Showing papers in "Structural and Multidisciplinary Optimization in 2011"
TL;DR: It is shown that simple projection methods do not ensure localMesh convergence and a modified robust topology optimization formulation based on erosion, intermediate and dilation projections is proposed that ensures both global and local mesh-convergence.
Abstract: Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence and propose a modified robust topology optimization formulation based on erosion, intermediate and dilation projections that ensures both global and local mesh-convergence.
1,047 citations
TL;DR: The paper presents an efficient 88 line MATLAB code for topology optimization using the 99 line code presented by Sigmund as a starting point, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops.
Abstract: The paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund (Struct Multidisc Optim 21(2):120---127, 2001) as a starting point. The original code has been extended by a density filter, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops. A speed improvement with a factor of 100 is obtained for a benchmark example with 7,500 elements. Moreover, the length of the code has been reduced to a mere 88 lines. These improvements have been accomplished without sacrificing the readability of the code. The 88 line code can therefore be considered as a valuable successor to the 99 line code, providing a practical instrument that may help to ease the learning curve for those entering the field of topology optimization. The paper also discusses simple extensions of the basic code to include recent PDE-based and black-and-white projection filtering methods. The complete 88 line code is included as an appendix and can be downloaded from the web site www.topopt.dtu.dk .
998 citations
TL;DR: The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate.
Abstract: The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.
354 citations
TL;DR: This forum article discusses the practical and scientific relevance of publishing papers that use immense computational resources for solving simple problems for which there already exist efficient solution techniques.
Abstract: Topology optimization is a highly developed tool for structural design and is by now being extensively used in mechanical, automotive and aerospace industries throughout the world. Gradient-based topology optimization algorithms may efficiently solve fine-resolution problems with thousands and up to millions of design variables using a few hundred (finite element) function evaluations (and even less than 50 in some commercial codes). Nevertheless, non-gradient topology optimization approaches that require orders of magnitude more function evaluations for extremely low resolution examples keep appearing in the literature. This forum article discusses the practical and scientific relevance of publishing papers that use immense computational resources for solving simple problems for which there already exist efficient solution techniques.
353 citations
TL;DR: In this article, a multiobjective robust optimization methodology is presented to address the effects of parametric uncertainties on multiple crashworthiness criteria, where several different sigma criteria are adopted to measure the variations.
Abstract: Although deterministic optimization has to a considerable extent been successfully applied in various crashworthiness designs to improve passenger safety and reduce vehicle cost, the design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on multiple crashworthiness criteria, where several different sigma criteria are adopted to measure the variations. As an example, a full front impact of vehicle is considered with increase in energy absorption and reduction of structural weight as the design objectives, and peak deceleration as the constraint. A multiobjective particle swarm optimization is applied to generate robust Pareto solution, which no longer requires formulating a single cost function by using weighting factors or other means. From the example, a clear compromise between the Pareto deterministic and robust designs can be observed. The results demonstrate the advantages of using multiobjective robust optimization, with not only the increase in the energy absorption and decrease in structural weight from a baseline design, but also a significant improvement in the robustness of optimum.
196 citations
TL;DR: In this paper, a Heaviside projection based topology optimization method with a scalar function that is filtered by a Helmholtz type partial differential equation is proposed, where the optimality can be strictly discussed in terms of the KKT condition.
Abstract: This paper deals with topology optimization based on the Heaviside projection method using a scalar function as design variables. The scalar function is then regularized by a PDE based filter. Several image-processing based filtering techniques have so far been proposed for regularization or restricting the minimum length scale. They are conventionally applied to the design sensitivities rather than the design variables themselves. However, it causes discrepancies between the filtered sensitivities and the actual sensitivities that may confuse the optimization process and disturb the convergence. In this paper, we propose a Heaviside projection based topology optimization method with a scalar function that is filtered by a Helmholtz type partial differential equation. Therefore, the optimality can be strictly discussed in terms of the KKT condition. In order to demonstrate the effectiveness of the proposed method, a minimum compliance problem is solved.
187 citations
TL;DR: Two multi-material interpolation schemes as direct generalizations of the well-known SIMP and RAMP material interpolation scheme originally developed for isotropic mixtures of two isotropIC material phases are presented.
