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## Optimal non-homogeneous composites for dynamic loading

### Structural and Multidisciplinary Optimization (2005-08-01) 30: 101-112 , August 01, 2005

An algorithm is proposed to optimize the performance of a two-phase composite under dynamic loading. The goal is to determine a series of different layouts of the two base materials in a three-dimensional region such that the time-averaged stress energy is minimized. Four cases with different boundary conditions and ratios of mass density are considered and solved numerically. The resulting optimal designs are compared to the static case to illustrate the effect of the dynamic loading. Furthermore, a qualitative comparison is done to indicate the difference between the optimization of eigenfrequencies and the present formulation.

## Large deformations and stability in topology optimization

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 459-476 , December 01, 2005

The present contribution focuses on the influence of geometrical nonlinearities on the structural behavior in the design process. The notion of the stiffest structure loses its clear definition in the case of nonlinear kinematics; here we will discuss this concept on the basis of different objectives. Apparently topology optimization is often a generator of slender struts, which tend to buckle before the structure is completely loaded. To include the instability phenomena into the design process, the critical load level will be determined directly; this condition will be included as an inequality constraint. Further on, to reduce the imperfection sensitivity, a geometrically modified structure including the imperfection shape is also introduced. The present optimization procedures are demonstrated by examples showing rather the principal effects of the enhancements than real practical design problems.

## Topology optimization of trusses with stress and local constraints on nodal stability and member intersection

### Structural and Multidisciplinary Optimization (2005-03-01) 29: 190-197 , March 01, 2005

A truss topology optimization problem under stress constraints is formulated as a Mixed Integer Programming (MIP) problem with variables indicating the existence of nodes and members. The local constraints on nodal stability and intersection of members are considered, and a moderately large lower bound is given for the cross-sectional area of an existing member. A lower-bound objective value is found by neglecting the compatibility conditions, where linear programming problems are successively solved based on a branch-and-bound method. An upper-bound solution is obtained as a solution of a Nonlinear Programming (NLP) problem for the topology satisfying the local constraints. It is shown in the examples that upper- and lower-bound solutions with a small gap in the objective value can be found by the branch-and-bound method, and the computational cost can be reduced by using the local constraints.

## An efficient evolutionary optimisation framework applied to turbine blade firtree root local profiles

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 382-390 , May 01, 2005

In this paper, an efficient evolutionary optimisation of a turbine blade firtree root local profile is presented. The firtree geometry is designed using an intelligent rule-based computer-aided design system (ICAD) and analysed using an industrial-strength finite element code. A large number of geometric and mechanical constraints drawn from past experience are incorporated in the design of the model. The high computational cost associated with finding optimal designs using high-fidelity codes is addressed using a surrogate-assisted genetic algorithm. The initial surrogate model is first built based on points sampled with a design-of-experiment method. A database of designs analysed using the high-fidelity code is built and augmented while the genetic algorithm progresses. In the procedure for deciding whether the high-fidelity code should be run, a simple 3σ principle is used instead of searching for the point with maximum expected improvement. This is combined with an appropriate ranking of the design points within the database. Some benchmark test problems are first used to illustrate the effectiveness and efficiency of the framework. When applied to the problem of local shape optimisation of a turbine blade firtree root, significant improvement is achieved using a limited computational budget.

## On optimal geometrically non-linear trusses

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 113-124 , February 01, 2005

The paper is devoted to the investigation of regularities inherent to optimal geometrically non-linear trusses. The single static loading case is considered, a single structural material is used (except specially indicated cases) and buckling effects are neglected. The so-called small strains and large rotations case is investigated.

Some regularities inherent to the kinematic and static variational principles for geometrically non-linear trusses are considered. Then the strain compatibility conditions resulting from the static variational principle are obtained and explored. It is shown that 1) the conditions are linear with respect to subcomponents of rod Green strains such as rotations and geometrically linear strains, 2) strains (in particular, rotations and geometrically linear strains) within rods which are not members of the so-called basic structure are fully determined by geometrically linear strains in rods of the basic structure.

Extensions of some theorems (Maxwell’s theorem, Michell’s theorem, theorems on the stiffness properties of equally-stressed structures, etc.) known for geometrically linear structures are proved.

Conditions assuring better or worse quality of equally-stressed geometrically non-linear truss as compared to geometrically linear ones are obtained.

It is shown that in numerical optimization of geometrically non-linear trusses in the case of negligible rotations of compressed rods some updated analytical optimization algorithms (derived earlier for geometrically linear case) are monotonic. A simple numerical example confirming the features is presented.

## Optimal rapid multidisciplinary response networks: RAPIDDISK

### Structural and Multidisciplinary Optimization (2005-03-01) 29: 213-231 , March 01, 2005

The role of uncertainty in information rich design systems is critical to the development of advanced propulsion systems. Future turbine engines would have lower lifetime operating costs similar to current evolving automotive systems. Detailed multi-physics models (thermo-fluid, structural and mechanical systems) and an operational environment are enablers of rapid correction and model-based predictive analyses. Bayesian machine learning paradigms are developed to identify the behavior of turbo-machinery components for preliminary design. The embedded models approach enables systematic evolution from individual components level to the advanced engine. The embedded models approach enables systematic evolution from individual components level to the advanced engine. A rapid response strategy is proposed, for design of turbine disks by using multidisciplinary optimization and neural networks. iSIGHT optimization software is interfaced with ANSYS to find optimum designs for a given set of design boundary conditions (rpm, live rim load, thermals, etc.). The optimum designs obtained from iSIGHT for different set of design conditions are used for machine learning and design knowledge recognition using the neural network technique. The trained network is used to predict responses for design boundary conditions. Responses predicted by the neural network are validated using ANSYS. Discrete design points are chosen from the wide design space of turbine disks. A hierarchical neural network approach provides an ability to quickly train the network and predict responses (weight, stresses, burst margin, etc.) for applied design conditions. This basic building process involves four steps starting from identifying design boundary conditions to the prediction of design shape for the disk. Sensitivity-based scaling rules are developed, to accommodate different materials for the disk. The technique is developed in RAPIDDISK, which provides an optimal preliminary shape and design attributes for a turbine disk.

## A new methodology to establish upper bounds on open-cell foam homogenized moduli

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 257-271 , April 01, 2005

The methodology to determine upper bounds on homogenized linear elastic moduli of cellular solids, described for the two-dimensional case in Dimitrovová and Faria (1999), is extended to three-dimensional open-cell foams. Besides the upper bounds the methodology provides necessary and sufficient conditions on the optimal media. These conditions are written in terms of generalized internal forces and geometrical parameters. In some cases dependence on internal forces can be replaced by geometrical expressions. In such cases optimality of some medium under consideration can be verified directly from the microstructure, without any additional calculation. Some of the bounds derived in this paper have not yet been published along with a proof of their optimality.

## Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature of composite laminates

### Structural and Multidisciplinary Optimization (2005-08-01) 30: 89-100 , August 01, 2005

Design and control optimization is presented to minimize the thermal postbuckling dynamic response and to maximize the buckling temperature level of composite laminated plates subjected to thermal distribution varying linearly through the thickness and arbitrarily with respect to the in-plane coordinates. The total elastic energy of the laminates is taken as a measure of the dynamic response. The optimization control problem is solved under constraints on the laminate thickness and the control energy produced by a transverse dynamic load distributed over the upper surface of the laminate. The constrained control objective is expressed as the sum of the total elastic energy and penalty term involving the control force, which may be considered as a measure of the control energy. The thickness of layers and the fibers orientation angles are taken as optimization design variables. The design and control objectives are formulated based on shear deformation theory accounting for the von-Karman nonlinearity. The displacements are chosen as the sum of time-independent displacements due to the static thermal load and time-dependent displacements due to the initial disturbances and the applied control force. Liapunov–Bellman theory is used to obtain the optimal control force, buckled deflections and controlled elastic energy. Numerical examples are presented for angle-ply antisymmetric laminates with simply supported edges. Graphical studies are carried out to show the advantages of the present design and control procedures.

## Multi-criteria shape optimization of a funnel in cathode ray tubes using a response surface model

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 374-381 , May 01, 2005

One purpose of simulation describing the behaviors of structures is to optimize the performances within specific functional requirements and customers’ needs with respect to the design variables. For reduction of the volume of cathode ray tubes, the design of the glass geometry, especially funnel geometry, is essential while maintaining the internal vacuum pressure of the cathode ray tube. In order to describe the three-dimensional geometry of the funnel in cathode ray tubes, a higher-order response surface model is employed in the simulation model instead of non-uniform rational B-splines (NURBS) or Bezier curves because the response surface model is more robust for understanding the geometry change in finite element analysis. We formulate the design problem as a multi-criteria optimization because minimization of both volume and maximum stress is required. Using the response surface model of the geometry of the funnel and sequential quadratic programming within the process integration framework, the shape optimization of a funnel is successfully performed and the maximum stress level of the funnel is decreased by almost half.

## Minimization of the expected compliance as an alternative approach to multiload truss optimization

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 470-476 , June 01, 2005

We show that a problem of finding the truss of minimum expected compliance under stochastic loading conditions is equivalent to the dual of a special convex minimax problem, and therefore may be efficiently solved. This equivalence makes it possible to provide classic multiload compliance minimization problems with interpretations in a probabilistic setting. In fact, we prove that minimizing the expected compliance amounts to solving a multiload-like problem associated with a particular finite set of loading scenarios, which depend on the mean and the variance of the perturbations.

