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Showing papers in "Structural Optimization in 1997"


Journal ArticleDOI
TL;DR: In this article, the authors provide a geometrical argument as to why the Pareto curve is convex, and show that this is not the case for all parts of the set.
Abstract: A standard technique for generating the Pareto set in multicriteria optimization problems is to minimize (convex) weighted sums of the different objectives for various different settings of the weights. However, it is well-known that this method succeeds in getting points from all parts of the Pareto set only when the Pareto curve is convex. This article provides a geometrical argument as to why this is the case.

1,052 citations


Journal ArticleDOI
TL;DR: An analytically robust, globally convergent approach to managing the use of approximation models of varying fidelity in optimization, based on the trust region idea from nonlinear programming, which is shown to be provably convergent to a solution of the original high-fidelity problem.
Abstract: This paper presents an analytically robust, globally convergent approach to managing the use of approximation models of various fidelity in optimization. By robust global behavior we mean the mathematical assurance that the iterates produced by the optimization algorithm, started at an arbitrary initial iterate, will converge to a stationary point or local optimizer for the original problem. The approach we present is based on the trust region idea from nonlinear programming and is shown to be provably convergent to a solution of the original high-fidelity problem. The proposed method for managing approximations in engineering optimization suggests ways to decide when the fidelity, and thus the cost, of the approximations might be fruitfully increased or decreased in the course of the optimization iterations. The approach is quite general. We make no assumptions on the structure of the original problem, in particular, no assumptions of convexity and separability, and place only mild requirements on the approximations. The approximations used in the framework can be of any nature appropriate to an application; for instance, they can be represented by analyses, simulations, or simple algebraic models. This paper introduces the approach and outlines the convergence analysis.

651 citations


Journal ArticleDOI
TL;DR: This paper presents a so-called ε-relaxed approach for structural topology optimization problems of discrete structures that eliminates the singular optima from the problem formulation and thus unify the sizing and topology optimized within the same framework.
Abstract: This paper presents a so-called e-relaxed approach for structural topology optimization problems of discrete structures. The distinctive feature of this new approach is that unlike the typical treatment of topology optimization problems based on the ground structure approach, we eliminate the singular optima from the problem formulation and thus unify the sizing and topology optimization within the same framework. As a result, numerical methods developed for sizing optimization problems can be applied directly to the solution of topology optimization problems without any further treatment. The application of the proposed approach and its effectiveness are illustrated with several numerical examples.

445 citations


Journal ArticleDOI
TL;DR: The authors develop a formal proof of convergence for the response surface approximation based optimization algorithm and show that response surface approximations constructed from variable fidelity data generated during concurrent subspace optimizations (CSSOs) can be effectively managed by the trust region model management strategy.
Abstract: To date the primary focus of most constrained approximate optimization strategies is that application of the method should lead to improved designs. Few researchers have focused on the development of constrained approximate optimization strategies that are assured of converging to a Karush-Kuhn-Tucker (KKT) point for the problem. Recent work by the authors based on a trust region model management strategy has shown promise in managing the convergence of constrained approximate optimization in application to a suite of single level optimization test problems. Using a trust-region model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, the authors have shown in application studies that the approximate optimization process converges to a KKT point for the problem. The approximate optimization strategy sequentially builds a cumulative response surface approximation of the augmented Lagrangian which is then optimized subject to a trust region constraint. In this research the authors develop a formal proof of convergence for the response surface approximation based optimization algorithm. Previous application studies were conducted on single level optimization problems for which response surface approximations were developed using conventional statistical response sampling techniques such as central composite design to query a high fidelity model over the design space. In this research the authors extend the scope of application studies to include the class of multidisciplinary design optimization (MDO) test problems. More importantly the authors show that response surface approximations constructed from variable fidelity data generated during concurrent subspace optimizations (CSSOs) can be effectively managed by the trust region model management strategy. Results for two multidisciplinary test problems are presented in which convergence to a KKT point is observed. The formal proof of convergence and the successfull MDO application of the algorithm using variable fidelity data generated by CSSO are original contributions to the growing body of research in MDO.