Abstract: This paper presents two multi-material interpolation schemes as direct generalizations of the well-known SIMP and RAMP material interpolation schemes originally developed for isotropic mixtures of two isotropic material phases. The new interpolation schemes provide generally applicable interpolation schemes between an arbitrary number of pre-defined materials with given (anisotropic) properties. The method relies on a large number of sparse linear constraints to enforce the selection of at most one material in each design subdomain. Topology and multi-material optimization is formulated within a unified parametrization.
179 citations
TL;DR: The modified ABC algorithm is described, which is a meta-heuristic optimization technique that mimics the process of food foraging of honeybees that is very effective and robust for the discrete optimization designs of truss structural problems.
Abstract: Over the past few years, swarm intelligence based optimization techniques such as ant colony optimization and particle swarm optimization have received considerable attention from engineering researchers and practitioners. These algorithms have been used in the solution of various engineering problems. Recently, a relatively new swarm based optimization algorithm called the Artificial Bee Colony (ABC) algorithm has begun to attract interest from researchers to solve optimization problems. The aim of this study is to present an optimization algorithm based on the ABC algorithm for the discrete optimum design of truss structures. The ABC algorithm is a meta-heuristic optimization technique that mimics the process of food foraging of honeybees. Originally the ABC algorithm was developed for continuous function optimization problems. This paper describes the modifications made to the ABC algorithm in order to solve discrete optimization problems and to improve the algorithm's performance. In order to demonstrate the effectiveness of the modified algorithm, four structural problems with up to 582 truss members and 29 design variables were solved and the results were compared with those obtained using other well-known meta-heuristic search techniques. The results demonstrate that the ABC algorithm is very effective and robust for the discrete optimization designs of truss structural problems.
174 citations
TL;DR: The proposed adaptive-sparse polynomial chaos expansion method is highly efficient and accurate for reliability analysis and its sensitivity analysis, and it is capable of handling a nonlinear correlation.
Abstract: This paper presents an adaptive-sparse polynomial chaos expansion (adaptive-sparse PCE) method for performing engineering reliability analysis and design. The proposed method combines three ideas: (i) an adaptive-sparse scheme to build sparse PCE with the minimum number of bivariate basis functions, (ii) a new projection method using dimension reduction techniques to effectively compute the expansion coefficients of system responses, and (iii) an integration of copula to handle nonlinear correlation of input random variables. The proposed method thus has three positive features for reliability analysis and design: (a) there is no need for response sensitivity analysis, (b) it is highly efficient and accurate for reliability analysis and its sensitivity analysis, and (c) it is capable of handling a nonlinear correlation. In addition to the features, an error decomposition scheme for the proposed method is presented to help analyze error sources in probability analysis. Several engineering problems are used to demonstrate the three positive features of the adaptive-sparse PCE method.
165 citations
TL;DR: In this article, the authors proposed modifications to optimizer algorithms and/or Heaviside formulation that allow this continuation method to be eliminated, and tested on minimum compliance and compliant inverter benchmark problems and are shown to be effective and efficient.
Abstract: Projection methods and density filters based on the Heaviside step function are an effective means for producing discrete (0---1) solutions in continuum topology optimization. They naturally impose a minimum length scale on designed features and thereby prevent numerical instabilities of checkerboards and mesh dependence, as well as provide the designer a tool to influence solution manufacturability. A drawback of the Heaviside approach is that a continuation method must be applied to the continuous approximation so as not to approach the step function too quickly. This is achieved by gradually increasing a curvature parameter known as β as the optimization progresses. This is not only inefficient, but also causes slight, artificial perturbations to the topology. This note offers simple modifications to optimizer algorithms and/or Heaviside formulation that allow this continuation method to be eliminated. The modifications are tested on minimum compliance and compliant inverter benchmark problems and are shown to be effective and efficient.
157 citations
TL;DR: In this paper, a new parameterization of the mechanical properties is proposed for the optimal selection of materials, and the weights are based on the shape functions of a quadrangular first order finite element.
