Showing papers in "Structural and Multidisciplinary Optimization in 2009"
TL;DR: In this paper, the authors evaluate and compare established numerical methods of structural topology optimization that have reached the stage of application in industrial software, and they hope that their text will spark off a fruitful and constructive debate on this important topic.
Abstract: The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive debate on this important topic.
896 citations
TL;DR: This paper discussed how PRESS is employed to estimate the RMS error, and whether to use the best PRESS solution or a weighted surrogate when a single surrogate is needed, and found that PRESS as obtained through the k-fold strategy successfully estimates the RMSE.
Abstract: Surrogate models are commonly used to replace expensive simulations of engineering problems. Frequently, a single surrogate is chosen based on past experience. This approach has generated a collection of papers comparing the performance of individual surrogates. Previous work has also shown that fitting multiple surrogates and picking one based on cross-validation errors (PRESS in particular) is a good strategy, and that cross-validation errors may also be used to create a weighted surrogate. In this paper, we discussed how PRESS (obtained either from the leave-one-out or from the k-fold strategies) is employed to estimate the RMS error, and whether to use the best PRESS solution or a weighted surrogate when a single surrogate is needed. We also studied the minimization of the integrated square error as a way to compute the weights of the weighted average surrogate. We found that it pays to generate a large set of different surrogates and then use PRESS as a criterion for selection. We found that (1) in general, PRESS is good for filtering out inaccurate surrogates; and (2) with sufficient number of points, PRESS may identify the best surrogate of the set. Hence the use of cross-validation errors for choosing a surrogate and for calculating the weights of weighted surrogates becomes more attractive in high dimensions (when a large number of points is naturally required). However, it appears that the potential gains from using weighted surrogates diminish substantially in high dimensions. We also examined the utility of using all the surrogates for forming the weighted surrogates versus using a subset of the most accurate ones. This decision is shown to depend on the weighting scheme. Finally, we also found that PRESS as obtained through the k-fold strategy successfully estimates the RMSE.
349 citations
TL;DR: In this paper, a comparative study on the performances of several representative uncertainty propagation methods, including a few newly developed methods that have received growing attention, is performed, and the insights gained are expected to direct designers for choosing the most applicable uncertainty propagation technique in design under uncertainty.
Abstract: A wide variety of uncertainty propagation methods exist in literature; however, there is a lack of good understanding of their relative merits. In this paper, a comparative study on the performances of several representative uncertainty propagation methods, including a few newly developed methods that have received growing attention, is performed. The full factorial numerical integration, the univariate dimension reduction method, and the polynomial chaos expansion method are implemented and applied to several test problems. They are tested under different settings of the performance nonlinearity, distribution types of input random variables, and the magnitude of input uncertainty. The performances of those methods are compared in moment estimation, tail probability calculation, and the probability density function construction, corresponding to a wide variety of scenarios of design under uncertainty, such as robust design, and reliability-based design optimization. The insights gained are expected to direct designers for choosing the most applicable uncertainty propagation technique in design under uncertainty.
290 citations
TL;DR: The selection of weight factors in the general weighted-sum formulation of an ensemble is treated as an optimization problem with the desired solution being one that minimizes a selected error metric.
Abstract: Approximate mathematical models (metamodels) are often used as surrogates for more computationally intensive simulations. The common practice is to construct multiple metamodels based on a common training data set, evaluate their accuracy, and then to use only a single model perceived as the best while discarding the rest. This practice has some shortcomings as it does not take full advantage of the resources devoted to constructing different metamodels, and it is based on the assumption that changes in the training data set will not jeopardize the accuracy of the selected model. It is possible to overcome these drawbacks and to improve the prediction accuracy of the surrogate model if the separate stand-alone metamodels are combined to form an ensemble. Motivated by previous research on committee of neural networks and ensemble of surrogate models, a technique for developing a more accurate ensemble of multiple metamodels is presented in this paper. Here, the selection of weight factors in the general weighted-sum formulation of an ensemble is treated as an optimization problem with the desired solution being one that minimizes a selected error metric. The proposed technique is evaluated by considering one industrial and four benchmark problems. The effect of different metrics for estimating the prediction error at either the training data set or a few validation points is also explored. The results show that the optimized ensemble provides more accurate predictions than the stand-alone metamodels and for most problems even surpassing the previously reported ensemble approaches.
