Showing papers in "Structural Optimization in 1998"
TL;DR: The current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method are summarized and the methods with which they can be avoided are listed.
Abstract: In this paper we seek to summarize the current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method. The checkerboard problem refers to the formation of regions of alternating solid and void elements ordered in a checkerboard-like fashion. The mesh-dependence problem refers to obtaining qualitatively different solutions for different mesh-sizes or discretizations. Local minima refers to the problem of obtaining different solutions to the same discretized problem when choosing different algorithmic parameters. We review the current knowledge on why and when these problems appear, and we list the methods with which they can be avoided and discuss their advantages and disadvantages.
1,796 citations
TL;DR: The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical.
Abstract: The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which design application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approximations to the objective function and managing the use of these approximations as surrogates for optimization. The result is to obtain convergence to a minimizer of an expensive objective function subject to simple constraints. The approach is widely applicable because it does not require, or even explicitly approximate, derivatives of the objective. Numerical results are presented for a 31-variable helicopter rotor blade design example and for a standard optimization test example.
1,057 citations
TL;DR: In this article, an adaptive material topology optimization is extended to elastoplasticity, and the objective of the design problem is to maximize the structural ductility which is defined by the integral of the strain energy over a given range of a prescribed displacement.
Abstract: Material topology optimization is applied to determine the basic layout of a structure. The nonlinear structural response, e.g. buckling or plasticity, must be considered in order to generate a reliable design by structural optimization. In the present paper adaptive material topology optimization is extended to elastoplasticity. The objective of the design problem is to maximize the structural ductility which is defined by the integral of the strain energy over a given range of a prescribed displacement. The mass in the design space is prescribed. The design variables are the densities of the finite elements. The optimization problem is solved by a gradient based OC algorithm. An elastoplastic von Mises material with linear, isotropic work-hardening/softening for small strains is used. A geometrically adaptive optimization procedure is applied in order to avoid artificial stress singularities and to increase the numerical efficiency of the optimization process. The geometric parametrization of the design model is adapted during the optimization process. Elastoplastic structural analysis is outlined. An efficient algorithm is introduced to determine the gradient of the ductility with respect to the densities of the finite elements. The overall optimization procedure is presented and verified with design problems for plane stress conditions.
221 citations
TL;DR: In this article, the exact analytical truss solutions for some "benchmark" problems, which are often used as test examples in both discretized layout optimization of trusses and variable topology shape optimization of perforated plates under plane stress, are provided.
Abstract: The aim of this paper is to provide the exact analytical truss solutions for some “benchmark” problems, which are often used as test examples in both discretized layout optimization of trusses and variable topology (or generalized) shape optimization of perforated plates under plane stress.
117 citations
TL;DR: In this paper, an explicit finite element code, DYNA3D, is adopted for simulating complicated crushing behavior of tubular structures, and the response surface approximation technique is applied to construct an approximated design subproblem in the preassigned design space by using the technique of design-of-experiment.
Abstract: This paper concerns the development of crashworthiness maximization techniques for tubular structures, and the application to the axial crushing problem of cylindrical tubes as well as square tubes. In the program system presented in this study, an explicit finite element code, DYNA3D is adopted for simulating complicated crushing behaviour of tubular structures. The response surface approximation technique is applied to construct an approximated design subproblem in the preassigned design space by using the technique of design-of-experiment. The approximated subproblem is solved by the usual mathematical programming technique. These optimization processes are repeated until the given convergence conditions are satisfied. Moreover, a comparison of the crushing energy absorption between cylindrical tubes and square tubes is discussed.
101 citations
TL;DR: In this paper, a general approach that avoids the abovementioned problems of nonconvexity when ply-angles are used as design variables is proposed, which is based upon the fact that the design space for an optimization problem formulated in lamination parameters is proven to be convex, because the laminate stiffnesses are expressed linearly in terms of the lamin parameters.