Abstract: In this paper, a new parameterization of the mechanical properties is proposed for the optimal selection of materials. Recent parameterization schemes from multi-phase topology optimization (i.e. Discrete Material Optimization--DMO) are compared to this novel approach in the selection of conventional laminates including only 0°, ??45°, 45° and 90° plies. In the new parameterization the material stiffness is computed as a weighted sum of the candidate material properties, and the weights are based on the shape functions of a quadrangular first order finite element. Each vertex of the reference quadrangle then represents a candidate ply. Compared to DMO, this method requires fewer design variables, since the four pseudo-densities representing the presence or the absence of a given candidate ply in DMO are now replaced, in the weights, by two design variables, which are the two natural coordinates of the reference quadrangular element sufficient to identify each of the four vertices. Another advantage of the new parameterization scheme is to penalize, in a more convenient way, the intermediate values of the design variables, possibly avoiding any blending of materials at the solution. Three simple numerical applications with in-plane loadings are proposed and solved in order to demonstrate that the new approach is an interesting alternative to DMO, able to select the optimal orientations and to combine the material distribution with optimal orientation problems.
TL;DR: New efficiency and accuracy strategies such as a hyper-spherical local window for surrogate model generation, sample reuse, local window enlargement, filtering of constraints, and an adaptive initial point for the pattern search are proposed.
Abstract: This paper presents a sampling-based RBDO method using surrogate models. The Dynamic Kriging (D-Kriging) method is used for surrogate models, and a stochastic sensitivity analysis is introduced to compute the sensitivities of probabilistic constraints with respect to independent or correlated random variables. For the sampling-based RBDO, which requires Monte Carlo simulation (MCS) to evaluate the probabilistic constraints and stochastic sensitivities, this paper proposes new efficiency and accuracy strategies such as a hyper-spherical local window for surrogate model generation, sample reuse, local window enlargement, filtering of constraints, and an adaptive initial point for the pattern search. To further improve computational efficiency of the sampling-based RBDO method for large-scale engineering problems, parallel computing is proposed as well. Once the D-Kriging accurately approximates the responses, there is no further approximation in the estimation of the probabilistic constraints and stochastic sensitivities, and thus the sampling-based RBDO can yield very accurate optimum design. In addition, newly proposed efficiency strategies as well as parallel computing help find the optimum design very efficiently. Numerical examples verify that the proposed sampling-based RBDO can find the optimum design more accurately than some existing methods. Also, the proposed method can find the optimum design more efficiently than some existing methods for low dimensional problems, and as efficient as some existing methods for high dimensional problems when the parallel computing is utilized.
TL;DR: In this article, a methodology for automotive chassis design in involving optimization techniques is presented, in particular, topology, topometry and size optimizations are coupled with fem analyses and adopted in cascade for reaching an optimum chassis configuration.
Abstract: Automotive chassis design in view of car weight reduction is a challenging task due to the many performance targets that must be satisfied, in particular in terms of vehicle safety. In this paper a methodology for automotive chassis design in involving optimization techniques is presented. In particular, topology, topometry and size optimizations are coupled with fem analyses and adopted in cascade for reaching an optimum chassis configuration. The methodology is applied to the design process of a rear-central engine high performance vehicle chassis. The objective of the optimization process is the chassis weight reduction, yet in fulfilment of structural performance constraints as required by Ferrari standards. The results demonstrate the general applicability of the methodology presented for obtaining the general trusses layout and thicknesses distribution of the structure. The numerical model at this stage shows a significant weight reduction when compared to the chassis of the Ferrari F458 Italia.
TL;DR: The optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines and a variety of problems have been solved.
Abstract: Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.
TL;DR: In this paper, the use of manufacturing-type constraints, in particular pattern gradation and repetition, in the context of building layout optimization is explored, by placing constraints on the design domain in terms of number and variable size of repeating patterns along any direction.
Abstract: This paper explores the use of manufacturing-type constraints, in particular pattern gradation and repetition, in the context of building layout optimization. By placing constraints on the design domain in terms of number and variable size of repeating patterns along any direction, the conceptual design for buildings is facilitated. To substantiate the potential future applications of this work, examples within the context of high-rise building design are presented. Successful development of such ideas can contribute to practical engineering solutions, especially during the building design process. Examples are given to illustrate the ideas developed both in two-dimensional and three-dimensional building configurations.
TL;DR: In this paper, the geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen-Loeve expansion.
Abstract: Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen---Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.
