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

Authors: Qian Yin; Yizhong Wu; Boxing Wang
Journal: Mathematical Problems in Engineering
 (2017)

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## 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.


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## Body

## 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 aConverger 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 andFixedPointIterator 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. Section3 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).(1)y1=y1X,y2,y3,…,yny2=y2X,y1,y3,…,yn⋮yn=ynX,y1,y2,…,yn-1,where n is the number of disciplines in the multidisciplinary system, X is the independent input vector, and yi is the output variable of the ith discipline. The equation yi=yiX,yj represents the ith discipline in the coupled system, where the notation X represents the vector of independent input variables x1,x2,…,xmT; yj is the input variable depending on the jth 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.(2)yk+1=yX,yk,where X is the independent input vector, y(k)=y1k,y2k,…,ynkT is the output vector of the previous iteration, and y(k+1)=y1k+1,y2k+1,…,ynk+1T is the output vector of the current iteration.(3)y1k+1=y1X,y2k,y3k,…,ynky2k+1=y2X,y1k+1,y3k,…,ynk⋮ynk+1=ynX,y1k+1,y2k+1,…,yn-1k+1,where X is the independent input vector, yi(k) is the output of the ith discipline from the previous iteration, and yi(k+1) is the output of the ith 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].(4)R1y1,y1X,y2,y3,…,yn=0,R2y2,y2X,y1,y3,…,yn=0,⋮Rnyn,ynX,y1,y2,…,yn-1=0,where Ri represents the residual form of the ith discipline, n is the number of disciplines, X is the independent input vector, and yi represents the coupling variable.The Newton-Raphson iterative equations [16] are presented as formulation (5).(5)yk+1=yk-JX,yk-1RX,yk,where yk is the vector of coupling variables from the previous iteration, y(k+1) is the output vector of coupling variables of the current iteration, and JX,yk 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):(6)Findyitmin∑i=1nyit-yi22,where n represents the number of coupling variables, yi is the output variable of the ith discipline, and yit is the unknown design variable, corresponding to the input variable of a broken coupling relation depending on yi. Here, yi is determined by formulation (7).(7)yi=yiX,yjt,j≠i.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.


## 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.(2)yk+1=yX,yk,where X is the independent input vector, y(k)=y1k,y2k,…,ynkT is the output vector of the previous iteration, and y(k+1)=y1k+1,y2k+1,…,ynk+1T is the output vector of the current iteration.(3)y1k+1=y1X,y2k,y3k,…,ynky2k+1=y2X,y1k+1,y3k,…,ynk⋮ynk+1=ynX,y1k+1,y2k+1,…,yn-1k+1,where X is the independent input vector, yi(k) is the output of the ith discipline from the previous iteration, and yi(k+1) is the output of the ith 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].(4)R1y1,y1X,y2,y3,…,yn=0,R2y2,y2X,y1,y3,…,yn=0,⋮Rnyn,ynX,y1,y2,…,yn-1=0,where Ri represents the residual form of the ith discipline, n is the number of disciplines, X is the independent input vector, and yi represents the coupling variable.The Newton-Raphson iterative equations [16] are presented as formulation (5).(5)yk+1=yk-JX,yk-1RX,yk,where yk is the vector of coupling variables from the previous iteration, y(k+1) is the output vector of coupling variables of the current iteration, and JX,yk 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):(6)Findyitmin∑i=1nyit-yi22,where n represents the number of coupling variables, yi is the output variable of the ith discipline, and yit is the unknown design variable, corresponding to the input variable of a broken coupling relation depending on yi. Here, yi is determined by formulation (7).(7)yi=yiX,yjt,j≠i.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.(8)Y=Y11Y12⋯Y1nY21Y22⋯Y2n⋮⋮⋱⋮Yn1Yn2⋯Ynn,where n represents the number of disciplines, Yii represents the ith discipline, and Yij represents the dependence relationship of the jth discipline on the ith 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 Section2.3 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. Figure1 shows the DSM representation of an example system with three disciplines. With the initial DSM as Figure 1(a), there are two feedback variables, y2 and y3. With the reordered DSM as Figure 1(b), there is only one feedback variable, y1.Figure 1
The DSM representations of an example system.
(a)
The initial DSM
(b)
The reordered DSMFigure2 shows the Boolean DSM representation of the aforementioned example. Both the initial Boolean DSM and the reordered Boolean DSM have two feedback marks. However, less feedback variables could be selected with the reordered DSM.Figure 2
The Boolean DSM representations of an example system.
(a)
The initial Boolean DSM
(b)
The reordered Boolean DSMTo minimize the number of feedback variables in the lower triangle, an optimal sequence problem is stated as formulation (9).(9)min⋃i=2n⋃j=1i-1Yij,wheren is the number of disciplines in a complex system and Yij is a feedback variable set mapping from discipline i into discipline j. ⋃i=2n⋃j=1i-1Yij represents the union set of all the variable sets in lower triangle, and Y denotes the number of elements in set Y.


