Robust Structural Optimization Research Articles

Deterministic-based robust design optimization of composite structures under material uncertainty

We propose a new deterministic robust design optimization method for composite laminate structures under worst-case material uncertainty. The method is based on a simultaneous parametrization of topology and material and combines a design problem and a material uncertainty problem into a single min–max optimization problem which provides an efficient approach to handle variation of material properties in stiffness driven design optimization problems. An analysis is performed using a design problem based on a failure criterion formulation to evaluate the ability of the proposed method to generate robust composite designs. The design problem is solved using various loads, boundary conditions and manufacturing constraints. The designs generated with the proposed method have improved objective responses compared to the worst-case response of designs generated with nominal material properties and are less sensitive to the variation of material properties. The analysis indicates that the proposed method can be efficiently applied in a robust structural optimization framework.

Robust structural optimization in presence of manufacturing uncertainties through a boundary-perturbation method

Most manufacturing processes are inevitably characterized by process tolerances that ultimately affect the way a component behaves and complies with the design requirements. These uncertainties determine the real performance of a structure, with their impact growing with increasing deviations from the nominal values. This work introduces a simple approach, applicable to both static and dynamic cases, to conduct robust structural topology optimization in presence of manufacturing uncertainties. This approach, based on the level set method, makes use of a computationally efficient boundary-perturbation technique to describe over- and under-etching errors. Compared to the existing methods, it does not require a frequent re-initialization of the level set function, nor does it require a mapping between the etched structures and the nominal one. Moreover, compared to the standard case with uniform uncertainty, the technique presented in this work allows dealing with arbitrary spatially varying errors without increasing the computational cost.

Robust optimization of structures subjected to frictionless unilateral contact with uncertain initial gaps

Robust optimization of structures subjected to frictionless unilateral contact with uncertain initial gaps

Robust optimization of uncertain structures based on interval closeness coefficients and the 3D violation vectors of interval constraints

In this paper, a novel direct interval robust optimization approach is proposed so that the robust optimal design vectors for structures with interval uncertainties can be achieved. A new concept of interval closeness coefficient is proposed to describe the relative positional relationship between the boundaries of two intervals. Subsequently, the explicit formulae for calculating the four interval closeness coefficients between the boundaries of an interval constraint mechanical performance index and those of its corresponding given interval constant are put forward. Consequently, the 3D violation vectors of different interval constraints can be calculated, and the feasibility of a design vector can be evaluated by its total 3D violation vector of all interval constraints. Finally, various design vectors are directly ranked according to the preferential guidelines considering the robustness of all the mechanical performance indices of uncertain structures, which is realized by integrating the Kriging technique and nested genetic algorithm. Unlike the traditional robust optimization of structures involving interval uncertainties, the proposed method can avoid the complicated model transformation process and ensure the robustness of all the mechanical performance indices of the optimal solution. Two examples are thoroughly investigated, the results of which demonstrate the applicability and advantages of the proposed approach.

Redundancy Optimization of Finite-Dimensional Structures: Concept and Derivative-Free Algorithm

Redundancy is related to the amount of functionality that the structure can sustain in the worst-case scenario of structural degradation. This paper proposes a widely-applicable concept of redundancy optimization of finite-dimensional structures. The concept is consistent with the robust structural optimization, as well as the quantitative measure of structural redundancy based on the information-gap theory. A derivative-free algorithm is proposed based on the sequential quadratic programming (SQP) method, where we use the finite-difference method with adaptively varying the difference increment. Preliminary numerical experiments show that an optimal solution of the redundancy optimization problem possibly has multiple worst-case scenarios.