## Designing plates for minimum internal resonances

### Structural and Multidisciplinary Optimization (2005-10-01) 30: 297-307 , October 01, 2005

Internal resonance is a nonlinear phenomenon for a structure when the eigenfrequencies of the structure are commensurable or close to being commensurable. Using optimization we have the possibility to control the eigenfrequencies, i.e., move an eigenfrequency, maximize a given eigenfrequency, or maximize the gap between eigenfrequencies. It is therefore also possible to design a structure that is as free as possible of internal resonance up to mode of order n.

We consider plates made of two materials. The designs depend on the boundary conditions and on the frequency range within which the plate should be as free of internal resonance as possible. The two materials can either be two physical materials, or one can be a physical material and the other a weakening of the first material. By doing this we are in principle solving three different problems: a reinforcement problem, a problem of where to put holes in the structure, and, finally the more involved case (from a manufacturing point of view), of two different materials. The optimizations are performed using the finite element method for analysis and the topology optimization approach for design. The optimization problem is formulated using a bound formulation where the objective is to maximize a minimum detuning parameter. Special attention is given to the formulation of the conditions for internal resonance. Using the method presented in this paper it is possible to remove an unwanted nonlinear phenomenon without the use of a nonlinear model and without knowledge of the nonlinearities present in the system.

## Multi-objective robust optimization using a sensitivity region concept

### Structural and Multidisciplinary Optimization (2005-01-01) 29: 50-60 , January 01, 2005

In multi-objective design optimization, it is quite desirable to obtain solutions that are “multi-objectively” optimum and insensitive to uncontrollable (noisy) parameter variations. We call such solutions robust Pareto solutions. In this paper we present a method to measure the multi-objective sensitivity of a design alternative, and an approach to use such a measure to obtain multi-objectively robust Pareto optimum solutions. Our sensitivity measure does not require a presumed probability distribution of uncontrollable parameters and does not utilize gradient information; therefore, it is applicable to multi-objective optimization problems that have non-differentiable and/or discontinuous objective functions, and also to problems with large parameter variations. As a demonstration, we apply our robust optimization method to an engineering example, the design of a vibrating platform. We show that the solutions obtained for this example are indeed robust.

## Parameter optimization of the sheet metal forming process using an iterative parallel Kriging algorithm

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 498-507 , June 01, 2005

Different numerical optimization strategies were used to find an optimized parameter setting for the sheet metal forming process. A parameterization of a time-dependent blank-holder force was used to control the deep-drawing simulation. Besides the already well-established gradient and direct search algorithms and the response surface method the novel Kriging approach was used as an optimization strategy. Results for two analytical and two sheet metal forming test problems reveal that the new Kriging approach leads to a fast and stable convergence of the optimization process. Parallel simulation is perfectly supported by this method.

## Experience with approximate reliability-based optimization methods II: an exhaust system problem

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 488-497 , June 01, 2005

Traditional reliability-based design optimization (RBDO) requires a double-loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. This situation is even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test RBDO methods based on the first-order reliability method (FORM), including single- and double-loop methods. In addition to the number of constraints, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.

## Vibration reduction optimum design of a steam-turbine rotor-bearing system using a hybrid genetic algorithm

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 43-53 , July 01, 2005

This paper describes the vibration optimum design for the low-pressure steam-turbine rotor of a 1007-MW nuclear power plant by using a hybrid genetic algorithm (HGA) that combines a genetic algorithm and a local concentration search algorithm using a modified simplex method. This algorithm not only calculates the optimum solution faster and more accurately than the standard genetic algorithm but can also find the global and local optimum solutions. The objective function is to minimize the resonance response (Q-factor) of the second occurring mode in the excessive vibration. Under the constraints of shaft diameter, bearing length and clearance, these factors play a very important role in the design of a rotor-bearing system. In the present work, the shaft diameter, bearing length and clearance are chosen as the design variables. The results show that the HGA can reduce the excessive response at the critical speed and improve the stability.

## Adaptive probability analysis using an enhanced hybrid mean value method

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 134-148 , February 01, 2005

This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value (HMV+) analysis in the performance measure approach (PMA) for numerical stability and efficiency in search of the most probable point (MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the *interpolated* moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method.

## Topology optimization of planar and plate structures using conformal mapping

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 125-133 , February 01, 2005

This paper proposes a new method to determine the optimum topology of planar and plate structures using conformal mapping. First it is proved that two invariants of stresses, which are the sum and difference of principal stresses, satisfy the Laplace equation. Additionally, it is proved that two invariants of bending moments are the sum and difference of principal bending moments, and is proved that they satisfy the Laplace equation under certain condition. Finally, we show that corresponding relationships between fluid mechanics and electromagnetism are valid to the theory of planar elasticity and plate bending. From these considerations, a convenient design method to optimize topologies of planar and plate structures for complicate design domains is proposed using conformal mapping. Several numerical examples of optimum topologies are treated by the proposed method. Through numerical results the effectiveness and validity of proposed method are discussed quantitatively and qualitatively.

## Design optimization of simply supported concrete slabs by finite element modelling

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 76-88 , July 01, 2005

This paper outlines the finite element prediction process for the development of charts for accurate peak load determination of simply supported, reinforced concrete slabs under uniformly distributed loading. Through a series of parametric studies using a simple concrete model, the simulation of tests on four simply supported slabs was used as a basis for establishing a set of optimum parameter values and computational conditions, which guarantees acceptable solution. The reliability of the established parameter values for prediction purposes was verified by the direct simulation of 11 other slabs. Following the successful reliability check, the finite element model was used for analysing 270 “computer model” slabs, from which charts were developed. These charts serve for quick and reliable peak load determination of arbitrary simply supported slabs. A comparative study of the direct finite element and chart predictions, with values from analytical and design methods, reveals the superiority of the charts over the latter methods, with accuracy comparable to that of the optimised finite element model. The chart prediction is noted to be accurate to within 4% of test results. A strategy for displacement determination is also established, with the same degree of success and the paper discusses possible practical applications of the developed finite element system.

## On polynomial response surfaces and Kriging for use in structural optimization of crashworthiness

### Structural and Multidisciplinary Optimization (2005-03-01) 29: 232-243 , March 01, 2005

The accuracy of different approximating response surfaces is investigated. In the classical response surface methodology (CRSM) the true response function is usually replaced with a low-order polynomial. In Kriging the true response function is replaced with a low-order polynomial and an error correcting function. In this paper the error part of the approximating response surface is obtained from “simple point Kriging” theory. The combined polynomial and error correcting function will be addressed as a Kriging surface approximation.

To be able to use Kriging the spatial correlation or covariance must be known. In this paper the error is assumed to have a normal distribution and the covariance to depend only on one parameter. The maximum-likelihood method is used to find the latter parameter. A weighted least-square procedure is used to determine the trend before simple point Kriging is used for the error function. In CRSM the surface approximation is determined through an ordinary least-square fit. In both cases the D-optimality criterion has been used to distribute the design points.

From this investigation we have found that a low-ordered polynomial assumption should be made with the Kriging approach. We have also concluded that Kriging better than CRSM resolves abrupt changes in the response, e.g. due to buckling, contact or plastic deformation.

## Two-point mid-range approximation enhanced recursive quadratic programming method

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 398-405 , May 01, 2005

This research represents an attempt to combine good convergence properties of recursive quadratic programming methods with the benefits of mid-range approximations, initially developed in the field of structural optimization. In this paper, an optimization method based on Arora and coworkers’ PLBA (Pshenichny–Lim–Belegundu–Arora) algorithm is proposed in which, during the line search phase, cost and constraint functions are substituted by their two-point approximations using the Generalized Convex Approximation formulae of Chickermane and Gea. The results showed that the proposed optimization method preserves the reliability and accuracy of the recursive quadratic programming method while it might simultaneously reduce the computational effort for some problems. Therefore, the proposed optimization method may be taken as potentially suitable for general design optimization purposes.

## Nonlinear topology optimization of layered shell structures

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 349-360 , May 01, 2005

Topology stiffness (compliance) design of linear and geometrically nonlinear shell structures is solved using the SIMP approach together with a filtering scheme. A general anisotropic multi-layer shell model is employed to allow the formation of through-the-thickness holes or stiffening zones. The finite element analysis is performed using nine-node Mindlin-type shell elements based on the degenerated shell approach, which are capable of modeling both single and multi-layered structures exhibiting anisotropic or isotropic behavior. The optimization problem is solved using analytical compliance and constraint sensitivities together with the Method of Moving Asymptotes (MMA). Geometrically nonlinear problems are solved using iterative Newton–Raphson methods and an adjoint variable approach is used for the sensitivity analysis. Several benchmark tests are presented in order to illustrate the difference in optimal topologies between linear and geometrically nonlinear shell structures.

## On extending the range of Michell-like optimal topology structures

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 85-92 , February 01, 2005

Structural topological optimization continues to have a rich development with widespread application of computer-based search techniques. However, there is only a limited range of known theoretically absolute optimum (minimum-weight) or Michell structures unconstrained by geometric constraints with which to compare. Such solutions are of interest in the development of numerically generated optimal structures. In this paper it is proposed that some potential new forms of optimal plane trusses can be constructed from existing, known forms developed much earlier. The cases of practical interest range from a single offset point load to some cases with more than one load, applied at the same level as the reactions. The emphasis is on conceptual extensions rather than on mathematical proofs.

## Minutes of the meeting of the general assembly of ISSMO held in Rio de Janeiro on 31 May 2005

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 500-502 , December 01, 2005

## Hybrid evolutionary algorithm and application to structural optimization

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 219-226 , September 01, 2005

This paper introduces a new evolutionary algorithm with a globally stochastic but locally heuristic search strategy. It is implemented by incorporating a modified micro-genetic algorithm with two local optimization operators. Performance tests using two benchmarking functions demonstrate that the new algorithm has excellent convergence performance when applied to multimodal optimization problems. The number of objective function evaluations required to obtain global optima is only 3.5–3.7% of that of using the conventional micro-genetic algorithm. The new algorithm is used to optimize the design of an 18-bar truss, with the aim of minimizing its weight while meeting the stress, section area, and geometry constraints. The corresponding optimal design is obtained with considerably fewer computational operations than required for the existing algorithms.

