Gyung-Jin Park

Robust Design: An Overview

Robust design has been developed with the expectation that an insensitive design can be obtained. That is, a product designed by robust design should be insensitive to external noises or tolerances. An insensitive design has more probability to obtain a target value, although there are uncertain noises. Theories of robust design have been developed by adopting the theories of other fields. Based on the theories, robust design can be classified into three methods: 1) the Taguchi method, 2) robust optimization, and 3) robust design with the axiomatic approach. Each method is reviewed and investigated. The methods are examined from a theoretical viewpoint and are discussed from an application viewpoint. The advantages and drawbacks of each method are discussed, and future directions for development are proposed.

Robust optimization considering tolerances of design variables

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.

Structural optimization using equivalent static loads at all time intervals

A quasi-static structural optimization for elastic structures under dynamic loads is presented. An equivalent static load (ESL) set is defined as a static load set, which generates the same displacement field as that from a dynamic load at a certain time. Multiple ESL sets calculated at all the time intervals are employed to represent the various states of the structure under the dynamic load. They can cover all the critical states that might happen at arbitrary times. The continuous characteristics of a dynamic load are considered by multiple static load sets. The calculated sets of ESLs are utilized as a multiple loading condition in the optimization process. A design cycle is defined as a circulated process between an analysis domain and a design domain. The analysis domain gives the loading condition needed in the design domain. The design domain gives a new updated design to be verified by the analysis domain in the next design cycle. The design cycles are iterated until the design converges. Structural optimization with dynamic loads is tangible by the proposed method. Standard example problems are solved to verify the validity of the method.

Optimization of Flexible Multibody Dynamic Systems Using the Equivalent Static Load Method

Recently, an algorithm for dynamic response optimization transforming dynamic loads into equivalent static loads has been proposed. In later research, it was proved that the solution obtained by the algorithm satisfies the Karush-Kuhn-Tucker necessary conditions. In the present research, the proposed algorithm is applied to the optimization of flexible multibody dynamic systems. The equivalent static load is derived from the equations of motion for a flexible multibody dynamic system. The equivalent load is utilized in sequential static response optimization of the flexible mutibody dynamic system. In the end, the converged solution of the sequential static response optimization is the solution of the original dynamic response optimization. Some standard examples are solved to show the feasibility and efficiency of the proposed method. The control arm of an automobile suspension system is optimized as a practical problem. The results are discussed regarding the application of the proposed algorithm to flexible multibody dynamic systems.

Validation of a structural optimization algorithm transforming dynamic loads into equivalent static loads

Generally, structural optimization is carried out based on external static loads. However, all forces have dynamic characteristics in the real world. Mathematical optimization with dynamic loads is almost impossible in a large-scale problem. Therefore, in engineering practice, dynamic loads are often transformed into static loads via dynamic factors, design codes, and so on. Recently, a systematic transformation of dynamic loads into equivalent static loads has been proposed in Refs. 1–3. Equivalent static loads are made to generate at each time step the same displacement field as the one generated by the dynamic loads. In this research, it is verified that the solution obtained via the algorithm of Refs. 1–3 satisfies the Karush–Kuhn–Tucker necessary conditions. Application of the algorithm is discussed.

Transformation of dynamic loads into equivalent static loads based on modal analysis

All the forces in the real-world act dynamically on structures. Since dynamic loads are extremely difficult to handle in analysis and design, static loads are usually utilized with dynamic factors. Static loads are especially exploited well in structural optimization where many analyses are carried out. However, the dynamic factors are not determined logically. Therefore, structural engineers often come up with unreliable solutions. An analytical method based on modal analysis is proposed for the transformation of dynamic loads into Equivalent Static Loads (ESLs). The ESLs are calculated to generate an identical displacement field with that from dynamic loads at a certain time. The process is derived and evaluated mathematically by using the modal analysis. Since the exact solution is extremely expensive, some approximation methods are proposed. Error analyses have been conducted for the approximation methods. Standard examples for structural design are selected and solved by the proposed method. Applications of the method to structural optimization are discussed. Copyright

Robust design for unconstrained optimization problems using the Taguchi method

Engineering optimization has been developed for the economic design of engineering systems. The conventional optimum is determined without considering noise factors. Thus, applications to practical problems may be limited. Within current design practice, noises tend to be allowed for by specification of closer tolerances, or the use of safety factors. However, these approaches may be economically infeasible. Thus, the inclusion of design-variable noises is required for practical design in optimization. A method is developed to find robust solutions for unconstrained optimization problems. The method is applied to problems with discrete variables. The orthogonal array based on the Taguchi concept is utilized to arrange the discrete variables. Through several examples, it is verified that the solutions from the suggested method are more insensitive to noise than the conventional optimum within the range of variations for design variables.

Robust optimization in discrete design space for constrained problems

Robust design in discrete design space is defined as a discrete design that is insensitive to external uncertainties or variations. The application of robust discrete design is not prevalent yet due to high computational cost. A relatively simple method is proposed to select discrete and robust optimum. At first, the discrete design is achieved as the postprocess of conventional optimization. An orthogonal array is established around a conventional optimum, and the characteristic functions are evaluated. The characteristic function is defined by considering the robustness of the objective and constraints. The parameter design of the Taguchi method is introduced to obtain the robust solution in discrete space. The present method has insensitive performance to variations of the design variables within the selected discrete values enhancing the feasibility of constraints. To enhance feasibility, ranking the estimators of the characteristic function is developed. Several structural problems are solved to show the usefulness of the present method.

Robust design of a vibratory gyroscope with an unbalanced inner torsion gimbal using axiomatic design

Recently, there has been considerable interest in micro gyroscopes made of silicon chips. These can be applied to many microelectromechanical systems, such as devices for stabilization, general rate control, directional pointing, autopilot systems, and missile control. A decoupled vibratory gyroscope has been fabricated and tested. In this research, design improvement is obtained from numerical analyses as well as from a theoretical design point of view. The existing design is analyzed by using the axiomatic approach, which provides a general framework for design. For axiomatic design, the functional requirements (FRs) are twofold: firstly, the natural frequencies should have fixed values, and secondly the system should be robust to large tolerances. According to the independence axiom, the design parameters (DPs) are classified into the same number of groups as the FRs. Each group of DPs is separately determined according to the sequence indicated by axiomatic design. When a group of DPs should be determined to enhance robustness, the Taguchi concept is employed to maintain robust performance regardless of the tolerances. It is noted that the Taguchi method is used as a unit process in the sequence of the axiomatic design.

Dynamic Response Topology Optimization in the Time Domain using Equivalent Static Loads

Most topology optimization techniques find the optimal layout of a structure under static loads. Some studies are focused on dynamic response topology optimization because the dynamic forces act in the real world. Dynamic response topology optimization is solved in the time or frequency domain. A method for dynamic response topology optimization in the time domain is proposed using equivalent static loads. Equivalent static loads are static loads that generate the same displacement field as dynamic loads at each time step. The equivalent static loads are made by multiplying the linear stiffness matrix and the displacement field from dynamic analysis and used as multiple loading conditions for linear static topology optimization. The results of topology optimization are utilized in dynamic analysis again and a cyclic process is utilized until the convergence criterion is satisfied. The paradigm of the method was originally developed for size and shape optimizations. A new objective function is defined to minimize the peaks of the compliance in the time domain and a convergence criterion is newly defined considering that there are many design variables in topology optimization. The developed method is verified by solving some examples and the results are discussed.