254 citations
TL;DR: In this article, a technique for imposing maximum length scale on features in continuum topology optimization is presented, where the design domain is searched and local constraints prevent the formation of features that are larger than the prescribed maximum-length scale.
Abstract: This paper presents a technique for imposing maximum length scale on features in continuum topology optimization. The design domain is searched and local constraints prevent the formation of features that are larger than the prescribed maximum length scale. The technique is demonstrated in the context of structural and fluid topology optimization. Specifically, maximum length scale criterion is applied to (a) the solid phase in minimum compliance design to restrict the size of structural (load-carrying) members, and (b) the fluid (void) phase in minimum dissipated power problems to limit the size of flow channels. Solutions are shown to be near 0/1 (void/solid) topologies that satisfy the maximum length scale criterion. When combined with an existing minimum length scale methodology, the designer gains complete control over member sizes that can influence cost and manufacturability. Further, results suggest restricting maximum length scale may provide a means for influencing performance characteristics, such as redundancy in structural design.
210 citations
TL;DR: In this article, a quantified measure for non-probabilistic reliability based on the multi-ellipsoid convex model is proposed for topology optimization of continuum structures in the presence of uncertain-but-bounded parameters.
Abstract: Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.
203 citations
TL;DR: In this paper, a minimum weight with stress constraints Finite element formulation for the topology optimization of continuum structures is proposed, where the material mass is restricted to a predefined percentage of the maximum possible mass, while no displacement constraints are taken into account.
Abstract: Topology structural optimization problems have been usually stated in terms of a maximum stiffness (minimum compliance) approach. The objective of this type of approach is to distribute a given amount of material in a certain domain, so that the stiffness of the resulting structure is maximized (that is, the compliance, or energy of deformation, is minimized) for a given load case. Thus, the material mass is restricted to a predefined percentage of the maximum possible mass, while no stress or displacement constraints are taken into account. This paper presents a different strategy to deal with topology optimization: a minimum weight with stress constraints Finite Element formulation for the topology optimization of continuum structures. We propose two different approaches in order to take into account stress constraints in the optimization formulation. The local approach of the stress constraints imposes stress constraints at predefined points of the domain (i.e. at the central point of each element). On the contrary, the global approach only imposes one global constraint that gathers the effect of all the local constraints by means of a certain so-called aggregation function. Finally, some application examples are solved with both formulations in order to compare the obtained solutions.
200 citations
TL;DR: The experimental results show that integrating the hybrid evolutionary algorithm with the adaptive constraint-handling technique is beneficial, and the proposed method achieves competitive performance with respect to some other state-of-the-art approaches in constrained evolutionary optimization.
Abstract: A novel approach to deal with numerical and engineering constrained optimization problems, which incorporates a hybrid evolutionary algorithm and an adaptive constraint-handling technique, is presented in this paper. The hybrid evolutionary algorithm simultaneously uses simplex crossover and two mutation operators to generate the offspring population. Additionally, the adaptive constraint-handling technique consists of three main situations. In detail, at each situation, one constraint-handling mechanism is designed based on current population state. Experiments on 13 benchmark test functions and four well-known constrained design problems verify the effectiveness and efficiency of the proposed method. The experimental results show that integrating the hybrid evolutionary algorithm with the adaptive constraint-handling technique is beneficial, and the proposed method achieves competitive performance with respect to some other state-of-the-art approaches in constrained evolutionary optimization.
195 citations
TL;DR: In this paper, the geometric properties of design are embedded into the NURBS basis functions and the control points whose perturbation naturally results in shape changes are used in both response and shape sensitivity analyses, where normal vector and curvature are continuous over the whole design space.
Abstract: Finite element based shape optimization has some difficulties in the parameterization of design domain. In isogeometric approach, however, the geometric properties of design are embedded into the NURBS basis functions and the control points whose perturbation naturally results in shape changes. Thus, exact geometric models can be used in both response and shape sensitivity analyses, where normal vector and curvature are continuous over the whole design space so that enhanced shape sensitivity can be obtained. In the problems of shape optimal design, refinements and design changes are easily implemented within the isogeometric framework, which maintains exact geometry without subsequent communication with CAD description. The variation of control points results in shape changes and is continuous over the whole design space. Through numerical examples, the developed isogeometric sensitivity is verified to demonstrate excellent agreements with finite difference sensitivity. Also, the proposed method works very well in various shape optimization problems.