## Topology optimization of frame structures—joint penalty and material selection

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 193-200 , September 01, 2005

This paper deals with joint penalization and material selection in frame topology optimization. The models used in this study are frame structures with flexible joints. The problem considered is to find the frame design which fulfills a stiffness requirement at the lowest structural weight. To support topological change of joints, each joint is modelled as a set of subelements. A set of design variables are applied to each beam and joint subelement. Two kinds of design variables are used. One of these variables is an area-type design variable used to control the global element size and support a topology change. The other variables are length ratio variables controlling the cross section of beams and internal stiffness properties of the joints. This paper presents two extensions to classical frame topology optimization. Firstly, penalization of structural joints is presented. This introduces the possibility of finding a topology with less complexity in terms of the number of beam connections. Secondly, a material interpolation scheme is introduced to support mixed material design.

## Optimization of plastic spherical shells of von Mises material

### Structural and Multidisciplinary Optimization (2005-11-01) 30: 381-387 , November 01, 2005

An optimization procedure is developed for spherical shells pierced with a central hole. The outer edge of the shell is simply supported whereas the inner edge is absolutely free. The material of the shell is assumed to be an ideal plastic material obeying the von Mises yield condition. Resorting to the lower bound theorem of limit analysis, shells with constant and piece-wise constant thickness are considered. The designs of spherical shells corresponding to maximal load carrying capacity are established for a given weight. Necessary optimality conditions are derived with the aid of variational methods from the theory of optimal control. The obtained set of equations is solved numerically.

## A review of optimization of cast parts using topology optimization

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 491-497 , December 01, 2005

In recent years, there has been considerable progress in the optimization of cast parts with respect to strength, stiffness, and frequency. Here, topology optimization has been the most important tool in finding the optimal features of a cast part, such as optimal cross-section or number and arrangement of ribs. An optimization process with integrated topology optimization has been used very successfully at Adam Opel AG in recent years, and many components have been optimized. This two-paper review gives an overview of the application and experience in this area. This is the first part of a two-paper review of optimization of cast parts.Here, we want to focus on the application of the original topology optimization codes, which do not take manufacturing constraints for cast parts into account. Additionally, the role of shape optimization as a fine-tuning tool will be briefly analyzed and discussed.

## An application of multi-objective stochastic optimisation to structural design

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 272-284 , April 01, 2005

An application to structural design of an innovative method for optimising stochastic systems is introduced in the paper. The proposed method allows one to carry out both the multi-objective optimisation of a structural element and to improve the robustness of the design. The innovative method is rather general. To show its effectiveness, an ideal cantilever has been designed in order to minimise both mass and deflection. The cantilever is shaped as a beam and is subject to random loads acting at its free end. The beam geometrical dimensions and material properties vary stochastically due to manufacturing tolerances. Different beam cross sections and two different materials (aluminium alloy and steel) have been considered. From the optimisation, it turned out that the optimal solutions are the O and the I beam, depending on the required lightness and stiffness. Compared to steel, aluminium alloy beams have provided better (or at least equal) performance.

## Erratum for the Brief Note “Extended optimality in topology design” by G.I.N. Rozvany, O.M. Querin, Z. Gaspar, V. Pomezanski, (SMO 24, 257–261, 2002)

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 504 , December 01, 2005

## Determination of optimal actuator forces and positions in smart structures using adjoint method

### Structural and Multidisciplinary Optimization (2005-10-01) 30: 308-319 , October 01, 2005

The problem of optimal design of structures with active support is analyzed in the paper. The sensitivity expressions with respect to the generalized force and the position of actuator are derived by the adjoint structure approach. Next, the optimality conditions are formulated by means of an introduced Lagrangian function. The problem of introduction of a new actuator is also considered and the condition of modification is expressed by means of the topological derivative. The obtained sensitivity formula, optimality conditions and modification conditions are applied in the optimization algorithm with respect to the number, positions and generalized forces of the actuators. Numerical examples of optimal control of beams illustrate the procedure proposed in the paper.

## Multiobjective structural optimization using a microgenetic algorithm

### Structural and Multidisciplinary Optimization (2005-11-01) 30: 388-403 , November 01, 2005

In this paper, we present a genetic algorithm with a very small population and a reinitialization process (a microgenetic algorithm) for solving multiobjective optimization problems. Our approach uses three forms of elitism, including an external memory (or secondary population) to keep the nondominated solutions found along the evolutionary process. We validate our proposal using several engineering optimization problems taken from the specialized literature and compare our results with respect to two other algorithms (NSGA-II and PAES) using three different metrics. Our results indicate that our approach is very efficient (computationally speaking) and performs very well in problems with different degrees of complexity.

## A reevaluation of the SIMP method with filtering and an alternative formulation for solid–void topology optimization

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 428-436 , December 01, 2005

The most popular way to introduce the notion of topology into the structural analysis of the topology optimization problem is through the Solid Isotropic Material with Penalization (SIMP) method. The fundamental principle behind its use requires a density design variable dependent material constitutive law that penalizes intermediate density material in combination with an active volume constraint. Here, the SIMP method with filtering is reevaluated, and an alternative topology optimization problem formulation, called the SINH (pronounced “cinch”) method, is developed that exploits this principle. The main advantages of the SINH method are that the optimization problem is consistently defined, the topology description is unambiguous, and the method leads to predominantly solid–void designs.

## Shape optimization of stiffeners in stiffened composite plates with thermal residual stresses

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 38-42 , July 01, 2005

The finite element analysis method is used to examine the influence of manufacturing-induced thermal residual stresses on the optimal shape of stiffeners in stiffened, symmetrically laminated plates. Three stiffener arrangements are studied via an optimization process in which the objective is to maximize the first natural frequency of the stiffened plate. The optimization problem is solved using the method of moving asymptotes (MMA). The numerical simulations indicate that thermal residual stresses can either cause a dispersion of stiffeners along the perimeter or a concentration around the centre. Further, the optimum fundamental frequency tends to increase with increasing temperature difference.

## A note on the theoretical convergence properties of the SIMP method

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 319-323 , April 01, 2005

The solid isotropic material with penalization (SIMP) method is used in topology optimization to solve problems where the variables are 0 or 1. The theoretical convergence properties have not been exhaustively studied. In this paper a convergence theorem with weaker assumptions than earlier conditions is given.

## Optimization of fiber orientations near a hole for increased load-carrying capacity of composite laminates

### Structural and Multidisciplinary Optimization (2005-11-01) 30: 335-341 , November 01, 2005

In this paper, a procedure for optimizing the fiber orientations near a hole in a single layer of multilayer composite laminates for increased strength is presented. A symmetric six-layer [(C-0)/+45/-45]_{s} laminate with central hole is considered. Within the ±45^{°} layers, the fiber orientations are fixed. The C-0 layer is divided into two fields: a small near field around the hole and a relatively large far field away from the hole. In the far field, the fiber orientations are fixed at 0^{°}, and in the near field, the fiber orientations are assumed to have continuous distribution represented by piecewise bilinear interpolation functions. The Tsai–Wu-criterion-based failure load magnitudes of [(C-0)/+45/-45]_{s} and [C-0]_{6} designs are maximized by iteratively alternating between a gradient-based search and a genetic algorithm. The results show that the load-carrying capacity of composite laminates with hole can be greatly increased through the optimization of continuous fiber orientation distribution within a small area around the hole in the C-0 layer, and the optimum fiber orientation distribution pattern in the C-0 layer is similar to that of the corresponding principal stress orientation distribution. The [(C-0)/+45/-45]_{s} design is only a few percent weaker than the [C-0]_{6} design, which is important for carrying off-design loadings.

## Adaptive weighted-sum method for bi-objective optimization: Pareto front generation

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 149-158 , February 01, 2005

This paper presents a new method that effectively determines a Pareto front for bi-objective optimization with potential application to multiple objectives. A traditional method for multiobjective optimization is the weighted-sum method, which seeks Pareto optimal solutions one by one by systematically changing the weights among the objective functions. Previous research has shown that this method often produces poorly distributed solutions along a Pareto front, and that it does not find Pareto optimal solutions in non-convex regions. The proposed adaptive weighted sum method focuses on unexplored regions by changing the weights adaptively rather than by using a priori weight selections and by specifying additional inequality constraints. It is demonstrated that the adaptive weighted sum method produces well-distributed solutions, finds Pareto optimal solutions in non-convex regions, and neglects non-Pareto optimal solutions. This last point can be a potential liability of Normal Boundary Intersection, an otherwise successful multiobjective method, which is mainly caused by its reliance on equality constraints. The promise of this robust algorithm is demonstrated with two numerical examples and a simple structural optimization problem.

## An optimality criterion for shape optimization in eigenfrequency problems

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 457-469 , June 01, 2005

For a broad class of static problems an optimality criterion of constant energy density at the designed boundary is known. In the present paper we prove a similar criterion for eigenfrequency problems. This optimality criterion serves as the tool for more basic understanding and for idealized reference cases as well as the basis for recursive procedures. Eigenfrequencies for in-plane vibrations as well as for out-of-plane vibrations of plates are optimized. The focus is on simplicity and multiple eigenfrequencies are not considered.