120 citations


Journal ArticleDOI
TL;DR: In this article, a mixed approach for probabilistic structural durability design of mechanical systems is proposed, where a deterministic design optimization that considers structural crack initiation and crack propagation lives at critical points of the structural component as design constraints is performed first.
Abstract: In this paper, a mixed approach for probabilistic structural durability design of mechanical systems is proposed. In this approach, a deterministic design optimization that considers structural crack initiation and crack propagation lives at critical points of the structural component as design constraints is performed first. After an optimal design is obtained, a reliability analysis is performed to ascertain if the deterministic optimal design is reliable. If the probability of the failure of the deterministic optimal design is found to be acceptable, a reliability-based design approach that employs a set of interactive design steps, such as trade-off analysis and what-if study, is used to obtain a near-optimal design that is reliable with an affordable computational cost. A 3-D tracked vehicle roadarm is employed to demonstrate the feasibility of the proposed approach.

94 citations


Journal ArticleDOI
TL;DR: For shape optimization methods to be fully accepted by the engineering community they must first be integrated with CAD systems, and here the critical link between these CAD and FEM data is developed within an industry standard feature-based modelling environment.
Abstract: The unification of computer aided design (CAD) and the finite element method (FEM) has greatly enhanced the engineer's ability to evaluate potential designs (Finniganet al. 1989). However, analysis alone is not the answer to design, and thus shape optimization methods have become increasingly popular, particularly for structural problems (Yanget al. 1992; Botkin 1992). However, for shape optimization methods to be fully accepted by the engineering community they must first be integrated with CAD systems. To date, the difficulty of integrating CAD and shape optimization is the inability to relate the finite element nodal coordinates to the CAD solid model dimensions. Here, the critical link between these CAD and FEM data is developed within an industry standard feature-based modelling environment. A three-dimensional connecting rod model is optimized to exemplify the method.

64 citations


Journal ArticleDOI
TL;DR: In this article, the problem of optimal structural design with linked discrete variables is addressed, and three strategies that combine a continuous variable optimization method with a genetic algorithm, simulated annealing, and branch and bound method are presented and implemented into a computer program for their numerical evaluation.
Abstract: The problem of optimal structural design having linked discrete variables is addressed. For such applications, when a discrete value for a variable is selected, values for other variables linked to it must also be selected from a table. The design of steel structures using available sections is a major application area of such problems. Three strategies that combine a continuous variable optimization method with a genetic algorithm, simulated annealing, and branch and bound method are presented and implemented into a computer program for their numerical evaluation. Three structural design problems are solved to study the performance of the proposed methods. CPU times for solution of the problems with discrete variables are large. Strategies are suggested to reduce these times.

59 citations


Journal ArticleDOI
TL;DR: In this article, general continuous topology design formulations based on families of classical Voigt and Reuss mixing assumptions are developed and applied to solve the multiple material layout problem for the design of high stiffness/high strength composites.
Abstract: General continuous topology design formulations based on families of classical Voigt and Reuss mixing assumptions are developed and applied to solve the multiple material layout problem for the design of high stiffness/high strength composites. In the novel design framework, computational homogenization is employed to compute stiffness and strength characteristics of individual composite designs. Alternative design formulations for both high stiffness and high strength are investigated along with design sensitivity analysis algorithms. Demonstrative material design problems for boron-epoxy and graphite-epoxy composites are solved with robust sequential quadratic programming (SQP) techniques.

55 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of optimal layout in three-dimensional elasticity is solved using a moment formulation to characterize the optimal orthotropy of the material at each point of the domain.
Abstract: The problem of optimal layout in three-dimensional elasticity is solved using a moment formulation to characterize the optimal orthotropy of the material at each point of the domain. Design for minimum compliance under a single constraint on the amount of material is considered.

54 citations


Journal ArticleDOI
TL;DR: In this paper, a new formulation for mathematical modelling to predict material properties for the optimal design of continuum structures is presented, based on an extended form of an already established characterization for continuum design, where the material properties tensor for an arbitrary structural continuum appears as the design variable.
Abstract: A new formulation is presented for mathematical modelling to predict material properties for the optimal design of continuum structures. The method is based on an extended form of an already established characterization for continuum design, where the material properties tensor for an arbitrary structural continuum appears as the design variable. The extension is comprised of means to represent an independently specifiedunit relative cost factor, which appears simply as a weighting function in the argument of the isoperimetric (cost) constraint of the original model. A procedure is demonstrated where optimal black/white topology is predicted out of a sequence of solutions to material properties design problems having thisgeneralized cost formulation form. A systematic adjustment is made in the unit relative cost field for each subsequent solution step in the sequence, and at the stage identified with final topology, no more than a small fraction of a percent of the total element area in the system has material property density off the bounding “black” or “white” levels. This technique is effective for the prediction of optimal black/white topology design for design around obstacles of arbitrary shape, as well as the more unusual topology design problems. Results are presented for 2D examples of both types of problem. In addition to the treatment for (the usual) minimum compliance design, an alternate formulation of the design problem is presented as well, one that provides for the prediction of optimum topology with a generalized measure of compliance as the objective.

