Robust optimization considering tolerances of design variables

doi:10.1016/S0045-7949(00)00117-6

Computers & Structures

Volume 79, Issue 1, January 2001, Pages 77-86

https://doi.org/10.1016/S0045-7949(00)00117-6 Get rights and content

Abstract

Optimization techniques have been applied to versatile engineering problems for reducing manufacturing cost and for automatic design. The deterministic approaches of optimization neglect the effects from uncertainties of design variables. The uncertainties include variation or perturbation such as tolerance band. At optimum, the constraints must be satisfied within the tolerance ranges of the design variables. The variation of design variables can also give rise to drastic change of performances. The two issues are related to constraint feasibility and insensitive performance. Robust design suggested in the present study has been developed to obtain an optimum value insensitive to variations on design variables within a feasible region. This is performed by using a mathematical programming algorithm. A multiobjective function is defined to have the mean and the standard deviation of the original objective function, while the constraints are supplemented by adding a penalty term to the original constraints. This method has an advantage in that the second derivatives of the constraints are not required. Several standard problems for structural optimization are solved to check the usefulness of the suggested method.

Introduction

Optimization methods can be classified into deterministic approaches only considering nominal values of design variables and stochastic approaches considering variations on design variables. The optimum evaluated from deterministic optimization may violate the imposed constraints or cause the system performance defined as the objective function to be varied drastically [1], [2], [3]. They are generated from the uncertainties [4], [5], [6], which include the distributions of design variables, material properties, applied loadings, etc. In the suggested robust design, the variations are limited to the ones of the design variables while the rest of them are treated as constants. Comparing with stochastic optimization, it has the same characteristics which include variations on the design variables and different aspects in handling the constraints. Robust design implies the robustness of the objective and the constraint functions. The robustness of the objective function makes the system performance insensitive to the variations on the design variables. On the contrary, the robustness of the constraint function is defined by the feasibility condition which indicates that the optimum, considering the tolerances, always lies in the feasible region.

Balling et al. [7] have developed the robust design as the postprocess of the deterministic optimization, where gradients of the constraints at the optimum are utilized. Bennet and Lust [8] have defined the objective function with the weight of a structure and suggested a formulation where the linearized constraints are satisfied. Ramakrishnan and Rao [9] have treated the variance of an original objective function as the new objective function, and the mean of the original constraint functions as the new constraint functions. Lee et al. [1], [2] have developed the robust design in the discrete design space using the Taguchi method. Parkinson [10] has performed the robust design using the post optimality analysis and various approaches based on the classifications of controlled and uncontrolled design variables. Most existing studies have not offered the more reliable robust optima, since they evaluated the optima from the postprocess of the deterministic optimization or did not consider the robustness for the objective and the constraint functions simultaneously. For the constraint robustness, the reliability index is setup as the constraints by using reliability based optimization [11], [12], [13]. Reliability based optimization utilizes the probabilistic concepts while the suggested method the revised deterministic optimization.

The formulation revising the ones in deterministic optimization was developed to overcome the difficulties mentioned above. The robustness of the objective functions and constraint functions are defined, and the robust design algorithm is developed. The multiobjective function is introduced to consider the minimization and the robustness, which is composed of the mean and the standard deviation of the objective function due to variations on the design variables. It has an identical form with the multiobjective function utilized in stochastic optimization [5]. Generally, the objective function is setup as the weight in structural optimization. Especially, in the design of the truss structure, the multiobjective function is reduced to one term considering the minimization since the weight of the structure is linear with respect to the design variables of areas. However, the optimization of the beam structure utilizes the multiobjective function consisting of the mean and the standard deviation since its weight is nonlinear for design variables. For the constraint functions, the characteristics of the mathematical programming algorithm is exploited for the constraint robustness. The penalty terms with penalty coefficients are supplemented to the original constraint functions. They are proportional to the gradients of constraints and the tolerances of design variables. The original constraints are satisfied considering the tolerances of design variables by the revised constraint functions. The feasibility corresponds to the robustness of the constraint functions. The larger penalty factor enhances the constraint feasibility while the performance of the objective function become worse. In the subproblem, the second derivatives of the constraint functions are not required since the penalty terms are treated as constants. The formulations offer practical application to structural designs.