Structural robust optimization design based on convex model

There exist a great amount of uncertain factors in actual engineering. In order to involve these uncertain factors in analytical model, they have been expressed as the convex variables. In addition, the convex model was further classified into the hyper-ellipsoidal model and the interval model. After pointing out the intrinsic difference between these two kinds of models, the principle for applying which one of the models within the analysis has been indicated according to the available testing points. After standardizing the convex variables, the difference and relation between these two models for the optimization and solution process have been presented. With the analysis mentality available from the hyper-ellipsoidal model, the basic method about the robust optimization for the interval model was emphasized. After classification of the interval variables within the optimization process, the characteristics of the robust optimization were highlighted with different constraint conditions. Using the target-performance-based analytical scheme, the algorithm, the solution step and the convergence criteria for the robust optimization have been also presented with only one reliability index. Numerical examples and engineering problems were used to demonstrate the effectiveness and correctness of the proposed approach.

Multi-objective structural robust optimization under stress criteria based on mixed plate super-elements and genetic algorithms

This paper presents a methodology for the multi-objective (MO) robust optimization of plate structures under stress criteria, based on Mixed Super-Elements (MSEs). The optimization is performed with a classical Genetic Algorithm (GA) method based on Pareto-optimal solutions. It considers antagonist objectives among them stress criteria and thickness parameters distributed along the plate. This work aims at providing fast and efficient objective calculations. Our method is based on the implementation of MSEs for each zone of the plate featured by its own thickness. They are constructed with a Mixed Finite Element Model (MFEM) based on a displacement-stress mechanical formulation, and is enhanced with a sub-structuring modal reduction method in order to reduce the size of each constant thickness MSE. Those methods combined enable a fast and stress-wise efficient structure analysis, which improves the performance of the repetitive GA. A few cases minimizing the mass and the maximum Von Mises stress within a plate structure under dynamic loads put forward the relevance of our method with promising results. For the sake of robustness, both discrete frequencies and frequency bands are studied. The MO optimization is able to satisfy multiple damage criteria with different thickness distributions. It brings simplicity, saves computational time and the Pareto-front presentation with stress objective provides a good overview of the possibilities for the designers.

Development of a computational efficient tool for robust structural optimization

Purpose – Optimization under a deterministic approach generally leads to a final design in which the performance may degrade significantly and/or constraints can be violated because of perturbations arising from uncertainties. The purpose of this paper is to obtain a better strategy that would obtain an optimum design which is less sensitive to changes in uncertain parameters. The process of finding these optima is referred to as robust design optimization (RDO), in which improvement of the performance and reduction of its variability are sought, while maintaining the feasibility of the solution. This overall process is very time consuming, requiring a robust tool to conduct this optimum search efficiently. Design/methodology/approach – In this paper, the authors propose an integrated tool to efficiently obtain RDO solutions. The tool encompasses suitable multiobjective optimization (MO) techniques (encompassing: Normal-Boundary Intersection, Normalized Normal-Constraint, weighted sum method and min-max methods), a surrogate model using reduced order method for cheap function evaluations and adequate procedure for uncertainties quantification (Probabilistic Collocation Method). Findings – To illustrate the application of the proposed tool, 2D structural problems are considered. The integrated tool prove to be very effective reducing the computational time by up to five orders of magnitude, when compared to the solutions obtained via classical standard approaches. Originality/value – The proposed combination of methodologies described in the paper, leads to a very powerful tool for structural optimum designs, considering uncertainty parameters, that can be extended to deal with other class of applications.

Robust optimization of structural dynamic characteristics based on adaptive Kriging model and CNSGA

Dynamic characteristics greatly influence the comprehensive performance of a structure. But they are rarely included as objectives in traditional robust optimization of structures. In this study, a robust optimization model including both means and standard deviations of dynamic characteristic indices in the objective and constraint functions is constructed for improving the structural dynamic characteristics and reducing their fluctuations under uncertainty. Adaptive Kriging models are employed for the efficient computation of dynamic characteristics. An intelligent resampling technology is proposed to save computational costs and accelerate convergence of Kriging models, which takes full advantage of the test points for precision verification, the sample points within the local region of the biggest relative maximum absolute error and the near-optimal point to improve the global and local precision of Krigings. The high efficiency of proposed intelligent resampling technology is demonstrated by a numerical example. Finally, an efficient algorithm integrating adaptive Kriging models, Monte Carlo (MC) method, constrained non-dominated sorting genetic algorithm (CNSGA) is proposed to solve the robust optimization model of structural dynamic characteristics. Kriging models are interfaced with MC method to efficiently compute the fitness of individuals during CNSGA. The implementation of proposed methodology is explained in detail and highlighted by the robust optimization of a cone ring fixture with lots of circumferentially distributed holes in a large turbo generator aimed at moving its natural frequencies away from the exciting one. The comparison of the optimized design with the initial one demonstrates that the proposed methodology is feasible and applicable in engineering practice.