### 3.3. MDSA_MF Procedure

The optimal discipline sequence model described in previous subsection is used to obtain a reordered DSM with the minimal number of feedback variables. The feedback variables in the lower triangle are selected to construct a least-squares problem to minimize the sum of residuals of the feedback couplings to zero and achieve the multidisciplinary consistence. The problem is stated as follows.(10)Findyismin∑i=1nyis-yi22,wheren represents the number of the selected coupling variables, yi is the output variable of the ith discipline, and yis is the unknown design variable, passed on to corresponding discipline input with respect to the broken coupling.As the feedback is removed, the disciplines are executed sequentially. The outputs with respect to the selected feedback are determined by formulation (11).(11)yi=yiX,yjs,yk,j>i,k<i,where yjs is the selected design variable, yi is the output of the ith discipline, and yk is the output of thekth discipline ahead of the ith discipline in the execution sequence of disciplines.In addition, some coupled systems could be divided into several strongly connected components. Each strongly connected component is a directed cyclic subgraph of the system. However, there is no interdependence relationship between any two strongly connected components. Thus, the system denoting each strongly connected component as a block is a directed acyclic graph, and the strongly connected components can be solved sequentially. The method of searching strongly connected component could aid in the discipline ordering [48].Figure3(a) shows the initial DSM representation of an example coupled system. The system can be divided into two strongly connected components and four individual nodes. Figure 3(b) shows the reordered DSM. A discipline can be regarded as a strongly connected component if the discipline is not coupled to other disciplines.Figure 3
An example DSM of a coupled system.
(a)
The initial DSM
(b)
The reordered DSMThe procedure of MDSA_MF is presented as follows.Step 1.
Search strongly connected components, and divide the system into a series of strongly connected components. Tarjan depth first search algorithm [49] is used to search strongly connected components.Step 2.
Topologically order the strongly connected components to generate the solving sequence.Step 3.
For each strongly connected component, reorder the discipline sequence to obtain a sub-DSM with minimal number of feedback variables by formulation (9).Step 4.
Solve the strongly connected components sequentially, and if some components do not depend on each other, they could be executed in parallel. For each strongly connected component, it is solved as follows.
Step  4.1. Identify the feedback variables, break the feedback, and generate evaluation functions using the subsystem without feedback.
Step  4.2. Construct a least-squares objective with respect to the residuals of broken couplings.
Step  4.3. Initialize the inputs of broken couplings with given starting values, or the current values of the corresponding outputs.
Step  4.4. Solve the least-squares problem as formulation (10).
Step  4.5. Pass on the output values to the corresponding inputs of the next subsystem, and turn toStep  4.1.Step 5.
Complete solving, and present outputs.


## 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.(8)Y=Y11Y12⋯Y1nY21Y22⋯Y2n⋮⋮⋱⋮Yn1Yn2⋯Ynn,where n represents the number of disciplines, Yii represents the ith discipline, and Yij represents the dependence relationship of the jth discipline on the ith 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 Section2.3 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. Figure1 shows the DSM representation of an example system with three disciplines. With the initial DSM as Figure 1(a), there are two feedback variables, y2 and y3. With the reordered DSM as Figure 1(b), there is only one feedback variable, y1.Figure 1
The DSM representations of an example system.
(a)
The initial DSM
(b)
The reordered DSMFigure2 shows the Boolean DSM representation of the aforementioned example. Both the initial Boolean DSM and the reordered Boolean DSM have two feedback marks. However, less feedback variables could be selected with the reordered DSM.Figure 2
The Boolean DSM representations of an example system.
(a)
The initial Boolean DSM
(b)
The reordered Boolean DSMTo minimize the number of feedback variables in the lower triangle, an optimal sequence problem is stated as formulation (9).(9)min⋃i=2n⋃j=1i-1Yij,wheren is the number of disciplines in a complex system and Yij is a feedback variable set mapping from discipline i into discipline j. ⋃i=2n⋃j=1i-1Yij represents the union set of all the variable sets in lower triangle, and Y denotes the number of elements in set Y.