Abstract: This paper deals with optimization of laminated composite structures in which the ply angles are taken as design variables. One of the major problems when using ply-angles as design variables, is the lack of convexity of the objective function and thus the existence of local optima, which implies that usual gradient based optimization procedures may not be effective. Therefore, a new general approach that avoids the abovementioned problems of nonconvexity when ply-angles are used as design variables is proposed. The methodology is based upon the fact that the design space for an optimization problem formulated in lamination parameters [introduced by Tsai and Pagano (1968)] is proven to be convex, because the laminate stiffnesses are expressed linearly in terms of the lamination parameters. However, lamination parameters have at least two major shortcomings: as yet, for the general case involving membrane-bending coupling, the constraints between the lamination parameters are not completely defined; also, for a prescribed set of lamination parameters physically realizable composite laminates (e.g. laminates with equal thickness plies) may not exist. The approach here, uses both lamination parameters and ply-angles and thereby uses the advantages of both and eliminates the shortcomings of both. In order to illustrate this approach, several stiffness optimization examples are provided.
83 citations
TL;DR: In this paper, an energy-based method is proposed to determine the optimal orientation of orthotropic materials under static loading, instead of assuming that the strain or stress fields are fixed with respect to orientational variables, the dependency of strain and stress fields on material orientation is explored by introducing an energy factor in the inclusion model.
Abstract: In this paper, an energy based method is proposed to determine the optimal orientation of orthotropic materials under static loading. Instead of assuming that the strain or stress fields are fixed with respect to orientational variables, the dependency of strain and stress fields on material orientation is explored by introducing an energy factor in the inclusion model. From the derivations, the strain based method and the stress based method can be recovered and their limitations are discussed. Numerical examples from these methods are presented and compared.
60 citations
TL;DR: In this article, a topology optimization based approach is proposed to design the optimal stiffener of three-dimensional shell/plate structures for static and eigenvalue problems, where the stiffener location problem is solved by a microstructure-based design domain method and the orientation problem is modelled as an optimization orientation problem of equivalent orthotropic materials.
Abstract: A systematic topology optimization based approach is proposed to design the optimal stiffener of three-dimensional shell/plate structures for static and eigenvalue problems. Optimal stiffener design involves the determination of the best location and orientation. In this paper, the stiffener location problem is solved by a microstructure-based design domain method and the orientation problem is modelled as an optimization orientation problem of equivalent orthotropic materials, which is solved by a newly developed energy-based method. Examples are presented to demonstrate the application of the proposed approach.
59 citations
TL;DR: In this paper, the authors present consistent numerical techniques commonly used for treatment of coupled structural and acoustic dynamics, and use the structural optimization tool ODESSY for solution of several coupled problems, and compare the numerical efficiency of alternative techniques and the relevance of selected objective functions.
Abstract: This paper is devoted to problems of structuralacoustic coupling with emphasis on analysis, design sensitivity analysis and optimization. The paper is divided into two parts, and it is the aim of Part I to (i) give a brief survey of recent developments in sensitivity analysis and sound emission and NVH (Noise, Vibration and Harshness) design of acoustically loaded structures, and (ii) discuss alternative objective functions and optimization formulations for structural acoustics. The aims of Part II are to (i) present consistent numerical techniques commonly used for treatment of coupled structural and acoustic dynamics, (ii) use the structural optimization tool ODESSY for solution of several coupled problems, and (iii) compare the numerical efficiency of alternative techniques and the relevance of selected objective functions.
59 citations
TL;DR: In this article, the optimum microstructures derived in explicit analytical form by Gibianski and Cherkaev are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading.
Abstract: In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic complicance or minimum total elastic energy at equilibrium. The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem. Several examples of optimum topology designs of three-dimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual two-dimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.
56 citations
TL;DR: The ability of the method to store known information about the behaviour of the problem makes it well-suited for practical multidisciplinary optimization.
Abstract: A mathematical programming method particularly suited for structural and multidisciplinary optimization problems is presented. It is based on the idea that all available information about the problem should be used in the attempt to obtain rapid convergence. Thus, it uses information from previous iterations to produce steadily improving approximations of the implicit objective and constraint functions of the problem. In some cases, this creates a response surface type of approximation. The ability of the method to store known information about the behaviour of the problem makes it well-suited for practical multidisciplinary optimization.