Robust high-cycle fatigue stress threshold optimization under uncertain loadings

This paper proposes a strategy to achieve robust optimization of structures against high-cycle fatigue when a potentially large number of uncertain load cases are considered. The strategy is heavily based on a convexity property of some of the most commonly used high-cycle design criteria. The convexity property is rigorously proven for the Crossland fatigue criterion. The proof uses a perturbation technique and involves the principal stress components and analytical expressions for the applicable fatigue criteria. The multiplicity of load cases is treated using load ratios which are bounded but are otherwise free to vary within certain limits. The strategy is applied to a notched plate subject to traditional normal and shear loadings that possess uncertain or unspecified components.

Sensitivity importance‐based robust optimization of structures with incomplete probabilistic information

SUMMARYRobust design optimization (RDO) is usually performed by minimizing the nominal value of a performance function and its dispersion considering equal importance to each individual gradient of the performance function. However, it is well known that all gradients are not equally important. An efficient sensitivity importance‐based RDO technique is proposed in the present study for optimum design of structures characterized by bounded uncertain input parameters. The basic idea of the proposed RDO formulation is to improve the robustness of a performance function by using a new gradient index that utilizes the importance factors proportional to the importance of the gradients of the performance function. The same concept is also extended to the constraints. To enhance the robustness of the constraints, the constraint functions are also modified by using the importance factor proportional to the importance of the associated gradient of the constraint. Because all the variables are not equally important to capture the presence of uncertainty, an improved robust solution is obtained by the proposed approach compared with the conventional RDO approach. The present formulation is illustrated with the help of three informative examples. The results are compared with the conventional RDO results to study the effectiveness of the proposed RDO approach. Copyright © 2012 John Wiley & Sons, Ltd.

Robust optimization of structures subjected to stochastic earthquake with limited information on system parameter uncertainty

The optimization of structures subjected to stochastic earthquake and characterized by uncertain parameters is usually posed in the form of non-linear programming with stochastic performance measures where the uncertain parameters are modelled as random variables. Such an approach, however, cannot be adopted in many real life situations when the limited information about uncertainty can be only modelled as of the uncertain but bounded (UBB) type. A robust optimization strategy for stochastic dynamic systems characterized by UBB parameters is proposed in the present study in the framework of the response surface method (RSM). In evaluating the stochastic constraints, repeated computations of the dynamic responses are avoided by applying an adaptive RSM based on the moving least squares method. Numerical results are presented to highlight the effectiveness of the proposed procedure. The effect of parameter uncertainty is also studied by comparing the results obtained from the proposed optimization approach with the conventional stochastic optimization results.

Robust Topology Optimization Based on an Aggregated Linear System and Eigenvalue Analysis

This paper proposes a robust topology optimization method for a linear elasticity design problem subjected to an uncertain load. The robust design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of “aggregation” used in the field of control research is introduced to evaluate the value of the robust compliance. The aggregation is applied to provide the direct relationship between the uncertain input load and output displacement using a small linear system composed of these vectors and the reduced size of a symmetric matrix in the context of a discretized linear elasticity problem using the finite element method. The robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix subject to the constraint that the Euclidean norm of the uncertain load set is fixed. Moreover, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed robust structural optimization method is implemented using the topology optimization method, sensitivity analysis and the method of moving asymptotes (MMA). The numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures.