## Optimal design of plastic cylindrical shells of von Mises material

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 227-232 , September 01, 2005

Optimal design of plastic circular cylindrical shells of von Mises material is studied. The optimization problem is stated as the maximization problem of the load carrying capacity for given weight of the shell. Shells with constant and piecewise-constant thickness are considered. The maximization problem is performed under the requirement that the material volume of the stepped shell is equal to the case of the reference shell of constant thickness. The material of the shell is assumed to be an ideal rigid plastic obeying von Mises yield criterion. The considered nonlinear problems are solved by using the CASes method.

## Optimal design of shells against buckling by means of the simulated annealing method

### Structural and Multidisciplinary Optimization (2005-01-01) 29: 61-72 , January 01, 2005

In this paper, the problem of optimal design of shells against instability under combined state of loadings is considered. We look for the shape of a meridian as well as the thickness of a shell, which ensures the maximal critical value of the loading parameter. The equality constraining the volume of material and the capacity of a shell are considered. The concept of a shell of uniform stability is applied.

## Application of structure optimization technique to aluminum beverage bottle design

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 304-311 , April 01, 2005

This paper deals with shape optimization of two-piece aluminum beverage bottle bottoms by applying the structural optimization technique. Nonlinear finite element analyses were performed to study the influence of the design parameters on the buckling strength and the stiffness of the bottom under an axial load and internal pressure, respectively. On the basis of sensitivity analyses, the most effective dimensions are selected as design variables in the first stage design optimization, while the dimensions with relatively small influences are optimized in the secondary and following stages. In the design optimization of the bottom shape, the column strength is maximized subjected to the bottom growth and the axial displacement constraints, while the bulge strength is restricted to fall into a predetermined range. Optimizing the design variables progressively can avoid unrealistic geometrical connectivity problems during the arrangement of design points using the orthogonal array in the design-of-experiment, and can ensure the design space for the important design variables as well as the reduction of optimization cost.

## Polynomial genetic programming for response surface modeling Part 2: adaptive approximate models with probabilistic optimization problems

### Structural and Multidisciplinary Optimization (2005-01-01) 29: 35-49 , January 01, 2005

This is the second in a series of papers. The first deals with polynomial genetic programming (PGP) adopting the directional derivative-based smoothing (DDBS) method, while in this paper, an adaptive approximate model (AAM) based on PGP is presented with the partial interpolation strategy (PIS). The AAM is sequentially modified in such a way that the quality of fitting in the region of interest where an optimum point may exist can be gradually enhanced, and accordingly the size of the learning set is gradually enlarged. If the AAM uses a smooth high-order polynomial with an interpolative capability, it becomes more and more difficult for PGP to obtain smooth polynomials, whose size should be larger than or equal to the number of the samples, because the order of the polynomial becomes unnecessarily high according to the increase in its size. The PIS can avoid this problem by selecting samples belonging to the region of interest and interpolating only those samples. Other samples are treated as elements of the extended data set (EDS). Also, the PGP system adopts a multiple-population approach in order to simultaneously handle several constraints. The PGP system with the variable-fidelity response surface method is applied to reliability-based optimization (RBO) problems in order to significantly cut the high computational cost of RBO. The AAMs based on PGP are responsible for fitting probabilistic constraints and the cost function while the variable-fidelity response surface method is responsible for fitting limit state equations. Three numerical examples are presented to show the performance of the AAM based on PGP.

## A topology Nash game for tumoral antiangiogenesis

### Structural and Multidisciplinary Optimization (2005-11-01) 30: 404-412 , November 01, 2005

Tumoral angiogenesis plays a critical role in the process of growth of cancerous tumors. We consider activators and inhibitors of angiogenesis as players of a Nash game. From the activator’s point of view, the medium formed by existing blood vessels, surrounding tissue, and the tumor outer surface is seen as a porous medium. In contrast, the inhibitor sees the same medium as an elastic structural environment. We then define a competition between activator and inhibitor distributions. The aim of the activator is to minimize the pressure drop, while the inhibitor’s aim is to minimize the elastic compliance of the surrounding host tissue or, in a duel version, minimize the drainage of the tumoral neovascularization. Numerical results illustrate how (theoretical) tumors develop multiple channels as an optimal response to the optimal distribution of inhibitors.

## On globally stable singular truss topologies

### Structural and Multidisciplinary Optimization (2005-03-01) 29: 170-177 , March 01, 2005

We consider truss topology optimization problems including a global stability constraint, which guarantees a sufficient elastic stability of the optimal structures. The resulting problem is a nonconvex semi-definite program, for which nonconvex interior point methods are known to show the best performance.

We demonstrate that in the framework of topology optimization, the global stability constraint may behave similarly to stress constraints, that is, that some globally optimal solutions are singular and cannot be approximated from the interior of the design domain. This behaviour, which may be called a *global stability singularity phenomenon*, prevents convergence of interior point methods towards globally optimal solutions. We propose a simple perturbation strategy, which restores the regularity of the design domain. Further, to each perturbed problem interior point methods can be applied.

## Topology optimization of free vibrations of fiber laser packages

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 341-348 , May 01, 2005

The optimization problems described in the present paper are inspired by the problem of fiber laser package design for vibrating environments. The optical frequency of tuned fiber lasers glued to stiff packages is sensitive to acoustic or other mechanical vibrations. The paper presents a method for reducing this sensitivity by limiting the glue point movement on the package while using only a limited knowledge of vibrating external forces. By use of topology optimization a density distribution for the package is obtained, where the critical eigenmode of the package only effects a small elongation of the fiber laser.

## Coordination specification in distributed optimal design of multilevel systems using the χ language

### Structural and Multidisciplinary Optimization (2005-03-01) 29: 198-212 , March 01, 2005

Coordination plays a key role in solving decomposed optimal design problems. Several coordination strategies have been proposed in the multidisciplinary optimization (MDO) literature. They are usually presented as a sequence of statements. However, a precise description of the concurrency in the coordination is needed for large multilevel or non-hierarchic coordination architectures. This article proposes the use of communicating sequential processes (CSP) concepts from concurrency theory for specifying and implementing coordination strategies in distributed multilevel optimization rigorously. CSP enables the description of the coordination as a number of parallel processes that operate independently and communicate synchronously. For this purpose, we introduce elements of the language χ, a CSP-based language that contains advanced data modeling constructs. The associated software toolkit allows execution of the specified coordination. Coordination specification using χ is demonstrated for analytical target cascading (ATC), a methodology for design optimization of hierarchically decomposed multilevel systems. It is shown that the ATC coordination can be compactly specified for various coordination schemes. This illustrates the advantage of using a high-level concurrent language, such as χ, for specifying the coordination of distributed optimal design problems. Moreover, the χ software toolkit is useful in implementing alternative schemes rapidly, thus enabling the comparison of different MDO methods.

## Topology design of large displacement compliant mechanisms with multiple materials and multiple output ports

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 477-490 , December 01, 2005

Topology optimization of compliant mechanisms is presented in this paper wherein the layout design problem is addressed in its original binary or discrete (0-1) form. Design variables are modeled as discrete variables and allowed to assume values pertaining only to their void (0) or solid (1) states. Due to this discrete nature, a genetic algorithm is employed as an optimization routine. Using the barrier assignment approach, the search algorithm is extended to use with multiple materials. The layout design of compliant mechanisms is performed wherein displacements at multiple points (ports) in the design region are maximized along the respective prescribed directions. With multiple output ports and multiple materials, additional freedom in motion and force transduction can be achieved with compliant mechanisms. Geometrically large deformation analysis is employed to compute the displacement-based multiple objectives that are extremized using Nondominated Sorting in Genetic Algorithms (or NSGA). With genetic algorithms, buckling or snap through like issues with nonconvergent solutions in the population when computing nonlinear deformations can be implicitly circumvented.

## Integrating linear physical programming within collaborative optimization for multiobjective multidisciplinary design optimization

### Structural and Multidisciplinary Optimization (2005-03-01) 29: 178-189 , March 01, 2005

Multidisciplinary design optimization (MDO) is a concurrent engineering design tool for large-scale, complex systems design that can be affected through the optimal design of several smaller functional units or subsystems. Due to the multiobjective nature of most MDO problems, recent work has focused on formulating the MDO problem to resolve tradeoffs between multiple, conflicting objectives. In this paper, we describe the novel integration of linear physical programming within the collaborative optimization framework, which enables designers to formulate multiple system-level objectives in terms of physically meaningful parameters. The proposed formulation extends our previous multiobjective formulation of collaborative optimization, which uses goal programming at the system and subsystem levels to enable multiple objectives to be considered at both levels during optimization. The proposed framework is demonstrated using a racecar design example that consists of two subsystem level analyses — force and aerodynamics — and incorporates two system-level objectives: (1) minimize lap time and (2) maximize normalized weight distribution. The aerodynamics subsystem also seeks to minimize rearwheel downforce as a secondary objective. The racecar design example is presented in detail to provide a benchmark problem for other researchers. It is solved using the proposed formulation and compared against a traditional formulation without collaborative optimization or linear physical programming. The proposed framework capitalizes on the disciplinary organization encountered during large-scale systems design.

## Review of formulations for structural and mechanical system optimization

### Structural and Multidisciplinary Optimization (2005-10-01) 30: 251-272 , October 01, 2005

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.

## Topology optimization of electrostatically actuated microsystems

### Structural and Multidisciplinary Optimization (2005-11-01) 30: 342-359 , November 01, 2005

This study addresses the design of electrostatically actuated microelectromechanical systems by topology optimization. The layout of the structure and the electrode are simultaneously optimized. A novel, continuous, material-based description of the interface between the structural and electrostatic domains is presented that allows the optimization of the interface topology. The resulting topology optimization problem is solved by a gradient-based algorithm. The electromechanical system response is determined by a coupled high-fidelity finite element model and a staggered solution procedure. An adjoint formulation of the coupled electromechanical design sensitivity analysis is introduced, and the global sensitivity equations are solved by a staggered method. The proposed topology optimization method is applied to the design of mechanisms. The optimization results show the significant advantages of varying the interface topology and the layout of the electrode versus conventional approaches optimizing the structural layout only.

