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It is common for papers on surrogate fitting to select test functions for testing algorithms. This raises the issue of how well the algorithms generalize to other functions. This editorial proposes as a possible complement to use a Gaussian process (GP) for generating test functions. The GP is defined by spatial decay rates of correlation between function values at different locations. The selection of these decay rates is tantamount to selection of the wave lengths of the functions that can be approximated by the surrogate. A possible complement to a given test function is a GP with similar decay rates to those of the test function, which would generate random functions with similar waviness. Two approaches are considered. In the simpler one, the GP is used to generate random test functions at a predetermined set of points. The more elaborate scheme targets situations where some samples are already available, and the GP is restricted to generate random functions that interpolate the samples. The process is illustrated for a one-dimensional case, where we seek to measure the correlation between the 95% confidence interval of the GP fit and the actual error. It is hoped that the editorial will stimulate discussion of the problem of algorithms customized to particular test functions and other options besides the one proposed here.

This paper investigates the reliability design optimization of an aeronautical hydraulic pipeline system, in which the constraint locations are treated as design parameters. To reduce the size of the optimization problem, two non-probabilistic global sensitivity indices are introduced and modified to screen out those constraint locations which have no or little effect on the optimization target. Considering the rest of constraint locations as design variables, the complexity of the optimization problem is dramatically reduced. The optimization of the pipeline systems demonstrates that the proposed method is superior to the traditional direct optimization method in both the optimization efficiency and results. This work indicates that introduction of sensitivity analysis can greatly improve the efficiency and performance of optimization, especially in those complex engineering problems involving a large number of design variables.

Topology optimization has been used in many industries and applied to a variety of design problems. In real-world engineering design problems, topology optimization problems often include a number of conflicting objective functions, such to achieve maximum stiffness and minimum mass of a design target. The existence of conflicting objective functions causes the results of the topology optimization problem to appear as a set of non-dominated solutions, called a Pareto-optimal solution set. Within such a solution set, a design engineer can easily choose the particular solution that best meets the needs of the design problem at hand. Pareto-optimal solution sets can provide useful insights that enable the structural features corresponding to a certain objective function to be isolated and explored. This paper proposes a new Pareto frontier exploration methodology for multiobjective topology optimization problems. In our methodology, a level set-based topology optimization method for a single-objective function is extended for use in multiobjective problems, using a population-based approach in which multiple points in the objective space are updated and moved to the Pareto frontier. The following two schemes are introduced so that Pareto-optimal solution sets can be efficiently obtained. First, weighting coefficients are adaptively determined considering the relative position of each point. Second, points in sparsely populated areas are selected and their neighborhoods are explored. Several numerical examples are provided to illustrate the effectiveness of the proposed method.

A large variety of algorithms for multidisciplinary optimization is available, but for various industrial problem types that involve expensive function evaluations, there is still few guidance available to select efficient optimization algorithms. This is also the case for multidisciplinary vehicle design optimization problems involving, e.g., weight, crashworthiness, and vibrational comfort responses. In this paper, an approach for the development of Representative Surrogate Problems (RSPs) as synthetic test functions for a relatively complex industrial problem is presented. The work builds on existing sensitivity analysis and surrogate data generation methods to establish a novel approach to generate surrogate function sets, which are accessible (i.e. not resource demanding) and aim to generate statistically representative instances of specific classes of industrial problems. The approach is demonstrated through the construction of RSPs for multidisciplinary optimization problems that occur in the context of structural car body design. As a "proof of concept" the RSP approach is applied for the selection of suitable optimization algorithms, for several problem formulations and for a meta-optimization (i.e. an optimization of the optimization algorithm parameters) to increase optimization efficiency. The potential of the approach is demonstrated by comparing the efficiency of several optimization algorithms on an RSP and an independent simulation-based vehicle model. The results corroborate the potential of the proposed approach and significant performance gains in optimization efficiency are achieved. Although the approach is developed for the particular application presented, the approach is described in a general way, to encourage readers to use the gist of the concept.