AN IMPLICIT FORMULATION OF MATHEMATICAL PROGRAM WITH COMPLEMENTARITY CONSTRAINTS FOR APPLICATION TO ROBUST STRUCTURAL OPTIMIZATION

This paper discusses an implicit reformulation of a class of MPEC (mathematical program with equilibrium constraints) problems. We particularly focus on an MPEC problem arising from the robust optimization of elastic structures subjected to the uncertain external load. We first review the relation between the worst-case detection and robust constraint satisfaction of the structural responses, and then derive an MPEC formulation of the robust structural optimization. Since a standard constraint qualification is not satisfied at any feasible solution of an MPEC problem, we propose a reformulation based on the smoothed Fischer-Burmeister function, in which the smoothing parameter is treated as an independent variable. Numerical examples of robust structural optimization are presented to demonstrate that the presented formulation can be solved by using a standard nonlinear programming approach.

Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

AbstractStructural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.

Global optimization of robust truss topology via mixed integer semidefinite programming

This paper discusses a global optimization method of robust truss topology under the load uncertainties and slenderness constraints of the member cross-sectional areas. We consider a non-stochastic uncertainty of the external load, and attempt to minimize the maximum compliance corresponding to the most critical load. A design-dependent uncertainty model in the external load is proposed in order to consider the variation of truss topology rigorously. It is shown that this optimization problem can be formulated as a 0–1 mixed integer semidefinite programming (0–1MISDP) problem. We propose a branch-and-bound method for computing the global optimal solution of the 0–1MISDP. Numerical examples illustrate that the topology of robust optimal truss depends on the magnitude of uncertainty. The presented method can provide global optimal solutions for benchmark examples, which can be used for evaluating the performance of any other local optimization method for robust structural optimization.

Fuzzy-based robust structural optimization

Robust optimization is conventionally defined as the collection of the possible problem solutions that can ensure acceptable performances and sufficient immunity against the effects of uncertain parameter variability. Methods proposed until now use a probabilistic way to model uncertainty and to quantify the final sensitivity. In this work, a fuzzy uncertainty modellization is adopted for structural engineering. In particular, to define solution performance scattering, the fuzzy entropy is used as a global measure of variable dispersion. The final formulation of the problem deals with two antithetical objective functions, the fuzzy expected value of structural performance and its fuzzy entropy. This fuzzy-based approach in robust design is able to give a set of Pareto optimal solutions in terms of structural efficiency and sensitivities regarding uncertainty, and represents a suitable tool in supporting the decision maker. Finally, different applications have been developed to demonstrate the applicability of the proposed method.

Robustness analysis of structures based on plastic limit analysis with uncertain loads

This paper presents a method for computing an info-gap robustness function of structures, which is regarded as one measure of structural robustness, under uncertainties associated with the limit load factor. We assume that the external load in the plastic limit analysis is uncertain around its nominal value. Various uncertainties are considered for the live, dead, and reference disturbance loads based on the nonstochastic info-gap uncertainty model. Although the robustness function is originally defined by using the optimization problem with infinitely many constraints, we show that the robustness function is obtained as an optimal value of a linear programming (LP) problem. Hence, we can easily compute the info-gap robustness function associated with the limit load factor by solving an LP problem. As the second contribution, we show that the robust structural optimization problems of trusses and frames can also be reduced to LP problems. In numerical examples, the robustness functions, as well as the robust optimal designs, are computed for trusses and framed structures by solving LP problems.

Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty

A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of the external forces based on the info-gap model, the maximization of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-procedure, we reformulate the problem into a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has a global convergent property. It is shown through numerical examples that optimum designs of various linear elastic structures can be found without difficulty.

5308 コンプライアントメカニズムのロバスト最適設計法(J17-2 解析・設計の高度化・最適化(2)(設計と最適化),J17 解析・設計の高度化・最適化:解析の高度化)

5308 コンプライアントメカニズムのロバスト最適設計法(J17-2 解析・設計の高度化・最適化(2)(設計と最適化),J17 解析・設計の高度化・最適化:解析の高度化)