183 citations
TL;DR: A two-scale optimization method is presented at finding optimal configurations of macro structures and micro-structures of cellular material with maximum structural fundamental frequency to meet today's manufacture practice and reduce manufacturing cost.
Abstract: Ultra-light cellular materials exhibit high stiffness/strength to weight ratios and bring opportunity for multifunctional performance. One of their potential applications is to build structure with optimum dynamic performance, which is extremely important for some structural parts in vehicle engineering and attracts a great attention. This paper presents a two-scale optimization method and aims at finding optimal configurations of macro structures and micro-structures of cellular material with maximum structural fundamental frequency. In this method macro and micro densities are introduced as independent design variables for macrostructure and microstructure. Optimizations at two scales are integrated into one system through homogenization theory and base material is distributed between the two scales automatically with optimization model. Microstructure of materials is assumed to be homogeneous at the macro scale to meet today’s manufacture practice and reduce manufacturing cost. Plane structure with homogeneous cellular material and perforated plate are studied. Numerical experiments validate the proposed method and computational model.
180 citations
TL;DR: A PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula is developed, which can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications.
Abstract: The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a transformation from the correlated input random variables into independent standard normal variables. For the transformation with correlated input variables, the two most representative transformations, the Rosenblatt and Nataf transformations, are investigated. The Rosenblatt transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear transformation. Thus, the Nataf transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.
TL;DR: In this article, both elastic materials as well as piezoelectric materials are considered for the design of energy harvesting devices under the topology optimization formulation, and the objective function for this study is to maximize the energy conversion factor.
Abstract: Energy harvesting devices based on the piezoelectric effect that converts ambient energy to electric energy is a very attractive energy source for remote sensors and embedded devices. Although topology optimization has been applied to the design of piezoelectric transducers, the locations of piezoelectric materials are predefined and only the optimal layout of elastic materials is considered. In this paper, both elastic materials as well as piezoelectric materials are considered for the design of energy harvesting devices under the topology optimization formulation. The objective function for this study is to maximize the energy conversion factor. The sensitivities of both stored strain energy and electrical energy are derived by the adjoint method. Examples of energy harvesting devices are presented and discussed using the proposed method.
TL;DR: In this article, the authors formulated necessary conditions for optimality in optimal control problems with dynamics described by differential equations of fractional order (derivatives of arbitrary real order) and proposed a new solution scheme.
Abstract: We formulate necessary conditions for optimality in Optimal control problems with dynamics described by differential equations of fractional order (derivatives of arbitrary real order). Then by using an expansion formula for fractional derivative, optimality conditions and a new solution scheme is proposed. We assumed that the highest derivative in the differential equation of the process is of integer order. Two examples are treated in detail.
TL;DR: In this paper, the authors use a suite of test problems to assess several error estimation measures for polynomial response surfaces and kriging. But, they find that the (actual) errors for response surfaces are less sensitive to the choice of experimental designs than the Kriging errors, and that no single error measure outperforms other measures on all the problems, and demonstrate that the geometric means of several combinations of error measures improve the actual errors over individual error measures.
Abstract: Error estimation measures are useful for assessing uncertainty in surrogate predictions. We use a suite of test problems to appraise several error estimation measures for polynomial response surfaces and kriging. In addition, we study the performance of cross-validation error measures that can be used with any surrogate. We use 1,000 experimental designs to obtain the variability of error estimates with respect to the experimental designs for each problem. We find that the (actual) errors for polynomial response surfaces are less sensitive to the choice of experimental designs than the kriging errors. This is attributed to the variability in the maximum likelihood estimates of the kriging parameters. We find that no single error measure outperforms other measures on all the problems. Computationally expensive integrated local error measures (standard error for polynomials and mean square error for kriging) estimate the actual root mean square error very well. The distribution-free cross-validation error characterized the actual errors reasonably well. While the estimated root mean square error for polynomial response surface is a good estimate of the actual errors, the process variance for kriging is not. We explore a few methods of simultaneously using multiple error measures and demonstrate that the geometric means of several combinations of error measures improve the assessment of the actual errors over individual error measures.