TL;DR: Due to a linear programming formulation of the optimization problem the method presented in the paper assures finding the global optimum, hence it may be considered as the useful tool for verification of results obtained in other ways.
Abstract: The main purpose of the paper is to provide an easy-to-use code for topological optimization of the least weight trusses, written in the Mathematica programming language. The main idea of the presented approach consists in using a fixed ground structure and the linear programming formulation of the optimization problem. The solver is based on the fast interior point method. The strong effort is done to create the effective generator of the computational model utilizing the high regularity of the ground structure and the high sparsity of the geometric matrix. The efficiency and reliability of the algorithm is confirmed in several numerical tests. Due to a linear programming formulation of the optimization problem the method presented in the paper assures finding the global optimum, hence it may be considered as the useful tool for verification of results obtained in other ways. The appended complete Mathematica code of the program developed will be supplied by the Publisher on SpringerLink.
TL;DR: In this paper, the authors studied how castable designs can be obtained by using a Heaviside design parameterization in a specified casting direction, which reduces the number of design variables considerably and the approach is simple to implement.
Abstract: From a practical point of view it is often desirable to limit the complexity of a topology optimization design such that casting/milling type manufacturing techniques can be applied. In the context of gradient driven topology optimization this work studies how castable designs can be obtained by use of a Heaviside design parameterization in a specified casting direction. This reduces the number of design variables considerably and the approach is simple to implement.
TL;DR: An easy iterative algorithm, which introduces a “new” step length to control the convergence of the sequence and can be named as finite-step-length iterative algorithms, is present and results indicate that the proposed algorithm is effective and as simple as the HL-RF but more robust.
Abstract: In the "first-order reliability method" (FORM), the HL-RF iterative algorithm is a recommended and widely used one to locate the design point and calculate the reliability index. However it may fail to converge if the limit state surface at the design point is highly nonlinear. In this paper, an easy iterative algorithm, which introduces a "new" step length to control the convergence of the sequence and can be named as finite-step-length iterative algorithm, is present. It is proved that the HL-RF method is a special case of this proposed algorithm when the step length tends to infinity and the reason why the HL-RF diverges is illustrated. This proposed algorithm is much easier than other optimization schemes, especially than the modified HL-RF algorithm, because the process of line search for obtaining the step length is not needed. Numerical results indicate that the proposed algorithm is effective and as simple as the HL-RF but more robust.
TL;DR: The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities.
Abstract: This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.
TL;DR: The results show that the proposed ensemble of surrogates with recursive arithmetic average provides more ideal prediction accuracy than the stand-alone surrogates and for most problems even exceeds the previously presented ensemble techniques.
Abstract: Surrogate models are often used to replace expensive simulations of engineering problems. The common approach is to construct a series of metamodels based on a training set, and then, from these surrogates, pick out the best one with the highest accuracy as an approximation of the computationally intensive simulation. However, because the choice of approximate model depends on design of experiments (DOEs), the traditional strategy thus increases the risk of adopting an inappropriate model. Furthermore, in the design of complex product system, because of its feature of one-of-a-kind production, acquiring more samples is very expensive and intensively time-consuming, and sometimes even impossible. Therefore, in order to save sampling cost, it is a reasonable strategy to take full advantage of all the stand-alone surrogates and then combine them into an ensemble model. Ensemble technique is an effective way to make up for the shortfalls of traditional strategy. Motivated by the previous research on ensemble of surrogates, a new technique for constructing of a more accurate ensemble of surrogates is proposed in this paper. The weights are obtained using a recursive process, in which the values of these weights are updated in each iteration until the last ensemble achieves a desirable prediction accuracy. This technique has been evaluated using five benchmark problems and one reality problem. The results show that the proposed ensemble of surrogates with recursive arithmetic average provides more ideal prediction accuracy than the stand-alone surrogates and for most problems even exceeds the previously presented ensemble techniques. Finally, we should point out that the advantages of combination over selection are still difficult to illuminate. We are still using an "insurance policy" mode rather than offering significant improvements.
TL;DR: In this paper, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented, which is derived primarily on the Kirsch's combined approximations method, and is also highly accurate for case of repeated eigenvalues problem.