Mathematical optimization theories are employed for the design of structures in structural optimization. Structural optimization is being widely utilized for practical problems due to well-developed commercial software systems. Three representative structural optimization systems such as Genesis, MSC Nastran and OptiStruct are investigated and evaluated by solving various test examples in different scales. The design capabilities of three software systems are explored and the performances of the systems are compared. The performance of structural optimization depends on the quality of the optimum solution and the computational time, and these aspects are compared from an application viewpoint. For a fair comparison, the same formulations are utilized, and the same optimization methods are employed for each example. Also, the same system environment is prepared, and the same optimization parameters are used. Additionally, various design options of each software system are tested for the best performance. Linear static response size, shape, topology, topometry and topography optimizations are applied to the examples and the results are compared. No system seems to be the best in all cases and each system has advantages and disadvantages depending on the application. In general, Genesis is excellent in computational time while OptiStruct gives excellent optimum solutions, in size, topometry and topology optimizations. Meanwhile, MSC Nastran presents good solutions in shape and topography optimizations.

This article provides a comprehensive review of structural optimization employing topology methods for structures under vibration problems. Topology optimization allows creative and radical design modifications, compared to shape and size optimization techniques. Various works of structural topology optimization, which are subjected to vibration as the response function of the optimization process, are reviewed. Different types of calculus and numerical methods commonly used for solving structural topological optimization problems are briefly discussed. Moreover, different aspects of topology optimization related to vibration problems are explained. The articles reviewed are largely confined to linear systems that concern small vibration amplitudes. Accordingly, the works related to vibration topological optimization are classified according to the method employed (homogenization, evolutionary structural optimization, solid isotropic material with penalization, or level set). The reviewed works are tabulated according to their methodology, year, and the objective functions and applications of each work. Although the homogenization and evolutionary methods were common in the past, the solid isotropic material with penalization (SIMP) method is the most popular method applied in recent years. The advantages of the level set method show promise for future applications.

The structural optimization of the components in multibody systems is performed using a fully coupled optimization method. The design's predicted response is obtained from a flexible multibody system simulation under various service conditions. In this way, the resulting optimization process enhances most existing studies which are limited to weakly coupled (quasi-) static or frequency domain loading conditions. A level set description of the component geometry is used to formulate a generalized shape optimization problem which is solved via efficient gradient-based optimization methods. Gradients of cost and constraint functions are obtained from a sensitivity analysis which is revisited in order to facilitate its implementation and retain its computational efficiency. The optimizations of a slider-crank mechanism and a 2-dof robot are provided to exemplify the procedure.

Topology optimization has evolved rapidly since the late 1980s. The optimization of the geometry and topology of structures has a great impact on its performance, and the last two decades have seen an exponential increase in publications on structural optimization. This has mainly been due to the success of material distribution methods, originating in 1988, for generating optimal topologies of structural elements. Previous methods suffered from mathematical complexity and a limited scope for applicability, however with the advent of increased computational power and new techniques topology optimization has grown into a design tool used by industry. There are two main fields in structural topology optimization, gradient based, where mathematical models are derived to calculate the sensitivities of the design variables, and non gradient based, where material is removed or included using a sensitivity function. Both fields have been researched in great detail over the last two decades, to the point where structural topology optimization has been applied to real world structures. It is the objective of this review paper to present an overview of the developments in non gradient based structural topology and shape optimization, with a focus on evolutionary algorithms, which began as a non gradient method, but have developed to incorporate gradient based techniques. Starting with the early work and development of the popular algorithms and focusing on the various applications. The sensitivity functions for various optimization tasks are presented and real world applications are analyzed. The article concludes with new applications of topology optimization and applications in various engineering fields.

Robust optimization aims at producing designs which are less sensitive to uncertainties. Risk optimization looks for designs with optimal balance between performance and safety. In spite of the different objectives, robust and risk-based formulations have strong similitude, which has not been thoroughly explored before. This paper explores the similarities and differences between these formulations. It is shown that the alpha factors, which are employed in compromise solutions in robust optimization, are equivalent to the costs of failure in risk-based optimization. Moreover, it is shown that the robust objective function is often non-convex, with results being given by (often arbitrary) design constraints. In some sense, the robust objective function lacks objectiveness, with results largely dependent on arbitrary normalizing constants. On the other hand, when there is a critical limit to performance, which characterizes system failure, and when costs of failure can be defined, the risk-based optimization yields consistent results, and no normalizing constants are needed.

Solving large-scale PDE-constrained optimization problems presents computational challenges due to the large dimensional set of underlying equations that have to be handled by the optimizer. Recently, projection-based nonlinear reduced-order models have been proposed to be used in place of high-dimensional models in a design optimization procedure. The dimensionality of the solution space is reduced using a reduced-order basis constructed by Proper Orthogonal Decomposition. In the case of nonlinear equations, however, this is not sufficient to ensure that the cost associated with the optimization procedure does not scale with the high dimension. To achieve that goal, an additional reduction step, hyper-reduction is applied. Then, solving the resulting reduced set of equations only requires a reduced dimensional domain and large speedups can be achieved. In the case of design optimization, it is shown in this paper that an additional approximation of the objective function is required. This is achieved by the construction of a surrogate objective using radial basis functions. The proposed method is illustrated with two applications: the shape optimization of a simplified nozzle inlet model and the design optimization of a chemical reaction.