45 citations


Journal ArticleDOI
TL;DR: In this article, the selective dynamic rounding (SDRR) algorithm is used for the optimal sizing design of truss structures subject to linear buckling constraints, and a continuous design based on the regression analysis of section effectiveness vs. area is used as a starting point for the dual step discrete optimization phase.
Abstract: The selective dynamic rounding (SDR) algorithm previously developed by the authors, and based on a dual step rounding approach, is used for the optimal sizing design of truss structures subject to linear buckling constraints. The algorithm begins with a continuous optimum followed by a progressive freezing of individual variables while solving the remaining continuous problems. The allowable member stresses are predicted by the linear regression of the tabular section properties, while the exact allowable compressive stresses are back-substituted for those variables fixed on discrete values in each intermediate mixed-discrete nonlinear problem. It is shown that a continuous design based on the regression analysis of section effectiveness vs. area is effective as a starting point for the dual step discrete optimization phase. A range of examples is used to illustrate that with “conservative” regression, discrete designs can be achieved which are significantly lighter than those in which the variables have been rounded up.

Journal ArticleDOI
TL;DR: In this paper, a stochastic GA was used for the optimization of stiffeners on a plate by varying their positions, while having well-defined dimensions, considering the nonlinearity, the non-convexity and the discontinuity of this problem.
Abstract: This paper deals with the optimization of stiffeners on plates by varying their positions, while having well-defined dimensions. Considering the nonlinearity, the non-convexity and the discontinuity of this problem, we have chosen to use a stochastic method, that is the genetic algorithm (GA). This work is above all a study of feasibility. Some improvements in the GA have been used and their influence on the convergence of the GA, while varying a number of parameters is made obvious by application on numerical examples and mechanical ones. To our knowledge, the problem in question, that is the topology optimization of the stiffeners on a plate, has not been treated in depth up till now, a fact that gives this subject particular importance.

Journal ArticleDOI
TL;DR: The paper discusses the computer implementation of a class of interior point algorithms for the minimization of nonlinear functions with equality and inequality constraints and the algorithms employed for the constrained line search and also with the quasi-Newton matrix updating.
Abstract: The paper discusses the computer implementation of a class of interior point algorithms for the minimization of nonlinear functions with equality and inequality constraints. These algorithms consist of fixed point iterations to solve KKT firstorder optimality conditions. At each iteration a descent direction is defined by solving a linear system. Then, the linear system is perturbed in such a way as to deflect the descent direction and obtain a feasible descent direction. A line search is finally performed to obtain a new interior point with a lower objective. Newton, quasi-Newton, or first-order versions of the algorithm can be obtained. This paper is mainly concerned with the solution of the internal linear systems, the algorithms that are employed for the constrained line search and also with the quasi-Newton matrix updating. Some numerical results obtained with a quasi Newton algorithm are also presented. A set of test problems were solved very efficiently with the same values of the internal parameters.

Journal ArticleDOI
TL;DR: In this article, the shape, volume fraction, and spatial arrangement of the piezoceramic rods, and the structure of the matrix material that maximizes the hydrophone performance were investigated.
Abstract: An optimal design problem for piezoelectric composite hydrophones is considered. The hydrophone consists of parallel piezoelectric rods embedded in a poroustransversely isotropic polymer matrix. We find the shape, volume fraction, and spatial arrangement of the piezoceramic rods, and the structure of the matrix material that maximizes the hydrophone performance characteristics. We found that the optimal composite consists of a hexagonal array of rods with small volume fraction, in a highly anisotropic matrix that is characterized by negative Poisson's ratios in certain directions. The performance characteristics of hydrophones with such a matrix are significantly higher than those with anisotropic polymer matrix. The results can be viewed as theoretical upper bounds on the hydrophone performance.


