Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1526041, 14 pages

https://doi.org/10.1155/2017/1526041

## A Method for Multidisciplinary System Analysis Based on Minimal Feedback Variables

National CAD Supported Software Engineering Centre, Huazhong University of Science and Technology, Wuhan, China

Correspondence should be addressed to Boxing Wang

Received 7 July 2017; Revised 20 November 2017; Accepted 28 November 2017; Published 21 December 2017

Academic Editor: Carlo Cosentino

Copyright © 2017 Qian Yin 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

As modern engineering design usually involves dependence of one discipline on another, multidisciplinary system analysis (MDSA) plays an important role in the multidisciplinary simulation and design optimization on coupled systems. The paper proposes an MDSA method based on minimal feedback variables (MDSA_MF) to enhance the solving efficiency. There are two phases in the method. In phase 1, design structural matrix (DSM) is introduced to represent a coupled system, and each off-diagonal element is denoted by a coupling variable set; then an optimal sequence model is built to obtain a reordered DSM with minimal number of feedback variables. In phase 2, the feedback in the reordered DSM is broken, so that the coupled system is transformed into one directed acyclic graph; then, regarding the inputs depending on the broken feedback as independent variables, a least-squares problem is constructed to minimize the residuals of these independents and corresponding outputs to zero, which means the multidisciplinary consistence is achieved. Besides, the MDSA_MF method is implemented in a multidisciplinary platform called FlowComputer. Several examples of coupled systems are modeled and solved in the platform using several MDSA methods. The results demonstrate that the proposed method could enhance the solving efficiency of coupled systems.

#### 1. Introduction

Engineering design generally involves multidisciplinary dependence relationships of one discipline on another. For coupled system, these dependent relationships among the disciplines make up one or more loops. Thus, multidisciplinary system analysis (MDSA) is required to achieve the output-input consistence of all the dependence relationships by iteratively executing discipline analyses. Accordingly, an optimization on a coupled system using general nonlinear optimization methods could be time-consuming.

To enhance the solving efficiency, various multidisciplinary design optimization (MDO) frameworks [1–3] are proposed to handle the discipline couplings by decomposition and coordination strategies. Some MDO frameworks, for example, individual discipline feasible (IDF) [4] and collaborative optimization (CO) [5], eliminate the discipline couplings and enforce the multidisciplinary consistence at the final solution. These methods, however, could not obtain a multidisciplinary feasible solution when the optimization is interrupted. Other types of MDO frameworks, for example, multidisciplinary feasible (MDF) [4], concurrent subspace optimization (CSSO) [6], and bilevel integrated system synthesis (BLISS) [7], try to reduce the number of MDSA processes using different strategies while ensuring the multidisciplinary feasibility during the whole optimization process. Thus, the solving efficiency of MDSA could be essential to enhance the MDO process on coupled systems. Furthermore, different multidisciplinary analysis and optimization strategies under uncertainty are developed to handle the stochastic and/or epistemic uncertainties in coupled engineering problems [8–10]. A likelihood-based approach is proposed to estimate the probability density function of coupling variables [11] and is further extended to handle the model uncertainty [12] and the uncertainty propagation in high dimensional coupled systems [13]. Gibbs sampling and sequential importance resampling techniques are introduced to reduce the computational cost for decoupled multidisciplinary uncertainty analysis [14, 15]. These MDSA methods under uncertainty generally guarantee statistical multidisciplinary consistence, rather functional consistence. The present paper is focused on enhancing the solving efficiency of deterministic MDSA.

Several methods can be used to perform MDSA for coupled systems. Fixed point iteration (FPI) is a common-used method for MDSA [16]. When certain convergence condition is satisfied, a multidisciplinary feasible solution could be obtained. However, the FPI method converges slowly, which could lead to numerous discipline simulations [17, 18]. Newton-like methods [16] could achieve rapid convergence in that derivative information is used. The methods might fail to converge when solving from a bad starting point. Nonlinear least-squares (NLS) method [19] could be regarded as a generic MDSA method. It constructs a least-squares problem to minimize the sum of residuals of coupling relationships to zero and to obtain a multidisciplinary feasible solution. The constructed least-squares problem for MDSA can be solved flexibly by related NLS methods, or other general optimization algorithms.