## Design of band-gap grid structures

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 418-431 , June 01, 2005

This paper discusses issues related to designing band-gaps in periodic plane grid structures. Finite element analysis is used to solve the dynamic behavior of a representative unit cell and Bloch–Floquet theory is used to extend the results to the infinite structure. Particular attention is given to the addition of non-structural masses that are introduced as design variables. These are used to create desirable features in the dispersion diagram. Physical insight is presented into the optimal choice of locations where masses should be added and the results of several numerical examples are provided to highlight this and other features of how band-gaps can be created and located at desired frequency ranges. The effect of the skew angle of the underlying grid structure is also explored, as are mathematical refinements of the modelling of the beam elements and the rotational inertia of the added masses. A scaling feature between the size of the reducible and the irreducible reference cell is exploited and the manner in which this can simplify optimization approaches is discussed.

## Topology/shape optimisation of axisymmetric continuum structures – a metamorphic development approach

### Structural and Multidisciplinary Optimization (2005-01-01) 29: 73-83 , January 01, 2005

A novel topology/shape optimisation method for axisymmetric elastic solids, based on solid modeling and FE analysis, is presented. Optimal profiles of minimum-mass axisymmetric structures are sought by growing and degenerating simple initial structures subject to response constraints. The rates of the growth and degeneration are controlled based on the current objective and constraint functions of the optimisation problem under consideration. The optimal structures are developed metamorphically in specified infinite design domains using both quadrilateral and triangular axisymmetric finite elements that are ideally suited for modeling continua involving curved boundaries.

The robustness of this fully automatic method is studied and validated with the first example of seeking the optimal shape of a centrally suspended axisymmetric object with minimum strain energy caused by self-weight. Then the method is applied to a practical industrial design problem: the design of a turbine disk. The variations of load and boundary conditions caused by shape change in these problems, including the gravitational and centrifugal loads, and temperature distribution are accommodated in the optimisation procedures. Thus, the design model closely resembles the real design problem. The results demonstrate the success of the method in generating optimal but realistic solutions to practical design problems.

## A new editorial system and new format for Structural and Multidisciplinary Optimization

### Structural and Multidisciplinary Optimization (2005-03-01) 29: 169-170 , March 01, 2005

## A hierarchical neighbourhood search method for topology optimization

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 325-340 , May 01, 2005

This paper presents a hierarchical neighbourhood search method for solving topology optimization problems defined on discretized linearly elastic continuum structures. The design of the structure is represented by binary design variables indicating material or void in the various finite elements.

Two different designs are called *neighbours* if they differ in only one single element, in which one of them has material while the other has void. The proposed neighbourhood search method repeatedly jumps to the “best” neighbour of the current design until a local optimum has been found, where no further improvement can be made. The “engine” of the method is an efficient exploitation of the fact that if only one element is changed (from material to void or from void to material) then the new global stiffness matrix is just a low-rank modification of the old one. To further speed up the process, the method is implemented in a hierarchical way. Starting from a coarse finite element mesh, the neighbourhood search is repeatedly applied on finer and finer meshes.

Numerical results are presented for minimum-weight problems with constraints on respectively compliance, strain energy densities in all non-void elements, and von Mises stresses in all non-void elements.

## Synthesis of multistable equilibrium linkage systems using an optimization approach

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 477-487 , June 01, 2005

This paper introduces a methodology for designing multistable equilibrium (MSE) systems. The methodology derives design criteria that relate system equilibrium characteristics to a potential energy curve. These design criteria are then used in a performance index that guides a candidate’s potential energy to approach the desired potential energy curve. As an example, a four-bar linkage with linear translational springs attached demonstrates the design methodology and the difficulties inherent in synthesizing MSE systems. To solve for the unknown design variables of the example problem a genetic algorithm is used.

## Optimum design of truss topology under buckling constraints

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 169-180 , September 01, 2005

The present paper studies the optimum design of truss topology under buckling constraints based on a new formulation of the problem. Through the incorporation of a global system stability constraint into the problem formulation, isolated compressive bars are eliminated from the final optimal topology. Furthermore, by including overlapping bars in the initial ground structure, the difficulty caused by hinge cancellation as pointed out by Rozvany (1996) can be overcome. Also, the importance of inclusion of compatibility conditions in the problem formulation is demonstrated. Finally, several numerical examples are presented for demonstration of the effectiveness of the proposed approach.

## Structural optimization with member dimensional imperfections

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 1-10 , July 01, 2005

The paper deals with the effect of dimensional imperfections of truss members on the minimum weight design of a structure. It is assumed that for each element its imperfections cannot exceed given *a priori* maximum values, called tolerances. The incorporation of the considered imperfections into the design is achieved by diminishing the limit values of state functions by the product of assumed imperfections and appropriate sensitivities. Therefore, the given method allows the introduction of tolerances into the design in a relatively simple way. In the submitted paper both members’ cross-section and length imperfections are discussed. The paper is illustrated with several design examples, considering cases with multiple loading conditions and buckling analysis. The achieved optimum solutions for designs with admissible tolerances show significant differences in structural weight and material distribution compared to ideal structures (i.e. having nominal dimensions). The calculations for designs with buckling analysis also reveal changes in material distribution compared to the designs without buckling constraints.

## Minimum cost design of a welded stiffened square plate loaded by biaxial compression

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 298-303 , April 01, 2005

The optimized plate structure consists of a simply supported square base plate stiffened with an orthogonal grid of flat stiffeners welded to the base plate by fillet welds. The uniformly distributed compressive load acts biaxially in the plane determined by the centre of gravity of T-sections, which consist of a part of the base plate and of a stiffener. In the optimization process the number of stiffeners as well as the thicknesses of the base plate and flat stiffeners, which minimize the cost function and fulfil the design constraints, as sought. The cost function includes the cost of material, assembly, welding and painting. Constraints relate to the global buckling, local buckling of base plate parts and stiffeners as well as to the deflection due to shrinkage of welds. To illustrate the effectiveness of the mathematical methods, the problem is solved by the Rosenbrock’s hill-climb algorithm as well as by entropy-based unconstrained minimization.

## Imparting desired attributes in structural design by means of multi-objective optimization

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 432-444 , June 01, 2005

Commonly available optimization methods typically produce a single optimal design as a constrained minimum of a particular objective function. However, in engineering design practice it is quite often important to explore as much of the design space as possible, with respect to many attributes, to discover what behaviors are possible and not possible within the initially adopted design concept. This paper shows that the very simple method of the sum of weighted objectives is useful for such exploration. By geometrical argument it is demonstrated that if every weighting coefficient is allowed to change its magnitude and its sign then the method returns a set of designs that are all feasible, diverse in their attributes, and include the Pareto and non-Pareto solutions, at least for convex cases. Numerical examples in the paper include the case of an aircraft wing structural box with thousands of degrees of freedom and constraints, and over 100 design variables, whose attributes are structural mass, volume, displacement, and frequency. The weighted coefficients method is inherently suitable for parallel, coarse-grained implementation that enables exploration of the design space in the elapsed time of a single structural optimization.

## Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field

### Structural and Multidisciplinary Optimization (2005-11-01) 30: 360-367 , November 01, 2005

We explore a typical optimal design problem in 2D conductivity: given fixed amounts of two isotropic dielectric materials, decide how we are to mix them in a 2D domain so as to minimize a certain cost functional depending on the underlying electric field. We rely on a reformulation of the optimal design problem as a vector variational problem and examine its relaxation, taking advantage of the explicit formulae for the relaxed integrands recently computed in Pedregal (2003). We provide numerical evidence, based on our relaxation, that Tartar’s result Tartar (1994) is verified when the target field is zero (also for divergence-free fields) and optimal solutions are given by first-order laminates. This same evidence also holds for a general quadratic functional in the field.

## Nonparametric gradient-less shape optimization for real-world applications

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 201-218 , September 01, 2005

A nonparametric gradient-less shape optimization approach for finite element stress minimization problems is presented. The shape optimization algorithm is based on optimality criteria, which leads to a robust and fast convergence independent of the number of design variables. Sensitivity information of the objective function and constraints are not required, which results in superior performance and offers the possibility to solve the structural analysis task using fast and reliable industry standard finite element solvers such as ABAQUS, ANSYS, I-DEAS, MARC, NASTRAN or PERMAS. The approach has been successfully extended to complex nonlinear problems including material, boundary and geometric nonlinear behavior. The nonparametric geometry representation creates a complete design space for the optimization problem, which includes all possible solutions for the finite element discretization. The approach is available within the optimization system TOSCA and has been used successfully for real-world optimization problems in industry for several years. The approach is compared to other approaches and the benefits and restrictions are highlighted. Several academic and real-world examples are presented.

## Development of CFRP racing motorcycle rims using a heuristic evolutionary algorithm approach

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 54-65 , July 01, 2005

The scope of this paper is the application of evolutionary optimization methods to the development of composite fibre reinforced plastics (CFRP) racing motorcycle rims. The mass and the moment of inertia of a front and a rear CFRP rim are minimized subject to manufacturing, strength, and stiffness constraints. The stacking sequence of the composite laminates is optimized by applying a sophisticated parameterization concept making an excellent compromise between a huge variety of structure properties and a reasonable number of optimization parameters. The mechanical properties are simulated using the finite element analysis package ANSYS . Resulting displacement and Tsai–Wu index values are combined with the mass of the rim in order to assign a fitness value to each different design solution. The smart formulation of the fitness function allows the exploration of solutions close to the required strength and stiffness properties. The proprietary software DynOPS is utilized as an optimization engine. It links an evolutionary algorithm to arbitrary simulation programs and controls the entire optimization process. The sophisticated parameterization concept, together with the fitness function formulation, are the basis for the development of CFRP motorcycle rims decisively lighter than state-of-the-art magnesium alloy rims.