Journal ArticleDOI
J. Lellep1, J. Majak1
TL;DR: In this paper, the problem of minimizing the elastic energy density in the two-dimensional case for a nonlinear elastic solid is solved in the presence of a power law stress-strain relation.
Abstract: The problem of minimization (or maximization) of the elastic energy density is solved in the two-dimensional case for a nonlinear elastic solid. The material behaviour is simulated on the basis of a power law stress-strain relation. Closed-form solutions which include corresponding solutions for a linear elastic solid are obtained. The latter may give local as well as global maxima and minima, respectively.

Journal ArticleDOI
TL;DR: This work considers the minimum-compliance formulation of the truss topology problem with additional linear constraints on the displacements: the so-called displacement constraints, and proposes a new bilevel programming approach to this problem.
Abstract: We consider the minimum-compliance formulation of the truss topology problem with additional linear constraints on the displacements: the so-called displacement constraints. We propose a new bilevel programming approach to this problem. Our primal goal (upper-level) is to satisfy the displacement constraint as well as possible — we minimize the gap between the actual and prescribed displacement. Our second goal (lower level) is to minimize the compliance — we still want to find the stiffest structure satisfying the displacement constraints. On the lower level we solve a standard truss topology problem and hence we can solve it in the formulation suitable for the fast interior point alogrithms. The overall bilevel problem is solved by means of the so-called implicit programming approach. This approach leads to a nonsmooth optimization problem which is finally solved by a nonsmooth solver.

Journal ArticleDOI
TL;DR: In this article, the authors have developed a refined semianalytical method based on the fact that the contribution to the pseudo-load vector corresponding to rigid body motions can be evaluated by exact differentiation of the rigid body modes.
Abstract: In the recent past inaccuracy problems that arise when computing shape design sensitivities by the semianalytical method have been reported. Therefore, the authors have developed a refined semianalytical method. This method is based on the fact that the contribution to the pseudo-load vector corresponding to the rigid body motions can be evaluated by exact differentiation of the rigid body modes. In the present paper the effectiveness of this refined semianalytical method is studied by means of the beam model problem. This model problem has been investigated earlier by Barthelemyet al. (1988), Barthelemy and Haftka (1988), Pedersenet al. (1989) and Olhoff and Rasmussen (1991).

Journal ArticleDOI
TL;DR: This paper describes the construct and applications of a test simulator, CASCADE (Complex Application Simulator for the Creation of Analytical Design Equations), that is capable of randomly generating and then converging a system of coupled analytical equations, of user-specified size.
Abstract: Many of the method development efforts in the field of multidisciplinary design optimization (MDO) attempt to simplify the design of a large, complex system by dividing the system into a series of smaller, simpler, and coupled subsystems. A representative and efficient means of determining the feasibility and robustness of MDO methods is crucial. This paper describes the construct and applications of a test simulator, CASCADE (Complex Application Simulator for the Creation of Analytical Design Equations), that is capable of randomly generating and then converging a system of coupled analytical equations, of user-specified size (Hulme and bloebaum 1996). CASCADE-generated systems can be used for test sequencing and system reduction strategies, convergence strategies, optimization techniques, MDO methods, and distributed computing techniques (via Parallel Virtual Machine), among others.

Journal ArticleDOI
TL;DR: In this paper, the effect of adaptive mesh refinement and error control on the quality of the velocity fields is discussed, as well as their ability to yield accurate first-order predictions of constraint values.
Abstract: The design sensitivities generated with the mesh velocity method, used by the authors, are compared with those obtained by the boundary layer and boundary displacement methods. The effect of adaptive mesh refinement and error control on the quality of the velocity fields is discussed, as well as their ability to yield accurate first-order predictions of constraint values. Two numerical shape optimization examples of a 2D and a 3D component are presented. These examples are used to illustrate the benefits of integrating analytical methods of design sensitivity analysis with parametric capabilities supported by state-of-the-art CAD systems.

Journal ArticleDOI
TL;DR: In this article, a systematic exploration of Michell layouts for various combinations of line supports is continued, including those for reentrant corners, a line support and two free edges, line supports forming any convex polygon, supports consisting of two symmetric curves, and finally, rectangular domains bounded by one line support.
Abstract: In this paper, a systematic exploration of Michell layouts for various combinations of line supports is continued. The solutions discussed include those for reentrant corners, a line support and two free edges, line supports forming any convex polygon, supports consisting of two symmetric curves, and finally, rectangular domains bounded by one line support and three free edges.

