manifested by the appearance of small oscillations in the neighbourhood of the solution.     

Performance comparison of SAM and SQP methods for structural shape optimization

This paper presents a numerical performance comparison of a modern version of the well-established sequential quadratic programming (SQP) method and the more recent spherical approximation method (SAM). The comparison is based on the application of these algorithms to examples with nonlinear objective and constraint functions, among others: weight minimization problems in structural shape optimization. The comparison shows that both the SQP and SAM-algorithms are able to converge to accurate minimum weight values. However, because of the lack of a guaranteed convergence property of the SAM method, it exhibits an inability to consistently converge to a fine tolerance. This deficiency is manifested by the appearance of small oscillations in the neighbourhood of the solution.     

Direct versus indirect methods in structural optimization

Mathematical programming methods are among the most powerful optimization techniques. These techniques may be separated into direct and indirect methods. Of the direct methods of attack on general nonlinear inequality constrained problems, the largest class is the method of feasible directions. Of the indirect methods, the interior penalty function appears to be the most reliable one while the variable metric method seems to be an extremely powerful algorithm. This paper presents a comparison between the results obtained using Zoutendijk's method of feasible directions and the method of interior penalty function coupled with the variable metric method as a minimizing algorithm. A considerable improvement in convergence has been achieved by considering each push-off factor as a linear function of the corresponding active constraint. A comparison of the half-step vs full-step search procedure is presented. Also a comparison between the use of either the normalized or the non-normalized gradients is illustrated. A discussion of the linear vs quadratic interpolations of a constraint function in search for a bound point is presented. An initial step length based on a present decrement of objective function is used. The two algorithms are demonstrated with elastic design of a 25-bar space tower, a 3-bay single-storey frame and a double-bay double-storey rigid jointed plane frame. Data on the differences in the optimal designs obtained from different starting points are reported.     

A general and flexible method of problem definition in structural optimization systems

This paper describes the implementation of a general and flexible method of formulating problems of mathematical programming in structural optimization systems. The method enables the formulation and solution of problems involving scalar, integral, min/max, max/min and possibly non-differentiable user defined functions in any conceivable mix. The mathematical formulation is based on the bound formulation, and the implementation specific details involve a parser capable of interpreting and performing symbolic differentiation of the user defined functions.     

On design sensitivities in the structural analysis and optimization of flexible multibody systems

Multibody System Dynamics - The structural analysis and optimization of flexible multibody systems become more and more popular due to the ability to efficiently compute gradients using...     

Minimization of sound radiation in fully coupled structural-acoustic systems using FEM-BEM based topology optimization

This paper presents a method to locally constrain multiple material volume domains for structural optimization with the Level Set Method (LSM). Two different Lagrangian formulations and multiplier update methods are used, for both the global and local problem. The local volume domains can be constrained by both equality and inequality constraints. The optimization objective is compliance minimization for well-posed statically loaded structures. For validation, several example problems are established and solved using the proposed method. Results show that the volume ratios for user established sub-domains can be controlled successfully. The local constraint values are met accurately in the case of equality constraints and remain in their feasible domain in the case of inequality constraints. Optimization results are not significantly hindered by the introduction of local volume constraints for comparable problems.     

A critical comparative assessment of differential equation-driven methods for structural topology optimization

In recent years, differential equation-driven methods have emerged as an alternate approach for structural topology optimization. In such methods, the design is evolved using special differential equations. Implicit level-set methods are one such set of approaches in which the design domain is represented in terms of implicit functions and generally (but not necessarily) use the Hamilton-Jacobi equation as the evolution equation. Another set of approaches are referred to as phase-field methods; which generally use a reaction-diffusion equation, such as the Allen-Cahn equation, for topology evolution. In this work, we exhaustively analyze four level-set methods and one phase-field method, which are representative of the literature. In order to evaluate performance, all the methods are implemented in MATLAB and studied using two-dimensional compliance minimization problems.     