TL;DR: Results show that methods introduced in this paper provide an effective way of improving computational efficiency of CO based on high fidelity simulation models.
Abstract: This paper focuses on the metamodel-based collaborative optimization (CO). The objective is to improve the computational efficiency of CO in order to handle multidisciplinary design optimization problems utilising high fidelity models. To address these issues, two levels of metamodel building techniques are proposed: metamodels in the disciplinary optimization are based on multi-fidelity modelling (the interaction of low and high fidelity models) and for the system level optimization a combination of a global metamodel based on the moving least squares method and trust region strategy is introduced. The proposed method is demonstrated on a continuous fiber-reinforced composite beam test problem. Results show that methods introduced in this paper provide an effective way of improving computational efficiency of CO based on high fidelity simulation models.
TL;DR: In this article, a chaotic dynamics analysis on the AMV iterative procedure is performed, and the stability transformation method of chaos feedback control is suggested for the convergence control of AMV procedure in the parameter interval in which the iterative scheme fails.
Abstract: Performance measure approach (PMA) is a recently proposed method for evaluation of probabilistic constraints in reliability-based design optimization of structure. The advanced mean-value (AMV) method is well suitable for PMA due to its simplicity and efficiency. However, when the AMV iterative scheme is applied to search for the minimum performance target point for some nonlinear performance functions, the iterative sequences could fall into the periodic oscillation and even chaos. In the present paper, the phenomena of numerical instabilities of AMV iterative solutions are illustrated firstly. And the chaotic dynamics analysis on the iterative procedure of AMV method is performed. Then, the stability transformation method of chaos feedback control is suggested for the convergence control of AMV procedure in the parameter interval in which the iterative scheme fails. Numerical results of several nonlinear performance functions demonstrate that the control of periodic oscillation, bifurcation and chaos for AMV iterative procedure is achieved, and the stable convergence solutions are obtained.
TL;DR: In this article, the authors considered the aeroelastic optimization of a membrane micro air vehicle wing through topology optimization, where the low aspect ratio wing is discretized into panels: a two material formulation on the wetted surface is used, where each panel can be membrane (wing skin) or carbon fiber (laminate reinforcement).
Abstract: This work considers the aeroelastic optimization of a membrane micro air vehicle wing through topology optimization. The low aspect ratio wing is discretized into panels: a two material formulation on the wetted surface is used, where each panel can be membrane (wing skin) or carbon fiber (laminate reinforcement). An analytical sensitivity analysis of the aeroelastic system is used for the gradient-based optimization of aerodynamic objective functions. An explicit penalty is added, as needed, to force the structure to a 0–1 distribution. The dependence of the solution upon initial design, angle of attack, mesh density, and objective function are presented. Deformation and pressure distributions along the wing are studied for various load-augmenting and load-alleviating designs (both baseline and optimized), in order to establish a link between stiffness distribution and aerodynamic performance of membrane micro air vehicle wings. The work concludes with an experimental validation of the superiority of select optimal designs.
TL;DR: An efficient multiobjective differential evolution algorithm is presented for engineering design and is capable of yielding a wide spread of solutions with good coverage and convergence to true Pareto-optimal fronts.
Abstract: Solving engineering design and resources optimization via multiobjective evolutionary algorithms (MOEAs) has attracted much attention in the last few years. In this paper, an efficient multiobjective differential evolution algorithm is presented for engineering design. Our proposed approach adopts the orthogonal design method with quantization technique to generate the initial archive and evolutionary population. An archive (or secondary population) is employed to keep the nondominated solutions found and it is updated by a new relaxed form of Pareto dominance, called Pareto-adaptive ϵ-dominance (paϵ-dominance), at each generation. In addition, in order to guarantee to be the best performance produced, we propose a new hybrid selection mechanism to allow the archive solutions to take part in the generating process. To handle the constraints, a new constraint-handling method is employed, which does not need any parameters to be tuned for constraint handling. The proposed approach is tested on seven benchmark constrained problems to illustrate the capabilities of the algorithm in handling mathematically complex problems. Furthermore, four well-studied engineering design optimization problems are solved to illustrate the efficiency and applicability of the algorithm for multiobjective design optimization. Compared with Nondominated Sorting Genetic Algorithm II, one of the best MOEAs available at present, the results demonstrate that our approach is found to be statistically competitive. Moreover, the proposed approach is very efficient and is capable of yielding a wide spread of solutions with good coverage and convergence to true Pareto-optimal fronts.