TL;DR: In this paper, the authors extend these results to the nonlinear model classified as power-law clasticity, and show that the proportionality between elastic strain energy density and elastic stress energy density implies localized sensitivity analysis for the total elastic energy.
Abstract: Recent results on optimal design with anisotropic materials and optimal design of the materials themselves are in most cases based on the assumption of linear clasticity. We shall extend these results to the nonlinear model classified as powerlaw clasticity. These models return proportionality between elastic strain energy density and elastic stress energy density. This is shown to imply localized sensitivity analysis for the total elastic energy, and for a number of optimal design problems this immediately gives practical, general results.
TL;DR: In this paper, the genetic algorithm based on real variable coding was applied to the strain energy minimization of rectangular laminated composite plates, and the results for both a point load and uniformly distributed load compare well with those achieved using trajectory methods for continuous global optimization.
Abstract: The design of laminated structures is highly tailorable owing to the large number of available design variables, thereby requiring an optimization method for effective design. Furthermore, in practice, the design problem translates to a discrete global optimization problem which requires a robust optimization method such as the genetic algorithm. In this paper, the genetic algorithm, based on the real variable coding, is applied to the strain energy minimization of rectangular laminated composite plates. The results for both a point load and uniformly distributed load compare well with those achieved using trajectory methods for continuous global optimization.
TL;DR: In this paper, the authors present a structural topology optimization problem that accounts for the presence of loads capable of causing permanent damage to the structure, represented in the form of an internal variable model which is standard in continuum damage mechanics.
Abstract: In this paper we present a formulation of the wellknown structural topology optimization problem that accounts for the presence of loads capable of causing permanent damage to the structure. Damage is represented in the form of an internal variable model which is standard in continuum damage mechanics. Here we employ an interpretation of this model as an optimum remodeling problem for maximal compliance over all damage distributions, making also the analysis of the damage model a study in structural optimization. The damage criterion can be included in the optimal design model in a number of ways. We present results for finding the optimal topology of the reinforcement of an existing design with the goal of minimizing damage. Also, we treat the problem of finding the topology of a structure where we seek maximal stiffness under service loads with a constraint on the amount of damage which occurs under a separate set of damage loads.
TL;DR: In this paper, a generalized evolutionary method is proposed to define structures that utilize their construction material to the greatest effect in the finite element sense, by basing the successive element erosion upon the contribution of an element to the strain energy of a structure and a certain material efficiency indicator.
Abstract: Generalized evolutionary methods, which successively construct and solve static equilibrium problems with progressive mesh adaptation, are useful tools for defining structures that utilize their construction material to greatest effect in the finite element sense. By basing the successive element erosion upon (i) the contribution of an element to the strain energy of a structure and (ii) a certain material efficiency indicator of a structure, several weaknesses associated with previous methods have been overcome. Under static loading conditions, the strain energy contribution of an element is determined solely by the related stiffness and displacement vector. Consequently, the method is effective, and efficient when applied to problems involving such loading conditions. The efficacy of the method is demonstrated through numerical applications to the problem of optimizing the topologies of two structures, a cantilever structure and a Michell structure
TL;DR: In this article, a multiparameter optimization method is developed for use with terrestrial and space reflector antenna electromechanical systems and other metallic and composite engineering structures, which incorporates the objectives from various structural and electromagnetic (EM) performances of the system at many working/loading cases simultaneously.
Abstract: A novel multiparameter optimization method is developed for use with terrestrial and space reflector antenna electromechanical systems and other metallic and composite engineering structures. To satisfy extremely high design requirements, the proposed approach incorporates the objectives from various structural and electromagnetic (EM) performances of the system at many working/loading cases simultaneously. A finite element method is used for structural analysis. Optical ray tracing, spline function aperture field interpolation, geometric optics aperture integration, and FFT techniques are employed to analyse the EM performances of distorted reflector antennas. A systematic method is used for parameter profile analysis of the system. The optimization involves member size, structural geometric and material design variables. Various terrestrial and orbital working environments and loading cases which affect antenna performances can be included in the optimization model. The optimization of an 8 m antenna system, as an example, is discussed and the results are given.