A process of compromise that addresses conflicting objective functions such as performance and cost is often involved in real-world engineering design activities. If such conflicting relationships among objective functions exist in a multiobjective design optimization problem, no single solution can simultaneously minimize all objective functions, and the solutions of the optimization problem are obtained as a set of design alternatives called a Pareto optimal solution set. This paper proposes a new gradient-based multiobjective c that incorporates a population-based aggregative strategy for obtaining a Pareto optimal solution set. In this method, the objective functions and constraints are evaluated at multiple points in the objective function space, and design variables at each point are updated using information aggregatively obtained from all other points. In the proposed method, a multiobjective optimization problem is converted to a single objective optimization problem using a weighting method, with weighting coefficients adaptively determined by solving a linear programming problem. A sequential approximate optimization-based technique is used to update the design variables, since it allows effective use of design sensitivities that can be easily obtained in many engineering optimization problems. Several numerical examples, including a structural optimization problem, are provided to illustrate the effectiveness and utility of the proposed method.

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.

In civil and industrial engineering, structural design optimization problems are usually characterized by the presence of multiple conflicting objectives, as to get the minimum investment cost and the maximum safety of the final design. This issue makes these problems to have not only one single solution, but a set them. Such solutions represent the possible trade-offs among the different objectives to be optimized. This paper reviews the latest developments in the field of multi-objective metaheuristics for solving design problems focusing on the optimization of the topology, shape, and sizing of civil engineering structures. We review both the algorithms and the applications, and the most relevant features of the solvers and the design optimization problems are analyzed. The paper ends by addressing a number of relevant and open issues that can be the subject of further research.

This paper is dedicated to the structural optimization of flexible components in mechanical systems modeled as multibody systems. While most of the structural optimization developments have been conducted under (quasi-)static loadings or vibration design criteria, the proposed approach aims at considering as precisely as possible the effects of dynamic loading under service conditions. Solving this problem is quite challenging and naive implementations may lead to inaccurate and unstable results. To elaborate a robust and reliable approach, the optimization problem formulation is investigated because it turns out that it is a critical point. Different optimization algorithms are also tested. To explain the efficiency of the various solution approaches, the complex nature of the design space is analyzed. Numerical applications considering the optimization of a two-arm robot subject to a trajectory tracking constraint and the optimization of a slider-crank mechanism with a cyclic dynamic loading are presented to illustrate the different concepts.

The coupling of finite element simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a generally applicable strategy for modeling and efficiently solving robust optimization problems based on time consuming simulations. Noise variables and their effect on the responses are taken into account explicitly. The robust optimization strategy consists of four main stages: modeling, sensitivity analysis, robust optimization and sequential robust optimization. Use is made of a metamodel-based optimization approach to couple the computationally expensive finite element simulations with the robust optimization procedure. The initial metamodel approximation will only serve to find a first estimate of the robust optimum. Sequential optimization steps are subsequently applied to efficiently increase the accuracy of the response prediction at regions of interest containing the optimal robust design. The applicability of the proposed robust optimization strategy is demonstrated by the sequential robust optimization of an analytical test function and an industrial V-bending process. For the industrial application, several production trial runs have been performed to investigate and validate the robustness of the production process. For both applications, it is shown that the robust optimization strategy accounts for the effect of different sources of uncertainty onto the process responses in a very efficient manner. Moreover, application of the methodology to the industrial V-bending process results in valuable process insights and an improved robust process design.

In this article, the optimization problem of designing transonic airfoil sections is solved using a framework based on a multi-objective optimizer and surrogate models for the objective functions and constraints. The computed Pareto-optimal set includes solutions that provide a trade-off between maximizing the lift-to-drag ratio during cruise and minimizing the trailing edge noise during the aircraft's approach to landing. The optimization problem was solved using a recently developed multi-objective optimizer, which is based on swarm intelligence. Additional computational intelligence tools, e.g., artificial neural networks, were utilized to create surrogate models of the objective functions and constraints. The results demonstrate the effectiveness and efficiency of the proposed optimization framework when applied to simulation-based engineering design optimization problems.