## 3.3. MDSA_MF Procedure

The optimal discipline sequence model described in previous subsection is used to obtain a reordered DSM with the minimal number of feedback variables. The feedback variables in the lower triangle are selected to construct a least-squares problem to minimize the sum of residuals of the feedback couplings to zero and achieve the multidisciplinary consistence. The problem is stated as follows.(10)Findyismin∑i=1nyis-yi22,wheren represents the number of the selected coupling variables, yi is the output variable of the ith discipline, and yis is the unknown design variable, passed on to corresponding discipline input with respect to the broken coupling.As the feedback is removed, the disciplines are executed sequentially. The outputs with respect to the selected feedback are determined by formulation (11).(11)yi=yiX,yjs,yk,j>i,k<i,where yjs is the selected design variable, yi is the output of the ith discipline, and yk is the output of thekth discipline ahead of the ith discipline in the execution sequence of disciplines.In addition, some coupled systems could be divided into several strongly connected components. Each strongly connected component is a directed cyclic subgraph of the system. However, there is no interdependence relationship between any two strongly connected components. Thus, the system denoting each strongly connected component as a block is a directed acyclic graph, and the strongly connected components can be solved sequentially. The method of searching strongly connected component could aid in the discipline ordering [48].Figure3(a) shows the initial DSM representation of an example coupled system. The system can be divided into two strongly connected components and four individual nodes. Figure 3(b) shows the reordered DSM. A discipline can be regarded as a strongly connected component if the discipline is not coupled to other disciplines.Figure 3
An example DSM of a coupled system.
(a)
The initial DSM
(b)
The reordered DSMThe procedure of MDSA_MF is presented as follows.Step 1.
Search strongly connected components, and divide the system into a series of strongly connected components. Tarjan depth first search algorithm [49] is used to search strongly connected components.Step 2.
Topologically order the strongly connected components to generate the solving sequence.Step 3.
For each strongly connected component, reorder the discipline sequence to obtain a sub-DSM with minimal number of feedback variables by formulation (9).Step 4.
Solve the strongly connected components sequentially, and if some components do not depend on each other, they could be executed in parallel. For each strongly connected component, it is solved as follows.
Step  4.1. Identify the feedback variables, break the feedback, and generate evaluation functions using the subsystem without feedback.
Step  4.2. Construct a least-squares objective with respect to the residuals of broken couplings.
Step  4.3. Initialize the inputs of broken couplings with given starting values, or the current values of the corresponding outputs.
Step  4.4. Solve the least-squares problem as formulation (10).
Step  4.5. Pass on the output values to the corresponding inputs of the next subsystem, and turn toStep  4.1.Step 5.
Complete solving, and present outputs.


## 4. System Design and Implementation

In the section, the implementation of MDSA_MF in a visual and intuitive multidisciplinary platform, called FlowComputer, is presented to integrate different discipline models; define the dependence relationships and solve coupled systems.Component objects and link objects are developed to define data and process models of coupled systems. A component object represents a discipline, or a data processing node, and executes some discipline analysis or computes a series of outputs from given inputs. A link object describes the variable mappings from one component to another. And the proposed MDSA_MF method is implemented to solve the coupled system modeled in FlowComputer.