TL;DR: In this article, the authors present consistent numerical techniques commonly used for treatment of coupled structural and acoustic dynamics, and use the structural optimization tool ODESSY for solution of several coupled problems and compare the numerical efficiency of alternative techniques and the relevance of selected objective functions.
Abstract: This two-part paper is devoted to problems of structural-acoustic coupling with emphasis on analysis, design sensitivity analysis and optimization. Part II of the paper aims to (i) present consistent numerical techniques commonly used for treatment of coupled structural and acoustic dynamics, (ii) use the structural optimization tool ODESSY for solution of several coupled problems, and (iii) compare the numerical efficiency of alternative techniques and the relevance of selected objective functions.
TL;DR: In this paper, an algorithm of optimal design of supports including their number, position and stiffness is proposed, where the number of supports constitute topological design parameters, their positions correspond to configuration parameters.
Abstract: An algorithm of optimal design of supports including their number, position and stiffness is proposed. The number of supports constitute topological design parameters, their positions correspond to configuration parameters. Both, elastic and rigid supports are considered and the optimization is aimed to minimize the total structure cost. The topology bifurcation points correspond to generation of new supports. The topological sensitivity derivative is used in deriving the optimality conditions The optimization procedure provides number of supports, their position and stiffness of both supports and beam segments.
TL;DR: In this article, the robustness of aeroelastic design optimization with respect to uncertainties in material and structural properties is studied both numerically and experimentally, and the results indicate that robust and reliable optimization is achievable, but careful formulation of the optimization problem is essential.
Abstract: The robustness of aeroelastic design optimization with respect to uncertainties in material and structural properties is studied both numerically and experimentally. The model consists of thin orthotropic composite wings virtually without fuselage. Three different configurations with consistent geometry but varying orientation of the main stiffness axis of the material are investigated. The onset of aeroelastic instability, flutter, is predicted using finite element analysis and the doublet-lattice method for the unsteady aerodynamic forces. The numerical results are experimentally verified in a low-speed wind tunnel. The optimization problem is stated as to increase the critical air speed, above that of the bare wing by massbalancing. It is seen that the design goals are not met in the experiments due to uncertainties in the structural performance of the wings. The uncertainty in structural performance is quantified through numerous dynamic material tests. Once accounting for the uncertainties through a suggested reformulation of the optimization problem, the design goals are met also in practice. The investigation indicates that robust and reliable aeroelastic design optimization is achievable, but careful formulation of the optimization problem is essential.
TL;DR: In this paper, a method for transforming qualitative information obtained using the House of Quality (HQ) management tool into a constrained multi-objective optimization problem is presented, combining this qualitative design information with quantitative constraints used in typical design problems.
Abstract: There is a tremendous gap between the sophisticated optimization methods employed by engineers and the ad hoc decision-making methods employed by engineering managers. The goal of this paper is to bridge that gap by presenting a method for transforming qualitative information obtained using the “House of Quality” (a popular engineering management tool) into a constrained multiobjective optimization problem. Subjective multiatribute utility optimization is employed to combine this qualitative design information with quantitative constraints used in typical design problems. A structural dynamics example with conflicting objectives of structural cost, maintenance cost, service life, construction duration and occupant discomfort resulting from vibration illustrates the approach.
TL;DR: An analytical model is presented for the optimal design of linearly elastic continuum structures and a basis is introduced covering a general set of energy invariants to facilitate the expression of the combined analysis and design problem in general form.