Differential evolution (DE) is an efficient population based algorithm used to solve real-valued optimization problems. It has the advantage of incorporating relatively simple and efficient mutation and crossover operators. However, the DE operator is based on floating-point representation only, and is difficult to use when solving combinatorial optimization problems. In this paper, a modified binary differential evolution (MBDE) based on a binary bit-string framework with a simple and new binary mutation mechanism is proposed. Two test functions are applied to verify the MBDE framework with the new binary mutation mechanism, and four structural topology optimization problems are used to study the performance of the proposed MBDE algorithm. The experimental studies show that the proposed MBDE algorithm is not only suitable for structural topology optimization, but also has high viability in terms of solving numerical optimization problems.

In this paper, an optimization approach is adopted to obtain the 12 material parameters used in McGinty’s Model for AL6022 by minimizing the differences between simulation and experimental stress–strain curves. Since the differences between the two stress–strain curves are implicitly related to the change of material parameters, the metamodeling technique is utilized to create explicit, approximate functions of these relationships. Radial basis functions (RBFs), which are shown from previous studies to be effective for both low- and high-order nonlinear responses, are used for the metamodels that are adaptively updated in the optimization work. Two optimization formulation schemes are studied to address the issue of using inaccurate RBF models in optimization. The sampling, metamodeling, and optimization works are performed using the integrated optimization framework HiPPO.

Surrogate-based optimization (SBO) for engineering design, popular in the optimization of complex engineering systems (e.g., aerospace, automotive, oil industries), proceeds in design cycles. Each cycle consists of the analysis of a number designs, the fitting of a surrogate, optimization based on the surrogate, and exact analysis at the design obtained by the optimization. However, due to time and cost constraints, the design optimization is usually limited to a small number of cycles each with substantial number of simulations (short cycle SBO) and rarely allowed to proceed to convergence. This paper takes a first step towards establishing a statistically rigorous procedure for assessing the merit of investing in another cycle of analysis versus accepting the present best solution. The proposed approach assumes that the set of locations for the next cycle is given, and it relies on: (1) a covariance model obtained from available input/output data, (2) a Gaussian process-based surrogate model, and (3) the fact that the predictions in the next cycle are a realization of a Gaussian process with a covariance matrix and mean specified using (1) and (2). Its effectiveness was established using descriptive and inference statistical elements in the context of a well-known test function and the optimization of an alkali-surfactant-polymer flooding of petroleum reservoirs.

The occurrence of dynamic problems during the operation of machinery may have devastating effects on a product. Therefore, design optimization of these products becomes essential in order to meet safety criteria. In this research, a hybrid design optimization method is proposed where attention is focused on structures having repeating patterns in their geometries. In the proposed method, the analysis is decomposed but the optimization problem itself is treated as a whole. The model of an entire structure is obtained without modeling all the repetitive components using the merits of the Component Mode Synthesis method. Backpropagation Neural Networks are used for surrogate modeling. The optimization is performed using two techniques: Genetic Algorithms (GAs) and Sequential Quadratic Programming (SQP). GAs are utilized to increase the chance of finding the location of the global optimum and since this optimum may not be exact, SQP is employed afterwards to improve the solution. A theoretical test problem is used to demonstrate the method.

This paper presents a classification of for- mulations for distributed system optimization based on formulation structure. Two main classes are identi- fied: nested formulations and alternating formulations. Nested formulations are bilevel programming problems where optimization subproblems are nested in the func- tions of a coordinating master problem. Alternating formulations iterate between solving a master problem and disciplinary subproblems in a sequential scheme. Methods included in the former class are collaborative optimization and BLISS2000. The latter class includes concurrent subspace optimization, analytical target cas- cading, and augmented Lagrangian coordination. Al- though the distinction between nested and alternating formulations has not been made in earlier comparisons, it plays a crucial role in the theoretical and computa- tional properties of distributed optimization methods. The most prominent general characteristics for each class are discussed in more detail, providing valuable insights for the theoretical analysis and further devel- opment of distributed optimization methods.

Optimization problems concerning complex structures with many design variables may entail an unacceptable computational cost. This problem can be reduced considerably with a multilevel approach: A structure consisting of several components is optimized as a whole (global) as well as on the component level. In this paper, an optimization method is discussed with applications in the assessment of the impact of new design considerations in the development of a structure. A strategy based on fully stressed design is applied for optimization problems in linear statics. A global model is used to calculate the interactions (e.g., loads) for each of the components. These components are then optimized using the prescribed interactions, followed by a new global calculation to update the interactions. Mixed discrete and continuous design variables as well as different design configurations are possible. An application of this strategy is presented in the form of the full optimization of a vertical tail plane center box of a generic large passenger aircraft. In linear dynamics, the parametrization of the component interactions is problematic due to the frequency dependence. Hence, a modified method is presented in which the speed of component mode synthesis is used to avoid this parametrization. This method is applied to a simple test case that originates from noise control.