TL;DR: In this article, a planar slider-crank mechanism with revolute joints with clearances was modeled as a massless virtual link, and a multi-layered neural network (MLNN) structure was used for approximating the motion of this link with respect to the position of input link.
Abstract: In this study, kinematic analysis of a planar slider-crank mechanism having revolute joints with clearances was presented. Joint clearance was modelled as a massless virtual link, and Multi-Layered Neural Network (MLNN) structure was used for approximating the motion of this link with respect to the position of input link. Training and testing data sets for the neural network were obtained from mechanism simulation using the ADAMS software. A genetic algorithm was also used to optimize the design parameters for minimizing the deviations due to clearances. When two joint clearances at crank-pin and piston-pin centers were considered, the effects of these clearances on the kinematic characteristics and transmission quality of the mechanism were investigated using continuous contact model between the journal and bearing at a joint.
TL;DR: In this paper, an ant colony optimization algorithm for optimum design of symmetric hybrid laminates is described, where the objective is simultaneous maximization of fundamental frequency and minimization of cost.
Abstract: An ant colony optimization algorithm for optimum design of symmetric hybrid laminates is described. The objective is simultaneous maximization of fundamental frequency and minimization of cost. Number of surface and core layers made of high-stiffness and low-stiffness materials, respectively, and fiber orientations are the design variables. Optimal stacking sequences are given for hybrid graphite/epoxy-glass/epoxy laminated plates with different aspect ratios and number of plies. The results obtained by ant colony optimization are compared to results obtained by a genetic algorithm and simulated annealing. The effectiveness of the hybridization concept for reducing the weight and keeping the fundamental frequency at a reasonable level is demonstrated. Furthermore, it is shown that the proposed ant colony algorithm outperforms the two other heuristics.
TL;DR: In this article, a growing ground structure method is proposed for truss topology optimization, which effectively expands or reduces the ground structure by iteratively adding or removing bars and nodes.
Abstract: A new method called the growing ground structure method is proposed for truss topology optimization, which effectively expands or reduces the ground structure by iteratively adding or removing bars and nodes. The method uses five growth strategies, which are based on mechanical properties, to determine the bars and nodes to be added or removed. Hence, the method can optimize the initial ground structures such that the modified, or grown, ground structures can generate the optimal solution for the given set of nodes. The structural data of trusses are manipulated using C++ standard template library and the Boost Graph Library, which help alleviate the programming efforts for implementing the method. Three kinds of topology optimization problems are considered. The first problem is a compliance minimization problem with cross-sectional areas as variables. The second problem is a minimum compliance problem with the nodal coordinates also as variables. The third problem is a minimum volume problem with stress constraints under multiple load cases. Six numerical examples corresponding to these three problems are solved to demonstrate the performance of the proposed method.
TL;DR: The eigenvector dimension reduction (EDR) method plays a pivotal role in making RBRDO effective because the EDR method turns out to be very efficient and accurate for probability analysis.
Abstract: This paper presents an effective methodology for reliability-based robust design optimization (RBRDO). The eigenvector dimension reduction (EDR) method plays a pivotal role in making RBRDO effective because the EDR method turns out to be very efficient and accurate for probability analysis. The use of the EDR method provides three benefits to RBRDO. First, an approximate response surface facilitates sensitivity calculation of reliability and quality where the response surface is constructed using the eigenvector samples. Thus, sensitivity analysis becomes very efficient and simple. Second, one EDR execution evaluates a set of quality (objective) and reliability (constraint) functions. In general, the EDR requires 2N + 1 or 4N + 1 simulation runs where N is the total number of random variables. The EDR execution does not require an iterative process, so the proposed RBRDO methodology has a single-loop structure. Moreover, the EDR execution time can be much shorter by taking advantage of a parallel computing power, and RBRDO can be far more efficient. Third, the EDR method allows solving problems with statistically correlated and non-normally distributed random inputs. Three practical engineering problems are used to demonstrate the effectiveness of the proposed RBRDO method using the EDR method.
