Robust design of structures using optimization methods

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Abstract

The robust design of structures with stochastic parameters is studied using optimization techniques. The first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second-order perturbation method for the approximation of mean value and variance of the structural response. In this framework, the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task. The robust design of structures is formulated as a multi-criteria optimization problem, in which both the expected value and the standard deviation of the objective function are to be minimized. The robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints. The two-criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm. To demonstrate the applicability of the presented method, numerical examples are given, involving static and dynamic response.

Introduction

Methods of structural optimization have gained an increasing importance in the design of engineering structures for improving the structural performance and reducing costs. Under this trend, considerable progress has been made in the field, but mostly under the assumption of deterministic parameters. In the optimum obtained in deterministic structural optimization problems, the objective functions are minimized and the constraints are satisfied in a deterministic sense with reference to nominal values of design variables and other structural parameters. In engineering problems, randomness and uncertainties are inherent and may be involved in four stages, namely in the system design, in the manufacturing process, in the service time and in the aging process. Actually, the applied loads and structural parameters defining a real structure, such as geometrical dimensions, spatial positions of joints and material properties may be subject to random fluctuations or inaccuracies and thus give rise to performance variability. The design obtained by deterministic optimization may not achieve the desired optimal goal or may become unfeasible due to the scatter of the structural behavior. Therefore it is reasonable to explore the effect of randomness on the design. In particular, robust design, in which the structural performance is required to be less sensitive to the random variability of the structural parameters, has gained an ever increasing importance.

There have been different statements regarding randomness or uncertainty in structural design problems. In reliability based design optimization (RBDO) [1], [2], the cost function of the problem is to be minimized under observance of probabilistic constraints instead of conventional deterministic constraints. The applicability of RBDO relies on the availability of the precise probabilistic distribution of the stochastic parameters. In a similar way as in RBDO, some other studies based on the convex model (or interval set) [3] or fuzzy set [4] focus exclusively on the issue of structural safety with the purpose of avoiding system catastrophe in the presence of parameter uncertainties.

Unlike the above mentioned formulations usually considered in structural design problems, the robust structural design aims rather to reduce the variability of structural performance caused by regular fluctuations than to avoid occurrence of catastrophe in extreme events. The structural robustness is assessed by the measure of the performance variability around the mean (or expected) value, most often by its standard deviation, whereas reliability is based on the probability of failure occurrence (Fig. 1). For design optimization problems, the structural performance defined by design objectives or constraints may be subject to large scatter at different stages of the service life-cycle. Such scatters may not only significantly worsen the structural quality and cause deviations from the desired performance, but may also add to the structural life-cycle costs, including inspection, repair and other maintenance costs. This raises the task of reducing the scatter of the structural performance without eliminating the source of variability, as by robust design. The practical concept of robust design was initially proposed by Taguchi and has been proved effective in reducing the number of physical experiments for design improvement. A review of Taguchi's methodology was given by Tsui [5].

Though robust design relying on physical experiments has been prevalent in the design of industrial products or processes, numerical analysis methods for structural robust design are less well developed, despite the widely adopted sophisticated optimization techniques in conjunction with finite elements in the discipline of engineering structures. Chi and Bloebaum [6] used the Taguchi concept to solve structural optimization problems with continuous and discrete variables. Lee et al. [7] treated the unconstrained optimization problem with discrete design variables with applications to the robust design of truss structures. These approaches and other methods utilizing the concept of design-of-experiments exhibit some disadvantages regarding efficiency and simplicity. Lee and Park [8] defined the robust design problem as a revised deterministic optimization problem, in which the weighting factor is introduced to define a scalar multi-objective function and a penalty factor accounts for the variation of the constraint functions within the tolerance bands of design variables. A robust optimum of the minimum weight design problem was obtained by mathematical programming. Hereby, the random parameters other than the design variables were not considered. In the work of Lautenschlager and Eschenauer [9], the randomness of the structural parameters was modeled as leveled noise factors. Using the experimental design method, a response surface model was built for approximation of the structural performance and its variance. Based on the response surface model, robust structural design was obtained with an optimization algorithm. Monte Carlo simulation was applied by Sandgren and Cameron [10] in a genetic optimization algorithm to produce an output distribution for objective function and constraints in order to locate a design which was less sensitive to fluctuations. Gumbert et al. [11] presented reliability results for the robust design optimization of a flexible wing under geometric uncertainty. The robust design was conducted incorporating first-order approximations based on automatic differentiation in previous work of the authors. The paper also discussed the conceptual difference between robust design and RBDO underlining the utility of structural robust design.