In recent years, finite element simulation has been increasingly combined with optimization techniques and applied to optimization of various metal-forming processes. The robustness and efficiency of process optimization are critical factors to obtain ideal results, especially for those complicated metal-forming processes. Gradient-based optimization algorithms are subject to mathematical restrictions of discontinuous searching space, while nongradient optimization algorithms often lead to excessive computation time. This paper presents a novel intelligent optimization approach that integrates machine learning and optimization techniques. An intelligent gradient-based optimization scheme and an intelligent response surface methodology are proposed, respectively. By machine learning based on the rough set algorithm, initial total design space can be reduced to self-continuous hypercubes as effective searching spaces. Then optimization algorithms can be implemented more effectively to find optimal design results. An extrusion forging process and a U channel roll forming process are studied as application samples and the effectiveness of the proposed approach is verified.

Investigations on optimization of structures working in high-cycle load conditions were carried out and are presented in this paper. The development of a simple in application optimization algorithm for such structures was the main object of the authors. The work was concentrated on three principle areas: fatigue of material (with special regard to multiaxial criteria of high-cycle fatigue), parametric optimization of structures, and application of the finite element method. The investigations and numerical implementation of several high-cycle criteria were made and the most convenient one for optimization was selected. The main process of fatigue optimization was preceded by the testing of methods of structural optimization and the preparing the tools for improving the efficiency of the optimization algorithm. This stage includes preparation of software tools based on evolutionary algorithms. In addition, the decision variables were preselected through an investigation of the sensitivity of the objective function on small increments of these variables. The work was illustrated by examples of optimization of mechanical structures working in high-cycle load conditions. As observed in the computational examples, the proposed methodology of optimization allowed effectively lowering the mass of the studied structure while maintaining its durability on an established level. The tools and fatigue optimization methodology presented in this paper have universal character and can be applied to any case of a structure subjected to high-cycle loads.

A hybrid method for robust and efficient optimization process is developed by integrating a new response surface method and pattern search algorithm. The method is based on: (1) multipoint approximations of the objective and constraint functions, (2) a multiquadric radial basis function (RBF) for the zeroth-order function approximation and a new RBF plus polynomial-based moving least-squares approximation for the first-order enhanced function approximation, and (3) a pattern search algorithm to impose a descent condition and applied adaptive subregion management strategy. Several numerical examples are presented to illustrate accuracy and computational efficiency of the proposed method for both function approximation and design optimization. To demonstrate the effectiveness of the proposed hybrid method, it is applied to obtain optimum designs of a microelectronic packaging system. A two-stage optimization approach is proposed for the design optimization. The material properties of microelectronic packaging system and the shape parameters of solder ball are selected as design variables. Through design optimization, significant improvements of durability performances are obtained using the proposed hybrid optimization method.

Structural optimization is a field of research that has experienced noteworthy growth for many years. Researchers in this area have developed optimization tools to successfully design and model structures, typically minimizing mass while maintaining certain deflection and stress constraints. Numerous optimization studies have been performed to minimize mass, deflection, and stress on a benchmark cantilever truss problem. Predominantly traditional optimization theory is applied to this problem. The cross-sectional area of each member is optimized to minimize the aforementioned objectives. This Technical Publication (TP) presents a structural optimization technique that has been previously applied to compliant mechanism design. This technique demonstrates a method that combines topology optimization, geometric refinement, finite element analysis, and two forms of evolutionary computation: genetic algorithms and differential evolution to successfully optimize a benchmark structural optimization problem. A nontraditional solution to the benchmark problem is presented in this TP, specifically a geometrically refined topological solution. The design process begins with an alternate control mesh formulation, multilevel geometric smoothing operation, and an elastostatic structural analysis. The design process is wrapped in an evolutionary computing optimization toolset.

Various sequential derivative-free optimization algorithms exist for solving black-box optimization problems. Two important building blocks in these algorithms are the trust region and the geometry improvement. In this paper, we propose to incorporate the D-optimality criterion, well known in the design of experiments, into these algorithms in two different ways. Firstly, it is used to define a trust region that adapts its shape to the locations of the points in which the objective function has been evaluated. Secondly, it is used to determine an optimal geometry-improving point. The proposed trust region and geometry improvement can both be implemented into existing sequential algorithms.