### 4.1. Discipline Integration Based on COTS Wrapping

Commercial-off-the-shelf (COTS) wrapping techniques are developed to integrate commercial software tools, or legacy codes. One of the major COTS wrapping approaches to discipline integration is the In-Process-Out (IPO) method, in which input files (I), process program (P), and output files (O) are used to integrate discipline tools. Input variables and output variables are stored in the input files and the output files, respectively. Process program is typically a discipline analysis tool, or a BAT file including a serial of discipline tool command lines, which reads values of input variables from input files, executes the corresponding analysis, and writes values of outputs into output files. Then, FlowComputer is able to integrate discipline simulation codes by wrapping input variables and output variables from input files and output files and specifying process programs and other supporting files. Currently, most discipline analysis tools, likeAnsys,Nastran,Adams,Abqus,Fluent,Ansoft, and so on, can be integrated by this approach. And aGeneric Wrapper component is provided to integrate various discipline analysis tools, especially the legacy simulation codes.For some cases that input and output variables are embedded in discipline models; another approach, plug-in method, is used to extract input and output variables from the model file of a third-party software tool by its API interface. Similarly, FlowComputer sets inputs to the model file, updates the model, and extracts outputs by the API interface. Now, discipline tools, includingMATLAB,Pro/E,CATIA, andExcel, are integrated using this approach.In addition,Expression component is also provided to compute a set of explicit expressions from given inputs.


### 4.2. Feedback Based Representation of Couplings

Link objects are designed to represent the dependence of one discipline on another. A link object consists of a source component, a destination component, and a set of variable mappings from the former to the latter. Thus, the couplings in a multidisciplinary system could be presented as a collection of link objects.To represent the coupling relationships, a type of feedback links is introduced to facilitate the modeling and solving of coupled systems. During the process of creating variable links, if a dependence loop is detected, the last constructed link is marked as a feedback link. DSM representation could intuitively exhibit the coupling relationships in a complex system. In the present study, the collection of link objects is used to represent the couplings, and DSM representation is used to intuitively exhibit the dependence relationships. Each link object represents a dependence relationship corresponding to an off-diagonal element of the DSM representation.


### 4.3. MDSA Implementation

Searching algorithm of strongly connected components and discipline reordering algorithm are used to analyze coupled systems, and all the iterative methods for MDSA described in Section2 are available to solve coupled systems, or strongly connected components. By default, the MDSA_MF method is used to construct the least-squares objective, and thehybrd solver from Minpack package [50] is selected to solve the least-squares problem.


### 4.4. Introduction to the User Interface

The main user interface of FlowComputer, shown as Figure4, is composed of flow view, components tree view, components class view, and components list view. According to a selected component class in the component class view, available components are displayed in the component list view. The components can be dragged and dropped into the flow view to integrate discipline models, and the relationship between two components can be defined by dragging-and-dropping operations. All components are displayed as a tree in the components tree view. Once a coupled system model is well defined, it can be executed automatically and monitored visually, and results can be shown graphically.Figure 4
The main user interface of FlowComputer.Dependence relationships between any two components could be constructed visually. Mapping viewer shown as Figure5 is provided to define, edit, and display variable mappings between any two disciplines. Two component trees are listed on the left and the right of the mapping viewer, and link lines indicating mappings are drawn from the variables on the left tree to input variables on the right tree. Users can drag-and-drop any variable on the left tree onto an input variable on the right tree to define a map from the former variable to the latter. The default map relation is that the right variable is equal to the left one.Figure 5
The mapping viewer for dependence relationships.The DSM representations of a coupled system in FlowComputer are shown as Figure6. There are nine disciplines, represented by nine blocks on the diagonal. The solid connection lines in the upper triangle represent the feed-forward couplings, and the dashed lines in the lower triangle represent the feedback.Figure 6
The screenshot of the DSM representation of a coupled system.
(a)
The initial DSM
(b)
The reordered DSM


## 4.1. Discipline Integration Based on COTS Wrapping

Commercial-off-the-shelf (COTS) wrapping techniques are developed to integrate commercial software tools, or legacy codes. One of the major COTS wrapping approaches to discipline integration is the In-Process-Out (IPO) method, in which input files (I), process program (P), and output files (O) are used to integrate discipline tools. Input variables and output variables are stored in the input files and the output files, respectively. Process program is typically a discipline analysis tool, or a BAT file including a serial of discipline tool command lines, which reads values of input variables from input files, executes the corresponding analysis, and writes values of outputs into output files. Then, FlowComputer is able to integrate discipline simulation codes by wrapping input variables and output variables from input files and output files and specifying process programs and other supporting files. Currently, most discipline analysis tools, likeAnsys,Nastran,Adams,Abqus,Fluent,Ansoft, and so on, can be integrated by this approach. And aGeneric Wrapper component is provided to integrate various discipline analysis tools, especially the legacy simulation codes.For some cases that input and output variables are embedded in discipline models; another approach, plug-in method, is used to extract input and output variables from the model file of a third-party software tool by its API interface. Similarly, FlowComputer sets inputs to the model file, updates the model, and extracts outputs by the API interface. Now, discipline tools, includingMATLAB,Pro/E,CATIA, andExcel, are integrated using this approach.In addition,Expression component is also provided to compute a set of explicit expressions from given inputs.