Abstract: An analytical model is presented for the optimal design of linearly elastic continuum structures. To facilitate the expression of the combined analysis and design problem in general form, a basis is introduced covering a general set of energy invariants. Both internal (strain) energy and the expression of generalized cost are represented conveniently in terms of this basis, and as a result the optimality conditions for the design problem have a particularly simple form. Present developments comprise a reinterpretation and an extension of existing models where the design variable is the material modulus tensor, and where “cost” is represented in a general form. The conventional potential energy statement for linear continuum elastostatics is restated in the form of an isoperimetric problem, as a preliminary step. This interpretation of the mechanics is then incorporated in a max-min formulation applicable for the general design of linear continuum structures. To exemplify its application, the model is interpreted as it would apply for certain materials with particular geometric structure, e.g. crystalline forms. Also problems treated earlier where optimal material properties are predicted for the case where unit cost is proportional to the trace of the modulus tensor are identified as examples within the generalized formulation. The application of a recently developed technique to predict optimal black-white structures, i.e. designs having sharp topological features, is considered in the setting of the present generalized model.
TL;DR: The problem is addressed of growing least-volume trusses, starting from the simplest possible layout rather than from a complex ground structure, which has the advantage that it can produce much simpler, more realistic structures.
Abstract: The problem is addressed of growing least-volume trusses, starting from the simplest possible layout rather than from a complex ground structure. This approach to the optimal-layout problem is shown to be well suited to deflection-space methods of solution, which allow geometry and layout to be optimized simultaneously. The method has the advantage that it can produce much simpler, more realistic structures; allows joint-weight to be taken into account; and involves smaller computational problems. The key task at each stage is to generate least-volume linearly-elastic pin-jointed frames with prespecified numbers of joints. This problem is well-posed and is shown by examples to be solvable. The obstacles to be overcome in order to produce a practical computer implementation are analysed.
TL;DR: In this article, the authors studied the reliability-based structural optimization of the civil engineering in the seismic zone, where the objective is to minimize the sum of construction material cost and the expected failure loss under severe earthquake.
Abstract: The present paper studies the reliability-based structural optimization of the civil engineering in the seismic zone. The objective is to minimize the sum of construction material cost and the expected failure loss under severe earthquake, which is obtained by the sum of the products of the failure probability and its failure losses for the important failure modes. The set of constraints includes the deterministic constraints, and the constraints based on structural reliability—the reliability index constraints of structural element failure for the serviceability state under minor earthquake and the failure probability of the structural system for the ultimate limit state under severe earthquake. By introducing the load roughness index, the structural system reliability computation under hazard load can be greatly simplified, which is approximately determined by its weakest failure mode. Finally, the numerical example of high rising shear RC frame is calculated.
TL;DR: Multilevel iterative optimal design procedures, horrowed from the theory of structural optimization by means of homogenization, are used in this paper for the optimal material design of composite material structures.
Abstract: Multilevel iterative optimal design procedures, horrowed from the theory of structural optimization by means of homogenization, are used in this paper for the optimal material design of composite material structures. The method is quite general and includes materials with appropriate microstructure, which may lead eventually to phenomenological, overall negative Poisson's ratios. The benefits of optimal structural design gained by this approach, together with the first attempts to explain the taskoriented microstructure of natural structures, are investigated by means of numerical examples, and simulation of, among others, human bones.
TL;DR: In this article, a static and steady-state dynamic analysis of structures with flaw(s) is performed by the boundary element method, which is formulated as output (i.e. measurement) error minimization problems and solved by numerical optimization techniques.
Abstract: Single and multiple flaw identification problems are considered. Static and steady-state dynamic analysis of structures with flaw(s) is performed by the boundary element method. Inverse problems are formulated as output (i.e. measurement) error minimization problems and they are solved by numerical optimization techniques. As it is shown in this paper by means of numerical experiments, for elastostatic cases, an appropriate modelling of the structural analysis problem, a good choice of the error measure, and the use of established numerical optimization software are usually sufficient for the solution of the problem. Even multiple flaw identification is possible. Elastodynamic loadings lead to nonconvex problems which are solved here by means of global, genetic optimization algorithms.