Abstract: Structural optimization with frequency constraints is highly nonlinear dynamic optimization problems. Genetic algorithm (GA) has greater advantage in global optimization for nonlinear problem than optimality criteria and mathematical programming methods, but it needs more computational time and numerous eigenvalue reanalysis. To speed up the design process, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented. This reanalysis technique is derived primarily on the Kirsch's combined approximations method, which is also highly accurate for case of repeated eigenvalues problem. The required number of basis vectors at every generation is adaptively determined and the rules for selecting initial number of basis vectors are given. Numerical examples of truss design are presented to validate the reanalysis-based frequency optimization. The results demonstrate that the adaptive eigenvalue reanalysis affects very slightly the accuracy of the optimal solutions and significantly reduces the computational time involved in the design process of large-scale structures.
TL;DR: In this article, a normalized Moment Based Quadrature Rule (NMBQR) is proposed to solve the problem, which can reduce the condition number of the coefficient matrix of the linear equations considerably and improve the robustness and stableness.
Abstract: This paper presents a combined reliability analysis approach which is composed of Dimension Reduction Method (DRM) and Maximum Entropy Method (MEM). DRM has emerged as a new approach in this field with the advantages of its sensitivity-free nature and efficiency instead of searching for the most probable point (MPP). However, in some recent implementations, the Moment Based Quadrature Rule (MBQR) in the DRM was found to be numerically instable when solving a system of linear equations for the integration points. In this study, a normalized Moment Based Quadrature Rule (NMBQR) is proposed to solve this problem, which can reduce the condition number of the coefficient matrix of the linear equations considerably and improve the robustness and stableness. Based on the statistical moments obtained by DRM+NMBQR, the MEM is applied to construct the probability density function (PDF) of the response. A number of numerical examples are calculated and compared to the Monte Carlo simulation (MCS), the First Order Reliability Method (FORM), the Extended Generalized Lambda Distribution (EGLD) and Saddlepoint Approximation (SA). The results show the accuracy and efficiency of the proposed method, especially for the multimodal PDF problem and multiple design point problem.
TL;DR: In this article, a decoupled approach is proposed to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency.
Abstract: This paper proposes formulations and algorithms for design optimization under both aleatory (i.e., natural or physical variability) and epistemic uncertainty (i.e., imprecise probabilistic information), from the perspective of system robustness. The proposed formulations deal with epistemic uncertainty arising from both sparse and interval data without any assumption about the probability distributions of the random variables. A decoupled approach is proposed in this paper to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency. The proposed methods are illustrated for the upper stage design problem of a two-stage-to-orbit (TSTO) vehicle, where the information on the random design inputs are only available as sparse point data and/or interval data. As collecting more data reduces uncertainty but increases cost, the effect of sample size on the optimality and robustness of the solution is also studied. A method is developed to determine the optimal sample size for sparse point data that leads to the solutions of the design problem that are least sensitive to variations in the input random variables.
TL;DR: In this article, a reliability-based topology optimization (RBTO) for 3D structures was performed using bi-directional evolutionary structural optimization (BESO) and the standard response surface method (SRSM).
Abstract: In this paper, a reliability-based topology optimization (RBTO) for 3-D structures was performed using bi-directional evolutionary structural optimization (BESO) and the standard response surface method (SRSM). In order to get a stable optimal topology, the most recently-developed filter scheme was implemented with BESO, and SRSM was used to generate an approximate limit state function. These results were compared with the recently announced results of RBTO for 2-D structures, and the differences between the results for the 3-D and 2-D structures were examined. A cantilever beam and an MBB beam were selected as the numerical examples. The comparison showed that the optimal topologies of deterministic topology optimization (DTO) and RBTO for the 2-D and 3-D MBB beams, respectively, are very different. Specifically, the two-support member on the left hand side comes into being along the width for the 3-D case, but not for the 2-D case. This shows that RBTO for 3-D structures should be performed as part of the design process.
TL;DR: The minimum compliance, mass constrained multiple load case problem is formulated and solved for a number of examples which numerically confirm the sought properties of the new scheme in terms of convergence to a discrete solution.