## An interval algorithm for multi-objective optimization

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 27-37 , July 01, 2005

This paper presents an interval algorithm for solving multi-objective optimization problems. Similar to other interval optimization techniques, [see Hansen and Walster (2004)], the interval algorithm presented here is guaranteed to capture all solutions, namely all points on the Pareto front. This algorithm is a hybrid method consisting of local gradient-based and global direct comparison components. A series of example problems covering convex, nonconvex, and multimodal Pareto fronts is used to demonstrate the method.

## The effect of normalization norms in multiple attribute decision making models: a case study in gear material selection

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 312-318 , April 01, 2005

The main objective of this paper is to evaluate the effect of different normalization norms within multiple attribute decision making (MADM) models. The application of the work is dedicated to gear material selection for power transmission. To this end, the general scheme of the decision model is first presented, with close attention to the context of material selection. Subsequently, the entropy method and technique for order preference by similarity to ideal solution (TOPSIS) are employed to weigh the selected failure criteria and to rank the selected material IDs, respectively. Finally, by the introduction of different norms to the solution algorithm, the effect of normalization formalism on the material selection using the MADM models is studied. A simple multiaxial strategy is also recommended from which safer engineering decisions may be attained.

## Optimal loading conditions in the design and identification of structures. Part 1: discrete formulation

### Structural and Multidisciplinary Optimization (2005-01-01) 29: 1-18 , January 01, 2005

The paper is concerned with a class of structural optimization problems for which loading distribution and orientation are unspecified. The optimal loading conditions correspond to the extremal structural response, which can be used in assessment of structural safety or in generating the maximum structure stiffness or compliance. In identification problems the optimal load distribution is selected in order to minimize the distance norm between model prediction and experimental data. The sensitivity derivatives and optimality conditions are derived in the paper using discretized formulations. The generalized coaxiality conditions of loading and displacement or adjoint displacement vectors generate eigenvalue problems specifying stationary solutions. The paper is illustrated by examples of optimal loading distribution in structure design and identification.

## CO-SIMP: extended SIMP algorithm with direct COrner COntact COntrol

### Structural and Multidisciplinary Optimization (2005-08-01) 30: 164-168 , August 01, 2005

Methods for suppressing corner contacts between solid elements in topology optimization are briefly reviewed and then a modified SIMP algorithm based on direct corner contact penalization is presented.

## A communication technology for internet-based collaboration optimization for distributed structural systems

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 391-397 , May 01, 2005

Along with development of incorporation of world economy, collaborative design and collaborative manufacture of enterprises often occur. Subsystems of a complex product system are often analyzed and optimized individually by different design units distributed at different places. In order to find the optimum designs of overall product systems automatically, the internet-based global coordinating optimization is necessary. In this paper, an email-based communication technology of global coordinative optimization of distributed structural systems is proposed, and its realizing approach, communication, process control, practical software and application example are presented.

## Joint ISSMO executive committee/local organizing committee meeting

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 503 , December 01, 2005

## Comparison between Newton and response-surface methods

### Structural and Multidisciplinary Optimization (2005-11-01) 30: 368-380 , November 01, 2005

A supporting-point placement scheme is presented that is used for calculating function derivatives by the method of differences as well as a quadratic response-surface approximation. The placement scheme unifies the Newton (NM) and response-surface (RSM) methods in the limiting case when the point-set distance parameter for the RSM is chosen as small as that for obtaining the derivatives needed by the NM. Two new RSM minimization strategies with and without line searches are presented. The numerical performance of the algorithms is studied by using well-known test functions and the paths through the two-dimensional variables space are plotted for easier interpretation of the performance results. The results are compared with results of numerical experiments found in the literature.

## On optimum thin-walled closed cross section

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 233-235 , September 01, 2005

This paper deals with the problem of finding a minimum area thin-walled closed cross section with prescribed constant thickness and flexural rigidity. The cross section is supposed to be double symmetrical with respect to the Cartesian reference system (x0y), where 0 is the centroid of the cross section, and subjected to a bending moment *M*. The vector *M* is taken to be non-coincident with the x or y axis. This means that to represent the flexural rigidity both I_{x} and I_{y} moments of inertia are required. The function describing the centerline of the thin-walled closed cross section is taken as unknown.

The solution of this problem shows that such a centerline is an ellipse.

## Structure topology optimization: fully coupled level set method via FEMLAB

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 407-417 , June 01, 2005

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.

## On the characteristics of optimum beams with optimum length surrounded by a Winkler’s medium

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 243-250 , September 01, 2005

In this paper an analytical approach is used to optimize a beam surrounded by a Winkler’s medium and laterally loaded by a force at the top. The optimization regards both the distribution of the mass along the beam and the beam length in order to minimize the top displacement. Therefore, after having defined a transversality condition, we implement an algorithm to optimize the length of optimum beams. After having achieved the dimensionless extremals of optimum beams with optimum length, we show that the found solutions describe a central field of moment extremals defined along the beam and with the origin at the top. Having achieved the Jacobi’s condition and the strengthened Legendre’s condition for an extremal of optimum beam length, a sufficient condition for a weak minimum along the beam is also achieved. Besides, a given set of cross-sections whose moment of inertia divided by a dimensioned constant is obtained by raising their area to the same exponent. The optimized distribution of the dimensionless cross-sectional area as well as the dimensionless rigidity of the Winkler’s medium are intrinsic properties of all optimum beams with optimum length whose cross-sections belong to the same set.

## Shape optimization of structural parts in dynamic mechanical systems based on fatigue calculations

### Structural and Multidisciplinary Optimization (2005-05-01) 29: 361-373 , May 01, 2005

In this paper a new method for an automated shape optimization of dynamically loaded components in mechanical systems is presented. The optimization is carried out by means of the results of a durability analysis based on finite elements. Load time histories, which are necessary for durability analyses, are derived from a multibody simulation. The whole optimization loop, which is an iterative procedure, incorporates all these gradual analysis steps and is implemented by the authors in a straightforward, batch-oriented manner using well-known standard software. Since the whole process involves several different analysis types, such as multibody system simulation and durability analysis, the resulting setup is rather complex. Furthermore, the reader may not be familiar with all the terms arising within the context of every single analysis domain. Therefore, some essential aspects of each of the stages involved in the process are explained to provide the reader with the necessary background. In the following, the required software setup as well as the implementation are described. Finally, an academic example is discussed to illustrate and clearly outline the potential of this method.

## Material identification in structural optimization using response surfaces

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 93-102 , February 01, 2005

This paper evaluates the use of a response surface optimization algorithm for structural material or parameter identification. The algorithm used is the successive response surface method (SRSM) as implemented in LS-OPT. Two methods are used in the formulation of the optimization problem. The first is to minimize the maximum deviation of the distance function between the simulated and experimental results at selected points, while the second approach minimizes the more standard least squares residual form of the distance function, effectively providing a compromised match over all the parameters selected. SRSM uses a trust region that is adapted using a heuristic contraction and panning approach. The method has only one user-required parameter, the size of the initial trust region. To illustrate the robustness of SRSM as a material identification tool, three test cases are presented. The first concerns the identification of the power-law material parameters of a simple tensile test specimen. The second test case determines the leakage coefficient-pressure load curve of an airbag given experimental kinematic data of a chest form impacting the airbag. In the third test case the material identification of a rate-dependent low-density foam material is conducted. It is shown that SRSM essentially converges within 10 iterations for all the test cases, and that the two distance function minimization approaches produce similar results.

## Cost optimization of singly and doubly reinforced concrete beams with EC2-2001

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 236-242 , September 01, 2005

A model for the optimal design of rectangular reinforced concrete sections is presented considering the stress–strain diagrams described in EC2-2001 and MC90. The following expressions are developed: economic bending moment; optimal area of steel and optimal steel ratio between upper and lower steel. All the expressions are in nondimensional form. The present model is applied to four different classes of concrete described in MC90. It is concluded that in nondimensional form the equations are nearly coincident for both singly and doubly reinforcement. It is also concluded that the ultimate strain for concrete in the compression zone, ε_{cm}, lies between the strain for peak stress ε_{c1} and the ultimate strain ε_{cu}. This result is relevant once that the maximum moment is obtained for this value, and not the value ε_{cu}, as defined in EC2-2001. Cost optimization is implemented in the code and compared with other optimum models based on the ultimate design of ACI.

## The SRV constraint for 0/1 topological design

### Structural and Multidisciplinary Optimization (2005-10-01) 30: 320-326 , October 01, 2005

In density-based topological design, 0/1 solutions are often sought, that is, one expects that the final design includes either elements with full material or no material, excluding grey areas. The accepted technique for achieving binary values for the densities is to use a solid isotropic microstructure with penalization (SIMP) material for which Young’s modulus is a polynomial function of the otherwise continuous relative densities. This approach indeed enhances 0/1 solutions in a significant manner and as such it has achieved prominent status in topological design. Nevertheless, this paper proposes a possible alternative to the SIMP methodology for generating 0/1 structures. The design variables are still the densities of the finite elements but Young’s modulus is a linear function of these densities (in some sense, a SIMP material without penalty). In order to drive the solution to a 0/1 layout a new constraint, labeled the sum of the reciprocal variables (SRV), is introduced. The constraint stipulates that the SRV must be larger or equal to its value at a discrete design for a specified amount of material. It is understood that this implies a minimum gage on the design variables, a provision which is also present in the standard fixed-grid formulation to avoid singular stiffness matrices. The technique turned out to be very effective in conjunction with the method of moving asymptotes (MMA) when using topological design methods for finding optimal layouts of patches of piezo-electric (PZT) material in order to minimize the mechanical noise emanating from vibrating surfaces. It also performed satisfactorily in classical structural topological design instances, as can be seen in the numerical examples that illustrate this work.