TL;DR: In this paper, a rational, systematic and efficient fuzzy optimum design method for structural systems is developed by combining a suboptimization concept and fuzzy decision-making techniques at the discrete combinations of common design variables.
Abstract: In practical structural design problems, the designer should take into account several objectives, such as economics, safety, serviceability, aesthetic feeling and so on, and relative evaluations among these different characteristic objectives have some tolerances or fuzziness. For these reasons, the practical optimum structural design process concerning multiobjectives can be defined as a kind of compromising optimum decision-making process with fuzziness. In this paper, a rational, systematic and efficient fuzzy optimum design method for structural systems is developed by combining a suboptimization concept and fuzzy decisionmaking techniques at the discrete combinations of common design variables. The design method is applied to an optimum design problem of a large scale prestressed concrete bridge system considering two objectives; the total construction cost and aesthetic feeling. The rationality, the systematic design process and the efficiency of the proposed method are demonstrated.
TL;DR: In this article, the shape optimization of laminated structures via semianalytical sensitivity anslysis, based on linear programming, is studied. But the focus is focused on the concise and correct finite element formulation of the problem taking into account explicit differentiation with respect to the control parameters, and numerical examples show the influence of orthotropy parameters (treated as the level of anisotropy), stacking sequences, and the employed 2-D plate theories of structures on the resulting optimal shapes.
Abstract: The present work deals with the shape optimization of laminated structures via semianalytical sensitivity anslysis, based on linear programming Particular attention is focused on the concise and correct finite element formulation of the problem taking into account explicit differentiation with respect to the control parameters A series of numerical examples shows the influence of orthotropy parameters (treated as the level of anisotropy), stacking sequences, FE models and the employed 2-D plate theories of structures on the resulting optimal shapes
TL;DR: The paper concerns the optimal shakedown design of structures discretized by elastic perfectly plastic finite elements, and an alternative numerical approach devoted to obtaining the optimality conditions of the relevant original problems.
Abstract: The paper concerns the optimal shakedown design of structures discretized by elastic perfectly plastic finite elements. The design problem is formulated in four alternative versions, i.e. as the search for the minimum volume design whose shakedown limit load multiplier is assigned or as the search for the maximum shakedown limit load multiplier design whose volume is assigned; both problems are approached on the grounds of the shakedown lower bound and upper bound theorems. Correspondingly four computational methods, one for each original problem, are presented. These methods consist in solving iteratively new problems which are simpler than the original ones, but expressed in such a way that the obtained design and behavioural variables fulfill the optimality conditions of the relevant original problems, and thus they provide the true optimal design. Finally, an alternative numerical approach devoted to obtaining the optimal shakedown design is presented. Several numerical examples confirm the theoretical results.
TL;DR: A mixed approximation method called DQA-GMMA is presented, which uses the combination of Diagonal Quadratic Approximation and the Generalized Method of the Moving Asymptotes to solve constructed optimization problem by dual approach.
Abstract: A mixed approximation method called DQA-GMMA is presented in this paper. This approximation uses the combination of Diagonal Quadratic Approximation (DQA) and the Generalized Method of the Moving Asymptotes (GMMA). It has the flexibility to deal with both monotonic and non-monotonic design functions. The convexity and the separable form of this approximation ensures the efficient solution of constructed optimization problem by dual approach. Truss geometry and configuration problems are solved by this method.
TL;DR: In this paper, first and second-order shape sensitivity analyses in a fully nonlinear framework are presented using the fixed domain technique and the adjoint approach, integral expressions over the domain are obtained.
Abstract: First- and second-order shape sensitivity analyses in a fully nonlinear framework are presented in this paper. Using the fixed domain technique and the adjoint approach, integral expressions over the domain are obtained. The Guillaume-Masmoudi lemma allows these expressions to be rewritten as integrals over the domain boundary. The formalism is then applied to the steady creep of a bar in torsion, as an example of power-law nonlinearities that occur not only in creep problems but also in viscoplastic fluid flow. Finally, a problem with known analytical solution is presented in order to show the equivalence between exact differentiation and the shape sensitivity approach.