Journal ArticleDOI
TL;DR: In this article, it is shown that there are important exceptions to this rule and that the modification of this restriction enables us to obtain new classes of solutions, such as least-weight trusses for one load condition with a stress or compliance constraint.
Abstract: It is often stated, even in standard references, that in classical Michell trusses (i.e. least-weight trusses for one load condition with a stress or compliance constraint) a pair of intersecting compression and tensile bars must always be orthogonal. The aim of this brief note is to show that there are important exceptions to this rule and that the modification of this restriction enables us to obtain new classes of solutions.

Journal ArticleDOI
TL;DR: In this article, the authors present the influence of a new parameter involved in this repartition, i.e., the distribution of cuts in the elastic constraining layer, which is used to determine the most appropriate distribution of the shear in the layers.
Abstract: To damp the flexural vibrations of homogeneous beams or plates in a large frequency range, one of the most efficient methods is the use of constrained viscoelastic layers. Since most of the damping ability is due to shearing stresses in the viscoelastic layer, it is interesting to determine the most appropriate distribution of the shear in the layers. This paper presents the influence of a new parameter involved in this repartition, i.e. the distribution of cuts in the elastic constraining layer. It will be demonstrated that modal damping may be significantly modified in this way. The number and the locations of the cuts may vary and are determined to optimize the damping. The vibrating beam modal analysis is performed by a finite element analysis using special finite elements which have variable d.o.f. in order to take into account the lack of continuity of the viscoelastic constrained displacement field. Using a genetic algorithm, an optimal distribution of the cuts has been determined for a maximum damping of one or serval flexural modes.

Journal ArticleDOI
TL;DR: In this article, two new criteria, namely the material efficiency criterion and the smooth change criterion, are derived for solving this kind of evolutionary optimization problem, and the evolutionary optimization method has been further extended and applied to maximize the difference between the fundamental and the second natural frequencies of a structure under both plane stress and thin plate flexural bending conditions.
Abstract: This paper deals with the evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure. Two new criteria, namely the material efficiency criterion and the smooth change criterion, are derived for solving this kind of evolutionary optimization problem. Using these two new criteria, the evolutionary optimization method has been further extended and applied to maximize the difference between the fundamental and the second natural frequencies of a structure under both plane stress and thin plate flexural bending conditions. The related results demonstrated that the extended evolutionary structural optimization method is useful in and applicable to dealing with the evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure. Moreover, the results also indicated that owing to the different mechanism between plane stress and thin plate flexural bending conditions, the optimal topologies, the normalized difference between two natural frequencies and the normalized material efficiency are different for a vibrating structure under these two different situations.

Journal ArticleDOI
TL;DR: In this paper, an ultrasonic transducer, consisting of two piezoelectric ceramic disks and two resonance rods, is optimized with respect to the shape of the resonance rods in order to maximize the amplitude of the vibrating tip.
Abstract: An ultrasonic transducer, consisting of two piezoelectric ceramic disks and two resonance rods, is optimized with respect to the shape of the two resonance rods in order to maximize the amplitude of the vibrating tip. The mathematical model is presented and a finite element model of the transducer is set up. Harmonic analysis (forced vibration) is used for calculation of resonance frequency, vibration mode, amplitude and phase. The resulting optimal shape is presented. The numerical analysis shows that the new design improves the amplitude 4.0 times. The improved transducer has been tested against the standard transducer and experiments show good agreement with theoretical results. A special design of the ultrasonic transducer for sonar applications has also been investigated. Applications of the improved ultrasonic transducer are more efficient transducers in high energy applications such as ultrasonic welding, drilling, disruption, cleaning, and sonar and underwater communication.

Journal ArticleDOI
TL;DR: In this paper a more detailed description of fatigue within continuum damage mechanics (CDM) considering the loading history is introduced and two new cost functions are determined for shape optimization of dynamically loaded machine parts.
Abstract: Almost any machine part is dynamically loaded and fails in most cases by fatigue. Nevertheless many authors recommend just a static shape optimization based on the minimization of maximum stress to improve the lifetime. In this paper a more detailed description of fatigue within continuum damage mechanics (CDM) considering the loading history is introduced. On this basis, two new cost functions are determined for shape optimization of dynamically loaded machine parts. An optimization procedure is built up using the methods of mathematical programming (MP). The optimized parts show a remarkably increased lifetime in numerical results as well as in experiments.