## Joint ISSMO executive committee/local organizing committee meeting

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 498-499 , December 01, 2005

## Shape design of pin-jointed multistable compliant mechanisms using snapthrough behavior

### Structural and Multidisciplinary Optimization (2005-10-01) 30: 327-334 , October 01, 2005

A general approach is presented for generating pin-jointed multistable compliant mechanisms using snapthrough behavior. An optimization problem is formulated for minimizing the total structural volume under constraints on the displacements at the specified nodes, stiffnesses at initial and final states, and load factors to lead to snapthrough behavior. The design variables are cross-sectional areas and the nodal coordinates. It is shown in the numerical examples that several mechanisms can be naturally found as a result of optimization starting from randomly selected initial solutions. It is also shown that no local bifurcation point exists along the equilibrium path, and the obtained mechanism is not sensitive to initial imperfections.

## Multi-fidelity optimization of laminated conical shells for buckling

### Structural and Multidisciplinary Optimization (2005-08-01) 30: 128-141 , August 01, 2005

Optimum laminate configuration for minimum weight of filament-wound laminated conical shells is investigated subject to a buckling load constraint. In the case of a composite laminated conical shell, due to the manufacturing process, the thickness and the ply orientation are functions of the shell coordinates, which ultimately results in coordinate dependence of the stiffness matrices (*A*,*B*,*D*). These effects influence both the buckling load and the weight of the structure and complicate the optimization problem considerably. High computational cost is involved in calculating the buckling load by means of a high-fidelity analysis, e.g. using the computer code STAGS-A. In order to simplify the optimization procedure, a low-fidelity model based on the assumption of constant material properties throughout the shell is adopted, and buckling loads are calculated by means of a low-fidelity analysis, e.g. using the computer code BOCS. This work proposes combining the high-fidelity analysis model (based on exact material properties) with the low-fidelity model (based on nominal material properties) by using correction response surfaces, which approximate the discrepancy between buckling loads determined from different fidelity analyses. The results indicate that the proposed multi-fidelity approaches using correction response surfaces can be used to improve the computational efficiency of structural optimization problems.

## Robust structural optimization of plate wing corresponding to bifurcation in higher mode flutter

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 437-446 , December 01, 2005

We propose a robust structural optimization of a plate wing that considers the bifurcation in higher mode flutter, which affects the robustness of an optimum design and the convergence of an optimization. In the optimization, the thickness distribution of a delta wing is considered to improve the critical dynamic pressure of supersonic flutter under the constant total mass. The convergence is improved by detaching the distance of eigenvalues between adjacent modes by the positive constant value in the optimization. Consequently, the critical dynamic pressure of the optimum design is six times larger than that of the uniform design. Moreover, to improve the robustness, the constraints are proposed to keep eigenvalues between adjacent modes apart by the specified distance which is composed of the sensitivity of the distance of eigenvalues. Numerical results indicate that the robustness of the optimum design is quantitatively obtained with the constraints.

## Managing approximation models in collaborative optimization

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 11-26 , July 01, 2005

Collaborative optimization (CO), one of the multidisciplinary design optimization techniques, has been credited with guaranteeing disciplinary autonomy while maintaining interdisciplinary compatibility due to its bi-level optimization structure. However, a few difficulties caused by certain features of its architecture have been also reported. The architecture, with discipline-level optimizations nested in a system-level optimization, leads to considerably increased computational time. In addition, numerical difficulties such as the problem of slow convergence or unexpected nonlinearity of the compatibility constraint in the system-level optimization are known weaknesses of CO.

This paper proposes the use of an approximation model in place of the disciplinary optimization in the system-level optimization in order to relieve the aforementioned difficulties. The disciplinary optimization result, the optimal discrepancy function value, is modeled as a function of the interdisciplinary target variables, and design variables of the system level. However, since this approach is hindered by the peculiar form of the compatibility constraint, it is hard to exploit well-developed conventional approximation methods. In this paper, neural network classification is employed as a classifier to determine whether a design point is feasible or not. Kriging is also combined with the classification to make up for the weakness that the classification cannot estimate the degree of infeasibility.

In addition, for the purpose of enhancing the accuracy of the predicted optimum, this paper also employs two approximation management frameworks for single-objective and multi-objective optimization problem in the system-level optimization. The approximation is continuously updated using the information obtained from the optimization process. This can cut down the required number of disciplinary optimizations considerably and lead to a design (or Pareto set) near to the true optimum (or true Pareto set) of the system-level optimization.

## Benchmark case studies in optimization of geometrically nonlinear structures

### Structural and Multidisciplinary Optimization (2005-10-01) 30: 273-296 , October 01, 2005

A structural optimization algorithm is developed for truss and beam structures undergoing large deflections against instability. The method combines the nonlinear buckling analysis using the displacement control technique, with the optimality criteria approaches. Several benchmark case studies illustrate the procedure and the results are compared with examples reported in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass or overestimates the buckling load for these types of structures, so a design based on the linear buckling analysis may result in catastrophic failure. The effect of geometrical nonlinearities and element imperfections has also been studied.

## Shape optimization of the cavitator for a supercavitating torpedo

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 159-167 , February 01, 2005

The torpedo is a vital component of the naval arsenal, and efforts are continually directed toward improving the technology to make the torpedo more lethal and more stealthy. Recently, a new direction of research is considering the high-speed torpedo, which is capable of reaching up to 200 mph underwater. When a torpedo travels at this speed, the flow around the body separates and a cavity is formed. This cavity generation due to high speeds is called supercavitation. And the drag force acting on this supercavitating torpedo dictates the thrust requirements for the propulsion system, to maintain a required cavity at the operating speed. Therefore, any reduction in the drag force, obtained by modifying the shape of the cavitator or the nose of the torpedo, would result in lower propulsion requirements. In this work, shape optimization techniques were employed to determine the optimum (minimum-drag) shape of the cavitator given certain operating conditions. Shape optimization was also used to determine the shape of the cavity for any given cavitator, using potential flow theory. Analytical sensitivities were derived for various parameters in order to implement a gradient-based optimization algorithm. The developed methodology is an optimization process where the cavity and cavitator shapes are determined simultaneously. The cavitator shape that induces minimum drag and the corresponding cavity shape can be used to model a supercavitating torpedo that fits in the generated cavity and satisfies the required performance characteristics.

## Compliant mechanism design using multi-objective topology optimization scheme of continuum structures

### Structural and Multidisciplinary Optimization (2005-08-01) 30: 142-154 , August 01, 2005

Topology optimization problems for compliant mechanisms using a density interpolation scheme, the rational approximation of material properties (RAMP) method, and a globally convergent version of the method of moving asymptotes (GCMMA) are primarily discussed. First, a new multi-objective formulation is proposed for topology optimization of compliant mechanisms, in which the maximization of mutual energy (flexibility) and the minimization of mean compliance (stiffness) are considered simultaneously. The formulation of one-node connected hinges, as well as checkerboards and mesh-dependency, is typically encountered in the design of compliant mechanisms. A new hybrid-filtering scheme is proposed to solve numerical instabilities, which can not only eliminate checkerboards and mesh-dependency efficiently, but also prevent one-node connected hinges from occurring in the resulting mechanisms to some extent. Several numerical applications are performed to demonstrate the validity of the methods presented in this paper.

## Topology optimization of channel flow problems

### Structural and Multidisciplinary Optimization (2005-09-01) 30: 181-192 , September 01, 2005

This paper describes a topology design method for simple two-dimensional flow problems. We consider steady, incompressible laminar viscous flows at low-to-moderate Reynolds numbers. This makes the flow problem nonlinear and hence a nontrivial extension of the work of Borrvall and Petersson (2003).Further, the inclusion of inertia effects significantly alters the physics, enabling solutions of new classes of optimization problems, such as velocity-driven switches, that are not addressed by the earlier method. Specifically, we determine optimal layouts of channel flows that extremize a cost function which measures either some local aspect of the velocity field or a global quantity, such as the rate of energy dissipation. We use the finite element method to model the flow, and we solve the optimization problem with a gradient-based math-programming algorithm that is driven by analytical sensitivities. Our target application is optimal layout design of channels in fluid network systems. Using concepts borrowed from topology optimization of compliant mechanisms in solid mechanics, we introduce a method for the synthesis of fluidic components, such as switches, diodes, etc.

## Structural topology optimization using a genetic algorithm with a morphological geometric representation scheme

### Structural and Multidisciplinary Optimization (2005-08-01) 30: 113-127 , August 01, 2005

This paper describes a versatile methodology for solving topology design optimization problems using a genetic algorithm (GA). The key to its effectiveness is a geometric representation scheme that works by specifying a skeleton which defines the underlying topology/connectivity of a structural continuum together with segments of material surrounding the skeleton. The required design variables are encoded in a chromosome which is in the form of a directed graph that embodies this underlying topology so that appropriate crossover and mutation operators can be devised to recombine and help preserve any desirable geometry characteristics of the design through succeeding generations in the evolutionary process. The overall methodology is first tested by solving ‘target matching’ problems—simulated topology optimization problems in each of which a ‘target’ geometry is first created and predefined as the optimum solution, and the objective of the optimization problem is to evolve design solutions to converge towards this target shape. The methodology is then applied to design two path-generating compliant mechanisms—large-displacement flexural structures that undergo some desired displacement paths at some point when given a straight line input displacement at some other point—by an actual process of topology/shape optimization.

