机械设计外文参考文献

Multi-Objective Collaborative Optimization Based on Evolutionary Algorithms

Su Ruiyi"

Beijing System Design Institute of

Electromechanical Engineering,

No. 31 Yongding Road, Haidian District,

Beijing 100854, China

e-mail: sry@https://www.360docs.net/doc/de188488.html,

Gui Liangjin

e-mail: gui@https://www.360docs.net/doc/de188488.html,

Fan Zijie

e-mail: zjfan@https://www.360docs.net/doc/de188488.html,

State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering,

Tsinghua University,

Beijing 100084, China

This paper proposes a novel multi-objective collaborative optimi-zation (MOCO) approach based on multi-objective evolutionary algorithms for complex systems with multiple disciplines and objectives, especially for those systems in which most of the disci-plinary variables are shared. The shared variables will conflict when the disciplinary optimizers are implemented concurrently. In order to avoid the confliction, the shared variables are treated as fixed parameters at the discipline level in most of the MOCa approaches. But in this paper, a coordinator is introduced to handle the confliction, which allocates more design freedom and independence to the disciplinary optimizers. A numerical example is solved, and the results are discussed. [DOl: 10.1115/1.4004970]

Keywords: multidisciplinary design optimization, multi-objective optimization, collaborative optimization

1 Introduction

Multidisciplinary design optimization (MDO) was developed for large scale and complex engineering problems and has attracted much attention in recent years [1-3]. The two challenges of MDO are computational and organizational complexities [2]. The MDO problem involves large size of design variables, multiple objectives, interdisciplinary coupling, etc., which increase the computational expense. Moreover, the interdisciplinary coupling requests data transfer and decision interaction among different disciplinary codes and groups, which bring challenges to the organization of software modules and staffs. Several MDO approaches have been developed to deal with these challenges, such as concurrent subspace optimization [4], collaborative optimization (CO) [5], bi-level integrated system synthesis [6], and analytical target cascading [7].

Collaborative optimization [5] is one of the most popular MDO approaches, which decomposes the complex engineering problem into multiple disciplines, components, or subsystems. Each subsystem can be optimized concurrently by a different subject expert group employing appropriate codes. The interaction among disciplinary analysis codes is described by an interdisciplinary compat- 'Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 13, 2011; final manuscript received August 25, 2011; published online October 18, 2011. Assoc. Editor: Shapour Azarm.

Journal of Mechanical Design ibility function. Meanwhile, a system level optirruzer is introduced to minimize the design objectives and ensure the interdisciplinary compatibility.

One of the computational challenges in complex systems is raised

due to multiple objectives. The typical CO approach can be readily used to solve multi-objective problems by applying an aggregate function to convert multiple objectives to a single objective. For

example, Tappeta and Renaud [8] used the weighted sum method in the system level optimizer to handle multiple objectives. However, the

disadvantages of using the aggregate function in CO are that it cannot

find the Pareto optimal set in a single run and is unable to capture any Pareto solutions on the non convex part of the Pareto frontier [9].

These difficulties can be overcome by introducing the

population-based multi-objective evolutionary algorithms (MOEAs) to the CO framework. This multi-objective collaborative optimization

(MOCO) approach has been studied by Depince et al. [10], Aute and

Azarm [11], and Li and Azarm [12]. In their approaches, the system objectives are optimized at the system level and each is also

decomposed to be considered at the subsystem level, both system and

subsystem problems are solved by an MOEA. Their work shows that the combination of MOEAs and CO can obtain the Pareto optimal

solutions of multi-objective and multidisciplinary problems

effectively.

However, for complex systems where most of the variables are

shared and significant to more than one discipline, the previous approaches [1Q-12] have organizational and computational troubles,

because the shared design variables are considered as fixed

parameters at the subsystem level. For example, the window pillars of a bus body are sensitive to the rollover crash safety and significant to

the Noise, Vibration, and Harshness (NVH) performances. Both the

crash and NVH groups expect to design the pillars independently. However, this cannot be achieved as the pillar variables are treated as

fixed parameters in the disciplines. As such, it brings troubles to

organization. Moreover, as the shared variables are fixed during the optimization, the design freedom of disciplinary groups is reduced: If

most of the disciplinary variables are shared, there would be little

freedom at the subsystem level, which makes it difficult to find the feasible solutions. In this case, the disciplinary optimization is

meaningless and the MDO of the complex system will fail. This is the

computational trouble.

Both organizational and computational troubles, aforementioned,

will be solved in this paper by proposing a novel MOCO framework, where the shared variables can be varied at the subsystem level. Thus,

the disciplinary groups have the most design freedom to obtain the

Pareto optimal solutions effectively. In order to handle the difference of the shared variables from different disciplines, a coordinator (called

middle coordinator) is introduced. Consequently, the typical bi-level

CO framework is transformed to a tri-level framework, where the system level problem is solved by an MOEA, while both the

subsystem and middle level problems are solved by the sequential

quadratic programming (SQP) method.

The remainder of this paper is organized as follows: Sec. 2

describes the terminology of MDO problems; Sec. 3 gives the details

of the proposed approach; Sec. 4 solves a numerical example and discusses the results; and Sec. 5 concludes the paper.

2 Terminology

Figure 1 shows a fully coupled three-discipline nonhierarchic system, which was commonly used in the literature [6] and [8]. Each

box annotated with Di is a discipline or subsystem, which calculates

the outputs according to the inputs. For discipline i, the inputs include the design variable vector Xi and state variable vector Yji (j =I- i); the

outputs are composed by the objective vector fi, constraint vector gi and state variable vector Yij (i =I- j). The state variable vector Yu is calculated in discipline i and used in discipline j. The design variables

of discipline i comprise local variables Xli and shared variables Xshi' It is

seen that both state variables and shared variables are interdisciplinary coupling factors in an MDO problem.

Copyright ? 2011 by ASME OCTOBER2011, Vol.133 / 104502-1