Various engineering design platforms are developed to provide integrated multidisciplinary design environments. These platforms could integrate discipline design tools, define coupled system models, and perform multidisciplinary system analysis and design optimization on engineering problems [20, 21]. Commercial software tools, for example, ModelCenter [22], iSIGHT [23], and VisualDOC [24], are mainly focused on the integration of discipline tools and the capability of diverse design exploration methods [25, 26]. Some simple MDO frameworks, for example, IDF [4] and CO [5], can be implemented directly based on optimizer-like components and wrapped analysis components within some commercial tools [23, 27]. Several open source platforms, for example, DAKOTA [28], pyMDO [29], and openMDAO [30], could support automatic implementation of several MDO frameworks and their variants from specific problem descriptions [31, 32]. Some MDSA methods, or generic MDSA solvers, are provided in some of the platforms. ModelCenter provides a* Converger* component based on FPI method to achieve the convergence between the guessed variables and the calculated variables. The Gauss-Seidel iteration is the default algorithm to perform MDSA in pyMDO [29]. Within OpenMDAO,* BroydenSolver* and* FixedPointIterator* are provided to perform the iterative system analysis [31]. Within other MDO platforms, however, implementation of MDSA is generally provided by users. Furthermore, with the number of coupling variables increasing, the MDSA could be too large to be solved efficiently. These generic solvers might have difficulties in performing MDSA on the coupled systems with large number of couplings.

The paper proposed an MDSA method based on minimal feedback variables (MDSA_MF) to enhance the solving efficiency. The method includes two phases. In phase 1, design structural matrix (DSM) is introduced to represent a coupled system, and each off-diagonal element is denoted by a coupling variable set mapping from one discipline into another. Then, an optimal discipline sequence model is constructed to minimize the number of feedback variables by reordering the discipline sequence, and obtain a reordered DSM with minimal feedback variables. In phase 2, the feedback in the lower triangle of the reordered DSM is broken, so that the coupled system is transformed into a directed acyclic graph in terms of graph theory. Then, regarding the input variables depending on the broken feedback as independent variables, a least-squares problem with respect to these new independent inputs is constructed to minimize the sum of residuals of the independents and the corresponding outputs. When the objective of the least-squares problem is minimized to zero, the multidisciplinary consistence of the broken couplings is achieved. Besides, the implementation of the MDSA_MF method in a multidisciplinary design platform, called FlowComputer, is presented. Discipline integration based on Commercial-off-the-shelf (COTS) is provided to integrate discipline components, and a graphical user interface with dragging-and-dropping operations and visual data displayed is presented.

The rest of the paper is organized as follows. The next section lists the general MDSA methods used in the paper. Section 3 describes the DSM representation of coupled systems, proposes an optimal discipline sequence model to minimize the number of feedback variables, and presents the procedure of the MDSA_MF method. Section 4 describes the implementation of the MDSA_MF method in a multidisciplinary design platform. In Section 5, test cases of coupled systems are implemented in FlowComputer, and numerical results are investigated. Conclusions and future work are presented in the final section.

#### 2. Related Methods of Multidisciplinary System Analysis

A large engineering design system usually involves a series of disciplines depending on one another. Such a multidisciplinary system can be generally stated as formulation (1).where is the number of disciplines in the multidisciplinary system, is the independent input vector, and is the output variable of the th discipline. The equation represents the th discipline in the coupled system, where the notation represents the vector of independent input variables ; is the input variable depending on the th discipline and is usually called coupling variable.

As the disciplines are dependent on one another, one or more execution loops exist. For a nonlinear system, the multidisciplinary consistence of coupling variables could not be satisfied if all of the disciplines are executed only once. Therefore, some iterative system analysis process is required. In this section, several iterative methods for MDSA are described.

##### 2.1. Fixed Point Iteration

Fixed point iteration (FPI) method uses the original equations of the system as the iterative functions from a starting point of coupling variables [16]. Jacobi iteration and Gauss-Seidel iteration are the typical FPI methods. The former uses the values of the coupling variables from previous iteration to evaluate the outputs, and the disciplines could be run in parallel. The latter uses the recent evaluated values of other disciplines from current iteration as much as possible, and the disciplines are executed sequentially [33]. In most cases, the Gauss-Seidel iteration could converge faster than Jacobi iteration, for the newly updated values from current iteration might be more near to the solution. Formulations (2) and (3) state the iterative equations of Jacobi iteration and Gauss-Seidel iteration, respectively.where is the independent input vector, is the output vector of the previous iteration, and is the output vector of the current iteration.where is the independent input vector, is the output of the th discipline from the previous iteration, and is the output of the th discipline during the current iteration.