Abstract: Design of composite laminated lay-ups are formulated as discrete multi-material selection problems. The design problem can be modeled as a non-convex mixed-integer optimization problem. Such problems are in general only solvable to global optimality for small to moderate sized problems. To attack larger problem instances we formulate convex and non-convex continuous relaxations which can be solved using gradient based optimization algorithms. The convex relaxation yields a lower bound on the attainable performance. The optimal solution to the convex relaxation is used as a starting guess in a continuation approach where the convex relaxation is changed to a non-convex relaxation by introduction of a quadratic penalty constraint whereby intermediate-valued designs are prevented. The minimum compliance, mass constrained multiple load case problem is formulated and solved for a number of examples which numerically confirm the sought properties of the new scheme in terms of convergence to a discrete solution.
TL;DR: The concept of the “fields of forces” is used to enhance the recently developed meta-heuristic, the Charged System Search (CSS), and the enhanced CSS is then applied to determine the configuration optimum design of structures.
Abstract: The concept of the "fields of forces" is utilized as a general model of meta-heuristic algorithms from physics. This model is capable of representing the properties of different meta-heuristics and in this paper, it is used to enhance the recently developed meta-heuristic, the Charged System Search (CSS). The enhanced CSS is then applied to determine the configuration optimum design of structures. Comparison of the results for some examples, illustrates the efficiency of the enhanced CSS algorithm.
TL;DR: In this article, both epistemic and aleatory uncertainties are considered in reliability sensitivity analysis and modeled using P-boxes, and the proposed method is based on Monte Carlo simulation (MCS), weighted regression, interval algorithm and first order reliability method (FORM).
Abstract: Reliability sensitivity analysis is used to find the rate of change in the probability of failure (or reliability) due to the changes in distribution parameters such as the means and standard deviations. Most of the existing reliability sensitivity analysis methods assume that all the probabilities and distribution parameters are precisely known. That is, every statistical parameter involved is perfectly determined. However, there are two types of uncertainties, epistemic and aleatory uncertainties that may not be perfectly determined in engineering practices. In this paper, both epistemic and aleatory uncertainties are considered in reliability sensitivity analysis and modeled using P-boxes. The proposed method is based on Monte Carlo simulation (MCS), weighted regression, interval algorithm and first order reliability method (FORM). We linearize original non-linear limit-state function by MCS rather than by expansion as a first order Taylor series at most probable point (MPP) because the MPP search is an iterative optimization process. Finally, we introduce an optimization model for sensitivity analysis under both aleatory and epistemic uncertainties. Four numerical examples are presented to demonstrate the proposed method.
TL;DR: In this article, the authors studied the effect of different mechanical properties for the steel reinforcement and for the concrete on the emerging topology of strut-and-tie models using the Isolines Topology Design (ITD) method.
Abstract: The Strut-and-Tie Method is considered a basic tool for analysis and design of reinforced concrete structures and has been incorporated in different codes of practice such as: EC-2, BS 8110, ACI 318-08, EHE-08, etc. The stress trajectories or load path methods have been used to generate strut-and-tie models. However, the models produced by these methods are not unique, with the result depending on the intuition or expertise of the designer, specifically with regards to region D of the structure, where the load path distribution is non-linear. Topology optimization can offer new opportunities to eliminate the limitations of traditional methods. The aim of this work was to study the effect of using different mechanical properties for the steel reinforcement and for the concrete on the emerging topology of strut-and-tie models. The Isolines Topology Design (ITD) method was used for this research. Three examples are presented to show the effect of different mechanical properties used for the tensile (steel) and compressive (concrete) regions of the structure, the: (1) Single short corbel; (2) Deep beam with opening; and (3) Double-sided beam-to-column joint.
TL;DR: A new Multi-Objective Particle Swarm Optimization (MOPSO), with a different velocity equation, for the calculation of the free parameters in active control systems is proposed and tested and a fuzzy control system is considered.
Abstract: Smart structures include elements of active, passive or hybrid control. In this paper, a new Multi-Objective Particle Swarm Optimization (MOPSO), with a different velocity equation, for the calculation of the free parameters in active control systems is proposed and tested. A fuzzy control system is considered. Fuzzy control is a suitable tool for the systematic development of nonlinear active control strategies and can be fine tuned if no experience exists or if one designs more complicated control schemes. The usage of MOPSO with a combination of continuous and discrete variables for the optimal design of the controller is proposed. Numerical applications on smart piezoelastic beams are presented.