## A multi-level optimization methodology for determining the dextrous workspaces of planar parallel manipulators

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 422-427 , December 01, 2005

A numerical multi-level optimization methodology is proposed for determining dextrous workspaces of 3-degree-of-freedom (3-dof) planar parallel manipulators, in which it is required that at any point within the workspace, the manipulator is able to assume any orientation in a specified range. The method starts by finding a single initial point on the boundary of the dextrous workspace. This first stage requires the successive solution of three separate optimization sub-problems, where the evaluation of the objective function for the second problem and the constraint functions in the third problem are determined by the solution of appropriate optimization problems at a lower level. Once the boundary point is identified, further successive points along the dextrous workspace boundary are traced by the application of the so-called chord method. In the latter procedure, the determination of each successive boundary point is also obtained via a constrained optimization problem, where the constraint functions are again evaluated via the solution of an optimization problem at a lower level. The proposed method is illustrated by its successful application to three different manipulator design geometries, and for various ranges of dexterity.

## Study of a new local update scheme for cellular automata in structural design

### Structural and Multidisciplinary Optimization (2005-02-01) 29: 103-112 , February 01, 2005

This paper investigates an improved local update scheme for cellular automata (CA) applied to structural design. Local analysis and design rules are derived for equilibrium and minimum compliance design. The new update scheme consists of repeating analysis and optimality-based design rules locally. The benefits of this approach are demonstrated through a series of systematic experiments. Truss topology design problems of various sizes are used based on the Gauss–Seidel and the Jacobi iteration modes. Experiments show the robust convergence of the approach as compared to an earlier CA implementation. The approach is also extended to a plate problem.

## Combined topology and fiber path design of composite layers using cellular automata

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 413-421 , December 01, 2005

The popular Solid Isotropic Material Penalization (SIMP) technique of topology design is extended to simultaneous fiber-angle and topology design of composite laminae in a cellular automata (CA) framework. CA is a novel methodology to simulate a physical phenomenon based on iterative local updates of both field and design variables. Displacements are updated satisfying local equilibrium of CA cells. Fiber angles and density measures are updated based on the optimality criteria for the minimum compliance design. Numerical results for the design of 2D cantilever plates for single and multiple load cases are used to demonstrate the robustness of the proposed algorithm.

## Structural optimization as a harmony of design, fabrication and economy

### Structural and Multidisciplinary Optimization (2005-07-01) 30: 66-75 , July 01, 2005

The main requirements of load-carrying engineering structures are safety, producibility and economy. The role and importance of fabrication constraints and costs is shown in the case of a compression tubular strut, a tubular truss of parallel chords and a welded stiffened square plate. The evaluation of optimal versions of these structures shows that, neglecting the fabrication constraints and costs, the design cannot result in up-to-date solutions.

## Optimal design of aircraft structures with damage tolerance requirements

### Structural and Multidisciplinary Optimization (2005-08-01) 30: 155-163 , August 01, 2005

Damage tolerance analysis (DTA) was considered in the global design optimization of an aircraft wing structure. Residual strength and fatigue life requirements, based on the damage tolerance philosophy, were investigated as new design constraints. The global/local finite element approach allowed local fatigue requirements to be considered in the global design optimization. AFGROW fatigue crack growth analysis provided a new strength criterion for satisfying damage tolerance requirements within a global optimization environment. Initial research with the ASTROS program used this damage tolerance constraint to optimize cracked skin panels on the lower wing of a fighter/attack aircraft. For an aerodynamic and structural model of this type of aircraft, ASTROS simulated symmetric and asymmetric maneuvers during the optimization. Symmetric maneuvers, without underwing stores, produced the highest stresses and drove the optimization of the inboard lower wing skin. Asymmetric maneuvers, with underwing stores, affected the optimum thickness of the outboard hard points. Subsequent design optimizations included DTA and von Mises stress constraints simultaneously. In the configuration with no stores, the optimization was driven by the DTA constraint and, therefore, DTA requirements can have an active role to play in preliminary aircraft design.

## Optimal design of truss structures using parallel computing

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 285-297 , April 01, 2005

Parallel design optimization of large structural systems calls for a multilevel approach to the optimization problem. The general optimization problem is decomposed into a number of non-interacting suboptimization problems on the first level. They are controlled from the second level through coordination variables. Thus, the solutions of the independent first-level subsystems are directed towards the overall system optimum.

In the present paper, optimal design of truss structures using parallel computing technique is described. In this method, optimization of a large truss structure has been carried out by decomposing the structure into sub-domains and suboptimization tasks. Each sub-domain has independent design variables and a small number of behaviour constraints. The two-level sub-domain optimum design approach is summarized by several numerical examples with speedups and efficiencies of algorithms on message passing systems. It has been noticed that the efficiency of the algorithm for design optimization increases with the size of the structure.

## Variable chromosome length genetic algorithm for progressive refinement in topology optimization

### Structural and Multidisciplinary Optimization (2005-06-01) 29: 445-456 , June 01, 2005

This article introduces variable chromosome lengths (VCL) in the context of a genetic algorithm (GA). This concept is applied to structural topology optimization but is also suitable to a broader class of design problems. In traditional genetic algorithms, the chromosome length is determined a priori when the phenotype is encoded into the corresponding genotype. Subsequently, the chromosome length does not change. This approach does not effectively solve problems with large numbers of design variables in complex design spaces such as those encountered in structural topology optimization. We propose an alternative approach based on a progressive refinement strategy, where a GA starts with a short chromosome and first finds an ‘optimum’ solution in the simple design space. The ‘optimum’ solutions are then transferred to the following stages with longer chromosomes, while maintaining diversity in the population. Progressively refined solutions are obtained in subsequent stages. A strain energy filter is used in order to filter out inefficiently used design cells such as protrusions or isolated islands. The variable chromosome length genetic algorithm (VCL-GA) is applied to two structural topology optimization problems: a short cantilever and a bridge problem. The performance of the method is compared to a brute-force approach GA, which operates ab initio at the highest level of resolution.

## Design of cellular structures for optimum efficiency of heat dissipation

### Structural and Multidisciplinary Optimization (2005-12-01) 30: 447-458 , December 01, 2005

Metal cellular material is a new material attractive for its light weight and potential multifunctionality. In the present paper, we study cylindrical structures made of linear metal cellular material. The outer surface of the cylindrical structure is subjected to thermal boundary condition, and cooling fluid is forced through the cylinder to remove heat through the inner cell walls. Optimum design aims at maximization of heat dissipation efficiency under prescribed flow pressure. Two classes of design variables, relative density, and local aperture distribution of cellular material are to be determined by optimization under given total material volume constraints. Although similar to the structural topology optimization concept of material distribution, our formulation results in a structure with realistic cellular material of finite-sized aperture. Numerical results for different cross-sectional shapes and thermal boundary conditions are presented. Interestingly, our present formulation leads to optimum designs for cellular structures that mimic natural biomaterials. We discuss in general the guideline for cellular structure design to maximize heat dissipation efficiency based on insights from these optimization results.

## Note on topology optimization of continuum structures including self-weight

### Structural and Multidisciplinary Optimization (2005-04-01) 29: 245-256 , April 01, 2005

This paper proposes to investigate topology optimization with density-dependent body forces and especially self-weight loading. Surprisingly the solution of such problems cannot be based on a direct extension of the solution procedure used for minimum-compliance topology optimization with fixed external loads. At first the particular difficulties arising in the considered topology problems are pointed out: non-monotonous behaviour of the compliance, possible unconstrained character of the optimum and the parasitic effect for low densities when using the power model (SIMP). To get rid of the last problem requires the modification of the power law model for low densities. The other problems require that the solution procedure and the selection of appropriate structural approximations be revisited. Numerical applications compare the efficiency of different approximation schemes of the MMA family. It is shown that important improvements are achieved when the solution is carried out using the gradient-based method of moving asymptotes (GBMMA) approximations. Criteria for selecting the approximations are suggested. In addition, the applications also provide the opportunity to illustrate the strong influence of the ratio between the applied loads and the structural weight on the optimal structural topology.

## Polynomial genetic programming for response surface modeling Part 1: a methodology

### Structural and Multidisciplinary Optimization (2005-01-01) 29: 19-34 , January 01, 2005

The second-order polynomial is commonly used for fitting a response surface but the low-order polynomial is not sufficient if the response surface is highly nonlinear. Based on genetic programming (GP), this paper presents a method with which high-order smooth polynomials, which can model nonlinear response surfaces, can be built. Since in many cases small samples are used to fit the response surface, it is inevitable that the high-order polynomial shows serious overfitting behaviors. Moreover, the high-order polynomial shows infamous wiggling, unwanted oscillations, and large peaks. To suppress such problematic behaviors, this paper introduces a novel method, called directional derivative-based smoothing (DDBS) that is very effective for smoothing a high-order polynomial.

The role of GP is to find appropriate terms of a polynomial through the application of genetic operators to GP trees that represent polynomials. The GP tree is transformed into the standard form of a polynomial using the translation algorithm. To estimate the coefficients of the polynomial quickly the ordinary least-square (OLS) method that incorporates the DDBS and extended data-set method is devised.

Also, by using the classical Lagrange multiplier method, the modified OLS method enabling interpolation is presented.

Four illustrative numerical examples are given to demonstrate the performance of GP with DDBS.