##### 2.2. Newton-Like Method

Newton-like methods convert the original coupled system into its residual form as formulation (4) and determine the next iterative point using the residual values and the corresponding derivative from the current point [16].where represents the residual form of the th discipline, is the number of disciplines, is the independent input vector, and represents the coupling variable.

The Newton-Raphson iterative equations [16] are presented as formulation (5).where is the vector of coupling variables from the previous iteration, is the output vector of coupling variables of the current iteration, and is the Jacobi matrix of the discipline residuals to the coupling variables during the previous iteration.

##### 2.3. Nonlinear Least-Squares Methods

Nonlinear least-squares (NLS) methods [19] break the coupling relationships and construct a least-squares objective, which minimizes the sum of squares of the residuals of the broken couplings, to find a multidisciplinary feasible solution. The least-squares problem is as formulation (6):where represents the number of coupling variables, is the output variable of the th discipline, and is the unknown design variable, corresponding to the input variable of a broken coupling relation depending on . Here, is determined by formulation (7).

The NLS algorithms, or other optimization algorithms, could be used to solve the least-squares problem, which makes the multidisciplinary problem more flexible to be solved. As each least-squares term is constructed with respect to a coupling variable, the number of unknown design variables is equal to the number of the least-squares terms.

#### 3. The Framework of MDSA_MF

##### 3.1. Discipline Dependence Representation of Coupled Systems

Design Structure Matrix (DSM) [34] is usually used to represent the dependence of one discipline on another in a coupled system. In the matrix, diagonal elements represent the disciplines, which might be analytical functions, specific disciplines, subsystems, components, black-boxes, or other objects. Each element in upper triangle represents a feed-forward coupling relationship between associated disciplines, and each one in lower triangle represents a feedback. Because the DSM representation is the adjacency matrix of the discipline dependence graph, a coupled system is generally a directed cyclic graph. Formulation (8) shows the general matrix representation.where represents the number of disciplines, represents the th discipline, and represents the dependence relationship of the th discipline on the th discipline.

The coupling relationships represented by the off-diagonal elements can be different expressions. A Boolean value, that is, “1” or “0,” can represent whether one discipline depends on another [35, 36]. The Boolean DSM could also be employed to model and solve Boolean Dynamical Systems [37, 38]. Derivative information could quantitatively indicate the influence of one discipline on another at a given point [39, 40]. And a natural number can represent the number of variables mapping from one discipline into another [41].

In the present paper, each off-diagonal element is represented by a collection of variables mapping from an output of one discipline into an input of another discipline. To simplify the collection, a set of output variables is often used. This representation can be converted into a Boolean value, or the number of feedback variables. Also, the representation can be extended to include other information about the corresponding coupling.

##### 3.2. The Optimal Discipline Sequence Model

In engineering designs, the MDSA problem could be too large to be solved efficiently when the number of coupling variables is large. In this case, a part of the couplings, for example, the feedback couplings, could be selected to construct a least-squares problem as Section 2.3 to implement the MDSA. Accordingly, the selected couplings are broken and the coupled system is transformed into one without feedback couplings. In terms of graph theory, one directed acyclic graph of the system is obtained.

The size of the least-squares problem depends on the number of selected feedback variables. With different sequence of diagonal elements, the feedback couplings in the lower triangle could be different. Several DSM-based optimization methods are proposed to reorder the discipline sequence [39, 42–44]. These methods try to minimize the number of feedback coupling loops [45, 46], or minimize an integrated objective taking other factors, for example, time, cost, and modularity, into account [43, 47]. Partitioning a coupled system into several small subsystems is another objective to reduce the complexity of the problems [41, 44].

The paper is focused on minimizing the number of feedback variables to reduce the MDSA problem size. The objective is to reduce the number of all the feedback variables in the lower triangle of DSM. Each off-diagonal element of the DSM is represented by a variable set consisting of feed-forward or feedback variables between two disciplines. Figure 1 shows the DSM representation of an example system with three disciplines. With the initial DSM as Figure 1(a), there are two feedback variables, and . With the reordered DSM as Figure 1(b), there is only one feedback variable, .