TL;DR: In this paper, the Gauss-type quadrature formula was used for statistical moment estimation with arbitrary input distributions and further extended its use to robust design optimization, and a semi-analytic design sensitivity analysis with respect to the statistical moments was proposed for two different multi-dimensional integration methods, the tensor product quadratures (TPQ) formula and the univariate dimension reduction (UDR) method.
Abstract: In this paper, we present the Gauss-type quadrature formula as a rigorous method for statistical moment estimation involving arbitrary input distributions and further extend its use to robust design optimization. The mathematical background of the Gauss-type quadrature formula is introduced and its relation with other methods such as design of experiments (DOE) and point estimate method (PEM) is discussed. Methods for constructing one dimensional Gauss-type quadrature formula are summarized and the insights are provided. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed for two different multi-dimensional integration methods, the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. Through several examples, it is shown that the Gauss-type quadrature formula can be effectively used in robust design involving various non-normal distributions. The proposed design sensitivity analysis significantly reduces the number of function calls of robust optimization using the TPQ formulae, while using the UDR method, the savings of function calls are observed only in limited situations.
TL;DR: In this paper, a finite element model for laminated plate structures with viscoelastic core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach.
Abstract: Recent developments on the optimization of passive damping for vibration reduction in sandwich structures are presented in this paper, showing the importance of appropriate finite element models associated with gradient based optimizers for computationally efficient damping maximization programs. A new finite element model for anisotropic laminated plate structures with viscoelastic core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach. The complex modulus approach is used for the viscoelastic material behavior, and the dynamic problem is solved in the frequency domain. Constrained optimization is conducted for the maximization of modal loss factors, using gradient based optimization associated with the developed model, and single and multiobjective optimization based on genetic algorithms using an alternative ABAQUS finite element model. The model has been applied successfully and comparative optimal design applications in sandwich structures are presented and discussed.
TL;DR: An efficient decomposed multiobjective genetic algorithm is proposed to jointly determine optimal platform selection, platform design, and variant design in product family optimization, addressing limitations of prior restrictive component sharing definitions.
Abstract: Product family optimization involves not only specifying the platform from which the individual product variants will be derived, but also optimizing the platform design and the individual variants. Typically these steps are performed separately, but we propose an efficient decomposed multiobjective genetic algorithm to jointly determine optimal (1) platform selection, (2) platform design, and (3) variant design in product family optimization. The approach addresses limitations of prior restrictive component sharing definitions by introducing a generalized two-dimensional commonality chromosome to enable sharing components among subsets of variants. To solve the resulting high dimensional problem in a single stage efficiently, we exploit the problem structure by decomposing it into a two-level genetic algorithm, where the upper level determines the optimal platform configuration while each lower level optimizes one of the individual variants. The decomposed approach improves scalability of the all-in-one problem dramatically, providing a practical tool for optimizing families with more variants. The proposed approach is demonstrated by optimizing a family of electric motors. Results indicate that (1) decomposition results in improved solutions under comparable computational cost and (2) generalized commonality produces families with increased component sharing under the same level of performance.
TL;DR: In this article, the Wachspress-type hexagonal elements are used for topology optimization and three edge-based symmetry lines are added to the standard Lagrangian-type finite elements, such as linear quads and triangles.
Abstract: Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these conventional elements exhibit the well-known “checkerboard” pathology in the iterative solution of topology optimization problems. A feasible alternative to eliminate such long-standing problem consists of using hexagonal (honeycomb) elements with Wachspress-type shape functions. The features of the hexagonal mesh include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads can display one-node connections, which can lead to checkerboard; and only have two edge-based symmetry lines. In addition, Wachspress rational shape functions satisfy the partition of unity condition and lead to conforming finite element approximations. We explore the Wachspress-type hexagonal elements and present their implementation using three approaches for topology optimization: element-based, continuous approximation of material distribution, and minimum length-scale through projection functions. Examples are presented that demonstrate the advantages of the proposed element in achieving checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.