## 4.2. Feedback Based Representation of Couplings

Link objects are designed to represent the dependence of one discipline on another. A link object consists of a source component, a destination component, and a set of variable mappings from the former to the latter. Thus, the couplings in a multidisciplinary system could be presented as a collection of link objects.To represent the coupling relationships, a type of feedback links is introduced to facilitate the modeling and solving of coupled systems. During the process of creating variable links, if a dependence loop is detected, the last constructed link is marked as a feedback link. DSM representation could intuitively exhibit the coupling relationships in a complex system. In the present study, the collection of link objects is used to represent the couplings, and DSM representation is used to intuitively exhibit the dependence relationships. Each link object represents a dependence relationship corresponding to an off-diagonal element of the DSM representation.


## 4.3. MDSA Implementation

Searching algorithm of strongly connected components and discipline reordering algorithm are used to analyze coupled systems, and all the iterative methods for MDSA described in Section2 are available to solve coupled systems, or strongly connected components. By default, the MDSA_MF method is used to construct the least-squares objective, and thehybrd solver from Minpack package [50] is selected to solve the least-squares problem.


## 4.4. Introduction to the User Interface

The main user interface of FlowComputer, shown as Figure4, is composed of flow view, components tree view, components class view, and components list view. According to a selected component class in the component class view, available components are displayed in the component list view. The components can be dragged and dropped into the flow view to integrate discipline models, and the relationship between two components can be defined by dragging-and-dropping operations. All components are displayed as a tree in the components tree view. Once a coupled system model is well defined, it can be executed automatically and monitored visually, and results can be shown graphically.Figure 4
The main user interface of FlowComputer.Dependence relationships between any two components could be constructed visually. Mapping viewer shown as Figure5 is provided to define, edit, and display variable mappings between any two disciplines. Two component trees are listed on the left and the right of the mapping viewer, and link lines indicating mappings are drawn from the variables on the left tree to input variables on the right tree. Users can drag-and-drop any variable on the left tree onto an input variable on the right tree to define a map from the former variable to the latter. The default map relation is that the right variable is equal to the left one.Figure 5
The mapping viewer for dependence relationships.The DSM representations of a coupled system in FlowComputer are shown as Figure6. There are nine disciplines, represented by nine blocks on the diagonal. The solid connection lines in the upper triangle represent the feed-forward couplings, and the dashed lines in the lower triangle represent the feedback.Figure 6
The screenshot of the DSM representation of a coupled system.
(a)
The initial DSM
(b)
The reordered DSM


## 5. Case Studies

In this section, two coupled systems are modeled and solved in FlowComputer. The MDSA method based on all couplings (MDSA_AC), the MDSA method based on initial feedback variables (MDSA_IF), and the MDSA_MF method are investigated. Different iterative algorithms, including FPI, the Newton method, and thehybrd solver, are employed within each MDSA method. The number of function calls to disciplines is recorded to evaluate the solving efficiency, and the 2-norm of discipline residuals to the final solution, as formulation (12), is used to evaluate the accuracy of different methods.(12)r=y∗-yX,y∗2,wherer is the 2-norm of discipline residuals, X represents the independent input variable vector with given values, and y∗ represents the final solution of the coupling variables.