Global optimization of structural systems using two new methods

Two new methods for finding global minimum points are applied to several truss and frame structural design problems. Basic ideas and procedures of the methods are explained. One of the methods finds many local minimum solutions while the second method is concerned mainly with finding only the global solution (however, it may find other local minima as well). A set of twenty-eight structural design test problems is devised by using two materials (steel and aluminium) and three cross-sectional shapes. The problems are solved with the two methods, and the results are analysed. Comparisons are made for the results with the truss and frame model for the same structure, use of steel or aluminium as the construction material, and use of different cross-sectional shapes. It is shown that many structural optimization problems possess several local minima, and so the global optimization methods can be useful for their solution. However, the computational effort to find a global solution can be substantial because many local minimizations must be performed. It is concluded that both the methods are useful for finding global minimum solutions for structural optimization problems.Notation A _i Cross-sectional area of thei-th member - i A parameter relating the area and moment of inertia of thei-th member - i, iO Calculated displacement and its limit for thei-th degree of freedom - A reduction factor used in the zooming method - E Young's modulus - F _e ^l Euler stress divided by a factor of safety - F _G ^* The global minimum value for the cost function - F _W ^* Cost function value at the worst local minimum point - f _a,F _a Calculated and allowable axial stresses - f _b,F _b Calculated and allowable bending stresses - f _s,F _s Calculated and allowable shear stresses - F _y Yield stress - F _yc,F _yt,F _ys Yield stress in compression, tension and shear, respectively - h Total height of anI-beam - I _i Moment of inertia of thei-th member - L _i Length of thei-th member - R Mean radius of the tube - i Material density of thei-th member - t Thickness of the tube - t _f Flange thickness of anI-beam - F Target level for the cost function at the global minimum point - t _w Web thickness of anI-beam - , _L Natural frequency of the structure and its lower limit - Flange width of anI-beam - W Weight of the structure in kips - x Design variable vector of dimensionn - x^* A local minimum point for the cost function - x _G ^* A global minimum point for the cost function     

Application of methods of feasible directions to structural optimization problems

A general approach to structural optimization which has received much attention in recent years is that of using mathematical programming (numerical search) techniques. These techniques may be separated into direct and indirect methods. Of the direct methods of attack on general nonlinear inequality constrained problems, the largest class is called methods of feasible directions. This paper presents the application fo Zoutendijk's method of feasible directions [5] to structural optimization problems. The algorithm requires the analytic gradient of the objective function and the constraint functions which are active at a given stage in the design process. A considerable improvement in convergence has been achieved by considering each pushoff factor as a linear function of the corresponding active constraint. A comparison of the half-step versus full-step search procedure is presented. An initial step length based on a present decrement of objective function is used. A discussion of the linear versus quadratic interpolations of a constraint function in search for a bound point is presented. The algorithm is demonstrated with elastic design of a 25-bar space tower, a 3-bay single storey and a double bay double storey rigid jointed plane frames. Data on the differences in the optimum designs obtained from different starting points is reported.     

A critical review of established methods of structural topology optimization

The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive debate on this important topic. This article is an extended version of a paper presented at the WCSMO-7 in Seoul in 2007.     

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.     

A study of direct vs. approximation methods in structural optimization

This note discusses the performances and applications of two methods generally used in structural optimization. One is the direct method which applies a nonlinear programming (NLP) algorithm directly to the structural optimization problem. The other is the approximation method which utilizes the engineering sense very well. The two methods are compared through standard structural optimization problems with truss and beam elements. The results are analysed based on the convergence performances, the number of function calculations, the quality of the cost functions, etc. The applications of both methods are also discussed.     

Application of optimization methods within structural design. Practical design example

The use of the automated plane frame design program previously described is discussed. A special adaptation for tanker transverse frames includes more realistic frame support conditions than otherwise obtainable. For the rather complex structural design of a ‘Roll-On — Roll-Off’ carrier, the frame design program is utilized together with other, general structural analysis methods. Both the methods and some results are shown.     

Motion planning and tracking control for coupled flexible beam structures

A systematic flatness-based motion planning approach for a structure consisting of coupled flexible bending beams is introduced. The beams are equipped with in-domain piezoelectric macro-fiber composite (MFC) actuators and are coupled by an elastic string, modeled as spring, which connects the tip masses of adjacent beams. This setup provides a challenging benchmark experiment to evaluate control concepts. Hamilton’s principle leads to the equations of motion involving spatially varying parameters due to the different material characteristics of the carrier structure and the attached MFC-actuators. The flat output is constructed by exploiting the spectral system representation and enables a differential parameterization of the system states and inputs. This parameterization transfers the motion planning problem into a trajectory assignment problem for the flat output. The resulting flatness-based feedforward control law is analyzed and evaluated at an experimental setup. To increase the performance and the robustness w.r.t. model uncertainties and external disturbances the flatness-based feedforward control is amended by a feedback controller with observer in the spirit of the two-degrees-of-freedom control approach.     

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.     

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.