Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2759762, 9 pages

https://doi.org/10.1155/2017/2759762

## Performance Evaluation of MDO Architectures within a Variable Complexity Problem

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Daiyu Zhang; nc.ude.upwn.liam@uyiad

Received 27 October 2016; Accepted 7 February 2017; Published 23 February 2017

Academic Editor: Quang Phuc Ha

Copyright © 2017 Daiyu Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Though quite a number of multidisciplinary design optimization (MDO) architectures have been proposed for the optimal design of large-scale multidisciplinary systems, how their performance changes with the complexity of MDO problem varied is not well studied. In order to solve this problem, this paper presents a variable complexity problem which allows people to obtain a MDO problem with arbitrary complexity by specifying its changeable parameters, such as the number of disciplines and the numbers of design variables. Then four investigations are performed to evaluate how the performance of different MDO architectures changes with the number of disciplines, global variables, local variables, and coupling variables varied, respectively. Finally, the results supply guidance for the selection of MDO architectures in solving practical engineering problems with different complexity.

#### 1. Introduction

Multidisciplinary design optimization (MDO) is a growing field of research in the design of large-scale engineering systems that consist of a number of interacting subsystems. The main motivation for using MDO is that the performance of a multidisciplinary system is driven not only by the individual disciplines but also by their interactions. By coordinating each discipline and decoupling their interaction reasonably in the design cycle, designers can improve the design and reduce the time and cost simultaneously.

Until now, many MDO architectures have been proposed for the optimal design of large-scale engineering systems. Generally, the architectures can be divided into two classes: monolithic architectures and distributed architectures. Monolithic architectures solve the MDO problem by casting it as a single optimization problem, which is easy to be implemented for small MDO problems. But for complicated engineering systems where people in charge of each discipline work independently of one another, these architectures may encounter big difficulty in integrating all the disciplines together. The monolithic architectures commonly include multidisciplinary feasible (MDF) [1–3], Individual Discipline Feasible (IDF) [1, 2, 4], and Simultaneous Analysis and Design (SAND) [1, 5]. However, the distributed architectures solve the MDO problems by decomposing it into smaller and more manageable discipline optimization problems that have the same solution when reassembled. One big advantage of these architectures is that they promote discipline autonomy and make people in different groups and teams just be in charge of their own fields and use their own discipline legacy. The distributed architectures usually include Concurrent Subspace Optimization (CSSO) [6, 7], Collaborative Optimization (CO) [8–10], Bilevel Integrated Systems Synthesis (BLISS) [11], Bilevel Integrated Systems Synthesis 2000 (BLISS-2000) [12, 13], Analytical Target Cascading (ATC) [14, 15], and MDO based on independent subspaces (MDOIS) [16].

Though many MDO architectures are available, the big challenge is to determine which architecture is the most efficient for a given MDO problem. To solve the problem, a benchmarking study of different MDO architectures needs to be carried out. Though many researches [17–23] have done excellent work in comparing the performances of different MDO architectures and Martins et al. [24] have made an initial exploration in assessing the computational complexity of MDO architectures, there are still some limitations of them. Firstly, many of the test MDO problems are of low dimensionality with few disciplines and variables. They lack the ability to test the performance of different MDO architectures for the practical engineering problems which are often composed of hundreds of design variables, constraints, and coupling variables. Secondly, the MDO architectures may perform differently when the test problem has different number of disciplines, design variables, constraints, or coupling variables. How the performance of MDO architectures changes with the complexity of MDO problem varied is not well studied.

In order to deal with the problems above, two works will be done in this paper. The first one is to present a variable complexity problem which is a nonseparable nonlinear problem that allows people to specify its complexity, such as the number of disciplines, design variables, and coupling variables. This makes it feasible to test the performance of different MDO architectures not only for high complexity problem, but also for problem with arbitrary complexity. The second one is to implement different MDO architectures to solve the variable complexity problem in four cases to evaluate how the performance of MDO architectures changes with the complexity of MDO problem varied.

The remainder of this paper is organized as follows. In Section 2, we state the terminology and mathematical notation that will be used throughout this paper. In Section 3, we present a variable complexity problem whose complexity can be varied by the changeable parameters. In Section 4, we give the specific formulations of four MDO architectures associated with the variable complexity problem. In Section 5, we evaluate how the performance of MDO architectures changes with the complexity of MDO problem varied. Finally, some conclusions are presented in the last section.

#### 2. Terminology and Mathematical Notation

Before introducing the variable complexity problem, we need to describe the terminology and mathematical notation that will be used throughout this paper.

Firstly, we define and clarify several terms that are specific to the field of MDO. Discipline analysis is usually a simulation that models the performance of one discipline in a multidisciplinary system. Global variables are the design variables that are shared by multiple disciplines. Local design variables are the design variables that only apply to one discipline. Coupling variables are the outputs that one discipline passes to other disciplines. Local constraints are the constraints that are only decided by one discipline. Global constraints are the constraints that are determined by multiple disciplines.

Then the common mathematical notation that will be used in the following sections is listed as follows. Note that all vectors in this paper are assumed to be column vectors unless indicated otherwise.

*Mathematical Notation for MDO Problem* : the vector of global variables shared by more than one discipline : the th element of : the vector of local design variables included in discipline : the th element of : the vector of coupling variables included in discipline : the th element of : the vector of local constraints included in discipline : the th element of : the vector of global constraints depending on the variables of multiple disciplines : the th element of

#### 3. Variable Complexity Problem

In this section, a variable complexity problem is proposed to allow the investigation that how the performance of MDO architectures changes with its complexity varied. It is a nonseparable nonlinear MDO problem which provides the ability to specify the changeable parameters in the following. Those parameters can be used to change the problem complexity as the user desires.

*Changeable Parameters in Variable Complexity Problem* : the number of disciplines : the number of global variables : the number of local variables included in discipline : the number of coupling variables included in discipline : the number of local constraints included in discipline : the number of global constraints

The remaining part of this section is laid out as follows to introduce the variable complexity problem. Firstly, the discipline analysis, constraint functions, and objective function of the variable complexity problem are described in detail, respectively. Then the monolithic mathematical model of the variable complexity problem is built and a simple example consisting of three disciplines is given for a more vivid description.

##### 3.1. Discipline Analysis

A discipline analysis is a simulation that models the behavior of one discipline. Generally, performing a discipline analysis is to solve an equation group, such as the Navier-Stokes equation in fluid mechanics, the static equilibrium equation in structural mechanics, or the equations of motion in control simulations. So a set of equations are created to represent the analysis of discipline in the variable complexity problem as follows.where is the vector of coupling variables of discipline , is a coefficient matrix of size , is a coefficient matrix of size , and is a coefficient matrix of size . In addition, all elements of , , and are real values.

If we denote (1) by in residual form as follows, the values of can be obtained by solving the equation and they are the solution of the discipline analysis

For (2), it is obvious that different disciplines are coupled with each other through the global variables and the coupling variables. In order to make it more clearly, a simple example consisting of three disciplines is described in Figure 1 by the extended design structure matrix (XDSM) method [25] which makes the description of MDO problems and architectures very convenient and legible.