TL;DR: This paper presents a classification of formulations for distributed system optimization based on formulation structure, and identifies nested and alternating formulations, which play a crucial role in the theoretical and computational properties of distributed optimization methods.
Abstract: This paper presents a classification of for- mulations for distributed system optimization based on formulation structure. Two main classes are identi- fied: nested formulations and alternating formulations. Nested formulations are bilevel programming problems where optimization subproblems are nested in the func- tions of a coordinating master problem. Alternating formulations iterate between solving a master problem and disciplinary subproblems in a sequential scheme. Methods included in the former class are collaborative optimization and BLISS2000. The latter class includes concurrent subspace optimization, analytical target cas- cading, and augmented Lagrangian coordination. Al- though the distinction between nested and alternating formulations has not been made in earlier comparisons, it plays a crucial role in the theoretical and computa- tional properties of distributed optimization methods. The most prominent general characteristics for each class are discussed in more detail, providing valuable insights for the theoretical analysis and further devel- opment of distributed optimization methods.
TL;DR: In this work, a computational fluid dynamics code coupled with a structural model is run to obtain the pressures and displacements for different wing geometries controlled by the design variables, and a methodology is proposed for the optimization of coupled problems, and applied to a 3D flexible wing.
Abstract: In this work a methodology is proposed for the optimization of coupled problems, and applied to a 3D flexible wing. First, a computational fluid dynamics code coupled with a structural model is run to obtain the pressures and displacements for different wing geometries controlled by the design variables. Secondly, the data are reduced by Proper Orthogonal Decomposition (POD), allowing to expand any field as a linear combination of specific modes; finally, a surrogate model based on Moving Least Squares (MLS) is built to express the POD coefficients directly as functions of the design variables. After the validation of this bi-level model reduction strategy, the approximate models are used for the multidisciplinary optimization of the wing. The proposed method leads to a reduction of the weight by 6.6%, and the verification of the solution with the accurate numerical solvers confirms that the solution is feasible.
TL;DR: In this paper, the authors extend the material design method to obtain functionally graded material architectures, i.e., materials that are graded at the local level (e.g. microstructural level).
Abstract: The computational design of a composite where the properties of its constituents change gradually within a unit cell can be successfully achieved by means of a material design method that combines topology optimization with homogenization. This is an iterative numerical method, which leads to changes in the composite material unit cell until desired properties (or performance) are obtained. Such method has been applied to several types of materials in the last few years. In this work, the objective is to extend the material design method to obtain functionally graded material architectures, i.e. materials that are graded at the local level (e.g. microstructural level). Consistent with this goal, a continuum distribution of the design variable inside the finite element domain is considered to represent a fully continuous material variation during the design process. Thus the topology optimization naturally leads to a smoothly graded material system. To illustrate the theoretical and numerical approaches, numerical examples are provided. The homogenization method is verified by considering one-dimensional material gradation profiles for which analytical solutions for the effective elastic properties are available. The verification of the homogenization method is extended to two dimensions considering a trigonometric material gradation, and a material variation with discontinuous derivatives. These are also used as benchmark examples to verify the optimization method for functionally graded material cell design. Finally the influence of material gradation on extreme materials is investigated, which includes materials with near-zero shear modulus, and materials with negative Poisson’s ratio.
TL;DR: In this paper, genetic algorithm and generalized pattern search algorithm are used for optimal stacking sequence of a composite panel, which is simply supported on four sides and is subject to biaxial in-plane compressive loads.
Abstract: In this paper, genetic algorithm and generalized pattern search algorithm are used for optimal stacking sequence of a composite panel, which is simply supported on four sides and is subject to biaxial in-plane compressive loads. The problem has several global optimum configurations in the vicinity of local optima. The composite plate under consideration is 64-ply laminate made of graphite/epoxy. The laminate is taken to be symmetric and balanced, comprised of two-ply stacks with discrete fiber angles of 02, ± 45, 902 in the laminate sequence. The critical buckling loads are maximized for several combinations of load case and plate aspect ratio, and are compared with published results. Performance of both algorithms is compared in terms of capability of identifying global optima. It is found that genetic algorithm is efficient for problems with global optima.