Journal ArticleDOI
TL;DR: In this article, a conical representation of yield stresses and their sensitivities is introduced based on a coneY¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ o of admissible pairs of external loads/strength increaments, and its bounds or approximations with respect to deterministic input or design variables are obtained by applying the transformation method/stochastic completion techniques; the derivatives of the yield stresses are represented again by certain expectations or multiple integrals.
Abstract: Yield stresses, allowable stresses, moment capacities (plastic moments), external loadings, manufacturing errors, etc., are not fixed quantities in practice, but must be modelled as random variables with a certain joint probability distribution. In reliability-oriented structural optimization the violation of the random behavioural constraints are evaluated by means of the corresponding probabilityp s of survival. Hence, the approximative computation ofp s and its sensitivities is of utmost importance. After the consideration of lower bounds ofp s based on a selection of certain redundants in the vector of internal forces/bending moments, and the consideration of upper bounds ofp s based on an optimizational representation of the yield or safety constraints by a pair of dual linear programs, a conical representation ofp s is introduced based on a coneY o of admissible pairs of external loads/strength increaments. Approximations ofp s can be constructed then by replacing the (finitely generated) coneY o by more simple ones, e.g. spherical or ellipsoidal cones. For the direct numerical computation of sensitivities ofp s and its bounds or approximations by using e.g. sampling methods or asymptotic expansion techniques based on Laplace integral representation of multiple integrals, exact differentiation formulae — of arbitrary order — forp s and its bounds or approximations with respect to deterministic input or design variables are obtained by applying the transformation method/stochastic completion techniques; the derivatives ofp s are represented again by certain expectations or multiple integrals.

Journal ArticleDOI
TL;DR: In this paper, an adaptation of GA's in decomposition-based design of multidisciplinary systems is described, where the coupled multi-disciplinary design problem is adaptively deomposed into a number of smaller subproblems, each with fewer design variables, and the design in each subproblem allowed to proceed in parallel.
Abstract: The paper describes an adaptation of genetic algorithms (GA's) in decomposition-based design of multidisciplinary systems. The coupled multidisciplinary design problem is adaptively deomposed into a number of smaller subproblems, each with fewer design variables, and the design in each subproblem allowed to proceed in parallel. Fewer design variables allow for shorter string lengths to the used in the GA-based optimization in each subproblem, reducing the number of design alternatives to be explored, and hence also reducing the required number of function evaluations to convergence. A novel procedure is proposed to account for interactions between the decomposed subproblems, and is based on the modelling of the biological immune system. This approach also uses the genetic algorithm approach to update in each subproblem the design changes of all other subproblems. The design representation scheme, therefore, is common to both the design optimization step and the procedure required to account for interaction among the subproblems. The decomposition based solution of a dual structural-control design problem is used as a test problem for the proposed approach. The convergence characteristics of the proposed approach are compared against those available from a nondecomposition-based method.

Journal ArticleDOI
TL;DR: In this paper, the optimal design of torsional beams using an arbitrary number of materials is considered, and an optimization algorithm for multimaterial composites is described and computational results for both perturbations and asymptotical cases are presented.
Abstract: In this paper, we consider the optimal design of torsional beams using an arbitrary number of materials. The problem is initially ill-posed, and must be relaxed by the introduction of multimaterial composites. The optimization algorithm for multimaterial composites is described and computational results for both perturbations and asymptotical cases are presented. It is noticed that the case for three or more composites undergoes a phenomenon very much like a phase transition when certain conditions are satisfied.

Journal ArticleDOI
TL;DR: In this article, a discrete model for the design sensitivity analysis of thin laminated angle-ply composite structures using a plate shell element based on a Kirchhoff discrete theory for the bending effects is presented.
Abstract: This paper presents a discrete model for the design sensitivity analysis of thin laminated angle-ply composite structures using a plate shell element based on a Kirchhoff discrete theory for the bending effects. To overcome the nondifferentiability of multiple eigenvalues, which may occur during a structural optimization involving free vibrations or buckling design situations, a nonsmooth eigenvalue based criterion is implemented. Angle-ply design variables and vectorial distances from the laminated midle surface to the upper surface of each layer are considered as design variables. The design sensitivities and the directional derivatives are evaluated analytically. The efficiency and accuracy of the model developed is discussed with two illustrative cases which show the need to compute sensitivities of multiple eigenvalues as directional derivatives for laminated composite structures.