A main difficulty that prohibits the application of the current robust design optimization methods is the extensive computation not only in approximating the expected value and the variance of the performance function but also in setting up sensitivity expressions for use in gradient based optimization algorithms. Practically, it is not always possible to acquire the entire distribution characteristics of the stochastic parameters and to completely explore the scatter of the structural performance. Therefore, in the present study, the uncertain parameters and the structural performance are quantified by the basic statistical characteristics, namely expected values and variances/standard deviations, without a preliminary assumption on their probabilistic distributions. In other words, the scatter of the objective function and the constraints is described in a non-statistical way, which allows for employment of the analytical perturbation method.

The prevalent methods of stochastic structural analysis can be classified into two major categories: statistical methods such as direct Monte Carlo simulation [12] and its variants [13], and non-statistical methods such as stochastic finite element methods (SFEM) based on second-order perturbation techniques [14]. The former methods rely on the sampling of the statistics, whereby the probabilistic distribution of the stochastic input variables are required, while the latter methods use analytical functional expansion requiring smoothness and differentiability. A recent review on structural stochastic analysis can be found in the work of Schueller [15].

The perturbation based stochastic finite element method provides a powerful tool for the analysis of structures with moderate parameter variability. In the framework of this approach, the random input parameters are described with up to the second statistical moment, regardless of the actual distribution. By consecutively solving the perturbed equilibrium equations, the zeroth-, first- and second-order solutions of displacements are obtained and thus the mean value and the covariance matrix of structural response can be approximated. Frequently, in design practice in areas such as structural engineering, mechanical engineering and aerospace engineering, the deviations of the random parameters from their nominal values can be controlled within limits. Under such conditions, the accuracy of perturbation based finite element method is considered sufficient for the purpose of computational robust design optimization.

In the present study, the sensitivity of the mean and variance of structural response is evaluated on the basis of the perturbed equations for stochastic analysis, and so the sensitivity of the mean and variance of the structural performance functions. The task of robust design optimization of structures is formulated as a two-criteria optimization problem, in which both the expected value and the standard deviation of the goal performance are to be minimized. The variability of the structural response may be present also in the constraints. The linear combination of the two criteria is stated as the ultimate objective-, or desirability function and the problem is solved by a gradient based optimization algorithm. The method accounts inherently for the interactions between design variables and random parameters, as well as the interactions among design variables and random parameters themselves. To demonstrate the applicability of the presented method, numerical examples are given, involving static and dynamic response.

Section snippets

Formulation of robust design optimization

Before discussing robust design optimization, the mathematical statement of a deterministic (nominal) design optimization problem is given as a prerequisite:finddminimizingf(d)subjecttogi(d)⩽0(i=1,2,…,k),ddd+,where dRm×1 denotes the vector of the design variables, d and d+ are the lower and upper bounds of the design variables, respectively. The objective function is denoted by f(d) and gi(d) is the constraint function. The design variables can be structural design parameters such as the

Basic equations for stochastic response analysis

In this section, the perturbation method is reviewed for use in the robust design problem for response and sensitivity analysis of structures with stochastic parameters.

The finite element equation of a random structure subject to static loads is expressed asK(b)u(b)=p(b),where KRn×n is the global stiffness matrix, uRn×1 is the vector of nodal displacements and pRn×1 is the vector of the external loads. The symbol bRq×1 denotes the vector of random variables. In this research, the random

Structural compliance optimization of a 25-bar space truss structure

The structural compliance, defined as the inner product of the applied load vector and the nodal displacement vector (ptu), of a 25-bar truss structure (see Fig. 5) resembling a power transmission tower is to be minimized. The design variables are bar cross-sectional areas. Six independent design variables are selected by linking various member sizes. The nodal coordinates and the member grouping information are given in Table 1, Table 3, respectively. The mass density of the material is ρ=0.1.

Conclusions and remarks

Compared with other formulations of structural optimization problems under uncertainties such as RBDO, the robust design task aims at controlling the variability of structural performance effectively. In the present study, the robust design of structures has been formulated and solved by optimization techniques incorporating perturbation based stochastic finite element analysis.

The second-order perturbation based stochastic finite element analysis is used to evaluate the mean value and the

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