The utilized algorithm is an recursive quadratic programming (RQP) method built in an optimization system named idesign3 [14]. Various examples have been solved to illustrate the results of the suggested robust design. The examples consist of standard problems for the structural optimization such as three-bar truss and two-member frame and the design of the space frame in an electrical vehicle.

Section snippets

Robustness of objective function and constraint functions

The formulation of the deterministic optimization is represented as [15] $minimize f(x),$ $subject to g_{j} (x)⩽0 j=1,…,m,$ $x_{L} ⩽ x ⩽ x_{U},$ where x, x_L and x_U are vectors for design variables, lower bounds and upper bounds and f(x) and g_j(x) are objective function and the jth constraint function of m constraints, respectively. In this chapter, the robustness of the objective and constraint functions are explained.

The distribution of a design variable, x_i is assumed as Fig. 1. A design variable is normally distributed

Formulation of the robust design

An optimization is formulated for the robust design as follows: $minimize Φ (x)=α· μ_{f} μ_{f}^{*} +(1−α)· σ_{f} σ_{f}^{*},$ $0⩽α⩽1,$ $subject to g_{new}_{j} =g_{j} +k ∑ i=1 n ∂ g_{j} ∂ x_{i} Δx_{i},$ $j=1,…,m, x_{L} ⩽ x ±Δ x ⩽ x_{U},$ where the penalty factor with respect to each constraint is k which is determined by the user. The penalty term makes the constraints conservative. When a subproblem is established, the penalty term is considered as constants. The subproblem for Eq. (13) is represented as $find δ x,$ $minimize ∇ Φ · δ x,$ $subject to g_{j} +k ∑ i=1 n ∂ g_{j} ∂ x_{i} Δx_{i} + ∑ l=1 n ∂ g_{j} ∂ x_{l} δ x_{l}$ $j=1,…,m,$

Example problems

The designs of truss and beam structures are solved to illustrate the validity of the suggested method. The mean and the standard deviation of the multiobjective function are considered in the designs of a beam structure. Only the mean of the multiobjective function is considered in the design of a truss structure. For the design of a two-member frame, the optimum values are investigated by varying the weighting factor. For the designs of the three-bar truss and the two-member frame, effects on

Discussion and concluding remarks

The following statements are concluded from this study:

(1) The mathematical algorithm presented in this study offers an efficient approach to robust design by defining the robustness of the objective and the constraint functions. The optimum evaluated from robust design is the one insensitive to the variations of design variables satisfying the imposed constraints. The algorithm is practical for structural designs since it does not include the second derivatives of the responses.

(2) With the

Acknowledgements

This research was supported by center of innovative Design Optimization Technology, Korea Science and Engineering Foundation. The authors are thankful to Mrs. Mi Sun Park for correcting the manuscript.

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View full text

Robust optimization considering tolerances of design variables

Abstract

Introduction

Section snippets

Robustness of objective function and constraint functions

Formulation of the robust design

Example problems

Discussion and concluding remarks

Acknowledgements

Robust design for unconstrained optimization problems using Taguchi method

AIAA Journal

Structural optimization post-process using Taguchi method

JSME Int J Ser A

A design-for-manufacturing methodology for incorporating manufacturing uncertainties in the robust design of fibrous laminated composite structures

J Composite Mat

A robust optimization procedure with variations on design variables and constraints

Engng Optim

Consideration of worst-case manufacturing tolerances in design optimization

J Mech Trans Automat Design Trans ASME

Conservative methods for structural optimization

AIAA Journal

A robust optimization approach using Taguchi’s loss function for solving nonlinear optimization problems

ASME