### 5.1. Test Case  1

The test case is modified from the scalable problem [51] and is stated as follows:(13)y1=-9.7160z+1.3627x1-11.008y31/3-8.2846y166.4976,y2=-1.5717z+7.8294x2-72.238y81/3-5.4647y417.0438,y3=-2.0489z+4.5625x3-67.350y181/3-7.8362y202.2016,y4=-1.9935z+9.3701x4-43.968y91/3-6.9368y610.9294,y5=-8.2844z+3.2383x5-11.369y101/3-9.2195y159.7504,y6=-1.3453z+1.8450x6-35.783y191/3-6.1040y1413.8736,y7=-2.4311z+4.6650x7-11.369y101/3-3.2488y13.5614,y8=-2.5485z+6.2020x8-61.040y141/3-8.5219y214.4476,y9=-1.3537z+7.0370x9-85.219y21/3-1.7807y78.7936,y10=-9.6129z+1.8295x10-92.195y151/3-4.8752y52.2738,y11=-8.1890z+5.1827x11-11.008y31/3-6.1040y1416.7888,y12=-1.5986z+2.3169x12-28.841y171/3-9.2195y154.7212,y13=-6.1798z+1.9304x13-69.368y61/3-6.7350y188.9082,y14=-3.5043z+3.3825x14-23.606y121/3-4.4541y1312.2080,y15=-7.9848z+9.9501x15-48.752y51/3-2.3606y1218.4390,y16=-1.3627z+1.3782x16-78.362y201/3-8.3944y1116.5692,y17=-3.7686z+3.4848x17-85.219y21/3-6.7350y185.7682,y18=-4.9599z+9.0051x18-54.647y41/3-4.3968y913.4700,y19=-8.6044z+7.2126x19-44.541y131/3-7.2238y87.1566,y20=-7.7096z+2.8943x20-69.368y61/3-2.3606y1215.6724,where xi and z are the independent variables and yi is the output of the ith discipline.This coupled system with 20 disciplines has 21 independent inputs and 20 coupling variables. The DSM of the system is shown as Figure7. The disciplines are coded with ascend digital numbers from 1 to 20. Each diagonal box with a number represents one discipline, and each black dot in the off-diagonal represents a coupling variable, marked on the left or on the right. The system is a strongly connected graph, and the reordered DSM with minimal feedback variables is shown as Figure 8.Figure 7
The initial DSM of the coupled system in test Case  1.Figure 8
The reordered DSM with minimal feedback variables in test Case  1.Table1 shows the solving results starting from the independent variables z=1 and xi=1 and the coupling variables yi=10. Here, each discipline is treated as a black-box simulation. The gradients required by the Newton method and thehybrd solver are computed via forward difference method with a relative step size of 10−6. The Newton method and thehybrd solver would terminate if the 2-norm of discipline residuals is less than 10−10. For the FPI method, the termination criteria are satisfied when the 2-norm of the distance between consecutive solutions is less than 10−10. Starting from other values of the independent variables and other initial values of the coupling variables, similar results are obtained.Table 1
The result data for test Case  1.
Iterative method/solver
r
Evaluation number
D
1
~
D
20
MDSA_AC
FPI
4.95
×
10
-
11
65
Newton method
1.05
×
10
-
12
106
hybrd solver
4.94
×
10
-
11
37
MDSA_IF
FPI
1.79
×
10
-
11
36
Newton method
3.08
×
10
-
14
71
hybrd solver
8.44
×
10
-
11
26
MDSA_MF
FPI
1.07
×
10
-
11
31
Newton method
2.22
×
10
-
13
29
hybrd solver
2.80
×
10
-
11
16
r represents the 2-norm of discipline residuals and Di represents the ith discipline. The evaluation counts of different disciplines are the same.The data in Table1 indicate that the coupled system is successfully solved using all the MDSA methods. For a given iterative method, for example, thehybrd solver, the MDSA_MF uses the fewest function calls to disciplines, followed by the MDSA_IF. And for the MDSA method based on the same selected couplings, for example, MDSA_MF, thehybrd solver uses the fewest discipline evaluations. The coupled system has 20 coupling variables. As shown in Figures 7 and 8, 13 feedback variables, y3, y4, y6, y8, y9, y10, y3, y4, y6, y8, y9, y10, and y9, are selected for MDSA_IF, and six coupling variables, y2, y5, y6, y9, y12, and y14, are selected for MDSA_MF. Sequentially, MDSA_MF uses less function calls to compute the derivative information. As the Jacobi matrix of the discipline residuals to coupling variables at each iterate is computed by the finite difference method, the Newton method uses more function calls to the disciplines.Figure9 shows the convergence histories of the 2-norm of discipline residuals. Similarly, for a given iterative method, for example,hybrd solver, MDSA_MF uses the fewest iterates, followed by MDSA_IF. The Newton methods converge to the final solution by the fastest speed, for derivative information is used at each iterate. However, the discipline evaluations increase when the finite difference method is used to compute derivative information.Figure 9
Convergence histories of the 2-norm of discipline residuals on test case  1.