Alternative formulations for optimization and simulation of structural and mechanical systems and other related fields are reviewed. The material is divided roughly into two parts. Part 1 focuses on the developments in structural and mechanical systems, including configuration and topology optimization. Here the formulations are classified into three broad categories: (i) the conventional formulation where only the structural design variables are treated as optimization variables, (ii) simultaneous analysis and design (SAND) formulations where design and some of the state variables are treated as optimization variables, and (iii) a displacement-based two-phase approach where the displacements are treated as unknowns in the outer loop and the design variables as the unknowns in the inner loop. Part 2 covers more general formulations that are applicable to diverse fields, such as economics, optimal control, multidisciplinary problems and other engineering disciplines. In these fields, SAND-type formulations have been called mathematical programs with equilibrium constraints (MPEC), and partial differential equations (PDE)-constrained optimization problems. These formulations are viewed as generalizations of the SAND formulations developed in the structural optimization field. Based on the review, it is concluded that the basic ideas of the formulations presented in diverse fields can be integrated to conduct further research and develop alternative formulations and solution procedures for practical engineering applications. The paper lists 187 references on the subject.

In this paper a structural sizing and shape optimization procedure is used to obtain optimum designs for plates and shells under dynamic loads. The optimum designs are obtained by using an integrated system for structural optimization of shells that is formed by a set of independent ‘subsystems’. They are surface and mesh generation, finite element analysis, sensitivity analysis, and optimization algorithms. The description of the geometry and mesh generation tools, as well as the implementation and test of the structural dynamic and sensitivity analysis, were given in Part I of this paper. In this part the tools presented in Part I are used along with an optimization algorithm to perform structural shape and sizing optimization (SSO) of shell structures. The sequential quadratic programming algorithm is used to solve the optimization problem. In the present paper special attention is devoted to the problem of elimination of clusters of multiple eigenvalues in the optimal design of symmetrical shells. Several examples are considered that illustrate the potential of the developed tool.

Topology optimization has become very popular in industrial applications, and most FEM codes have implemented certain capabilities of topology optimization. However, most codes do not allow simultaneous treatment of sizing and shape optimization during the topology optimization phase. This poses a limitation on the design space and therefore prevents finding possible better designs since the interaction of sizing and shape variables with topology modification is excluded. In this paper, an integrated approach is developed to provide the user with the freedom of combining sizing, shape, and topology optimization in a single process.

In this paper, a two-level optimization approach is developed for the preliminary and conceptual design of airframe structures. The preliminary design, involving a single objective multidisciplinary optimization, constitutes the lower level where ASTROS (Automated STRuctural Optimization System) is employed for multidisciplinary optimization. The conceptual design, which is carried out at the upper level, aims mainly at configuration design. The multiple objectives are incorporated as a single objective function by using the K-S function formulation. The objective function and constraints at the upper level are modelled through response surface approximation. During the upper level optimization process, the branch and bound method is applied for solving the problem with discrete design variables. The proposed strategy is demonstrated by the optimization of an Intermediate Complexity Wing (ICW) model.

In this paper, implementation concepts of a structural optimization software using object-oriented programming (OOP) in C++ is presented. A brief mathematical formulation of structural optimization and continuum-based sensitivity analysis is presented. The requirements of a computational optimization environment are derived from this formulation. The OOP characteristics are analysed and this paradigm is employed in the implementation of design variables, structural performance functionals, velocity fields, design model and mathematical programming algorithms using C++. Finally, the program obtained is applied to 2D linear elastic examples of sizing and shape optimization.

In this paper the response surface methodology (RSM) and stochastic optimization (SO) are compared with regard to their efficiency and applicability in crashworthiness design. Optimization of simple analytic expressions and optimization of a front rail structure are the applications used to assess the respective qualities of both methods. A low detail vehicle structure is optimized to demonstrate the applicability of the methods in engineering practice. The investigations reveal that RSM is better compared to SO for fewer than 10–15 design variables. The convergence behaviour of SO improves compared to RSM when the number of design variables is increased. A novel zooming method is proposed which improves the convergence behaviour. A combination of both the RSM and the SO is efficient, stochastic optimization could be used in order to determine appropriate starting points for an RSM optimization, which continues the optimization. Two examples are investigated using this combined method.