### 5.2. Test Case  2

The second test case, from [39], is stated as formulation (14).(14)t1=23.521x112+45.122x121/2+0.9352y11/2-0.2835v2,t2=23.175x12+30.645x13-0.5392v1+0.0024z12,t3=12.944x11+43.846x12+18.345x132-0.0113v22-0.0021z22-0.1635u2,t4=12.347x112/3+21.478x12+7.745x132+0.563w21/2-0.7256v4+0.9476y3,u1=8.435x212+12.563x22+0.1577z1-0.9388w31/2,u2=24.547x21+24.356x22-1.457t12+0.3846t3+0.9862w21/2-0.7377z31/2,u3=9.367x222+0.0285t11/4-0.6758y11/2,v1=16.846x31+10.360x322-0.3367t2+0.157u21/2+0.0054z1-0.4783w21/2,v2=23.644x321/2+14.466x33-0.0064t12+0.5347y2,v3=8.846x312+44.467x331/2+0.3748u31/2-0.1094w1,v4=14.536x322-0.0949y1+0.4503t4+0.1003w31/2+0.0083u1-0.0776y31/2,w1=14.896x413+13.746x42+0.4567y1-0.0144y22+0.0944z2,w2=16.436x422-0.0113z12-0.0673z2+0.0356z3+0.0275u2,w3=8.638x41+26.693x42+6.536x432-0.6685t4-0.8467v4,y1=27.783x51+35.552x52+10.377x532-0.0059t12+0.0983u3+0.0633v31/2,y2=15.367x512+18.653x53+0.1079u3+0.2763w1-0.831v2,y3=22.561x52+35.649x53+0.6463z3-0.7366t4-0.0667u1,y4=9.882x512+0.0011t32-0.0359u2+0.2033z3,z1=8.629x61+7.932x622-0.0274t2-0.0235t3+0.0021v1,z2=14.367x612+48.837x62+0.0009v22-0.0328w11/2-0.0825v3,z3=29.663x62-0.0846w1+0.0024t21/2+0.0436u2+0.0875w2-0.0289y41/2,where xij represents the independent input and ti, ui, vi, wi, yi, and zi are the coupling variables.The system has 21 disciplines and 21 coupling variables. The DSM of the system is shown as Figure10. The disciplines are coded with ascend digital numbers from 1 to 21. The boxes on the diagonal and the black dots on the off-diagonal denote the same as Figure 7.Figure 10
The initial DSM of the coupled system in case 2.The coupled system could be divided into three strongly connected components, and the reordered DSM is shown as Figure11. The strongly connected components are denoted byGroup A,Group B, andGroup C, respectively. Within each strongly connected component, the discipline sequence with minimal number of feedback variables is presented.Figure 11
The reordered DSM of the system in case 2 with minimal feedback variables.Within FlowComputer, the coupled system is successfully solved using several methods starting from some given values of the independent variables and different initial values of the coupling variables. This coupled system could be solved by constructing an MDSA problem with 21 disciplines and could also be solved by constructing three sequential subproblems corresponding to the three strongly connected components in Figure11. Both of the two solving methods are investigated. The settings for algorithms, the step size of finite differences, and the termination criteria are the same as the first test case.Tables2 and 3 show the results starting from the independent variables xij=1 and the coupling variables (ti,uj,vk,wl,ym,zn)=(10,10,10,10,10,10). The evaluation number of the three groups of disciplines and the 2-norm of discipline residuals at the final solution are presented. Solving from other initial values of coupling variables presents similar data. However, solving from some given values of the independent variables far from the point xij=1 might fail in finding a multidisciplinary solution.Table 2
The result data for test case 2 solving as a whole coupled system.
Iterative method/solver
r
Evaluation number
D
1
~
D
21
MDSA_AC
FPI
3.00
×
10
-
11
33
Newton method
2.70
×
10
-
14
111
hybrd solver
1.03
×
10
-
11
33
MDSA_IF
FPI
5.61
×
10
-
12
17
Newton method
2.02
×
10
-
14
57
hybrd solver
1.57
×
10
-
12
21
MDSA_MF
FPI
4.01
×
10
-
12
17
Newton method
2.32
×
10
-
14
45
hybrd solver
1.56
×
10
-
12
18
r represents the 2-norm of discipline residuals and Di represents the ith discipline. The evaluation counts of different disciplines are the same.Table 3
The result data for test case 2 solving the three strong components sequentially.
Iterative method/solver
r
Evaluation number
Group A
Group B
Group C
MDSA_AC
FPI
5.01
×
10
-
11
33
32
20
Newton method
7.24
×
10
-
11
37
37
19
hybrd solver
3.94
×
10
-
11
20
15
11
MDSA_IF
FPI
6.35
×
10
-
12
17
16
12
Newton method
3.65
×
10
-
14
25
19
13
hybrd solver
1.04
×
10
-
11
13
11
8
MDSA_MF
FPI
6.51
×
10
-
12
17
16
7
Newton method
8.33
×
10
-
15
21
16
10
hybrd solver
8.31
×
10
-
12
11
9
6
r represents the 2-norm of discipline residuals.The data listed in Tables2 and 3 show that the coupled system is successfully solved with good accuracy using all of the methods. For each iterative method, MDSA_MF uses the fewest discipline evaluations, followed by the MDSA_IF method. If the system is solved as a whole coupled unit as in Table 2, the FPI method uses fewer function calls when the initial feedback, or the minimal feedback, is selected to be broken. However, thehybrd solver uses the fewest function calls when the three strongly connected components are solved sequentially. The possible reason is that the strategy solving the strongly connected components sequentially reduces the MDSA size. The strategy, however, does not influence the efficiency of Gauss-Seidel iteration significantly. Hence, MDSA_MF could use the fewest function calls, and solving the strongly connected components sequentially could further enhance the MDSA efficiency.Figure12 shows the convergence histories of the 2-norm of discipline residuals for solving the subsystem,Group A. The histories of solving the other two subsystems present the similar behavior. The convergence data indicate that MDSA_MF could use the fewest iterates for a given iterative method. The FPI iteration shows slow convergence. As derivative information is updated at each iterate, the Newton iterations converge to the final solution by fastest speed. However, computing the derivative information using the finite difference method increases the function calls. When the derivative information is easy to be obtained, the Newton iteration could generally converge to the final solution as the fastest speed.Figure 12
Convergence histories of the Group A in test case 2.


## 5.1. Test Case  1

The test case is modified from the scalable problem [51] and is stated as follows:(13)y1=-9.7160z+1.3627x1-11.008y31/3-8.2846y166.4976,y2=-1.5717z+7.8294x2-72.238y81/3-5.4647y417.0438,y3=-2.0489z+4.5625x3-67.350y181/3-7.8362y202.2016,y4=-1.9935z+9.3701x4-43.968y91/3-6.9368y610.9294,y5=-8.2844z+3.2383x5-11.369y101/3-9.2195y159.7504,y6=-1.3453z+1.8450x6-35.783y191/3-6.1040y1413.8736,y7=-2.4311z+4.6650x7-11.369y101/3-3.2488y13.5614,y8=-2.5485z+6.2020x8-61.040y141/3-8.5219y214.4476,y9=-1.3537z+7.0370x9-85.219y21/3-1.7807y78.7936,y10=-9.6129z+1.8295x10-92.195y151/3-4.8752y52.2738,y11=-8.1890z+5.1827x11-11.008y31/3-6.1040y1416.7888,y12=-1.5986z+2.3169x12-28.841y171/3-9.2195y154.7212,y13=-6.1798z+1.9304x13-69.368y61/3-6.7350y188.9082,y14=-3.5043z+3.3825x14-23.606y121/3-4.4541y1312.2080,y15=-7.9848z+9.9501x15-48.752y51/3-2.3606y1218.4390,y16=-1.3627z+1.3782x16-78.362y201/3-8.3944y1116.5692,y17=-3.7686z+3.4848x17-85.219y21/3-6.7350y185.7682,y18=-4.9599z+9.0051x18-54.647y41/3-4.3968y913.4700,y19=-8.6044z+7.2126x19-44.541y131/3-7.2238y87.1566,y20=-7.7096z+2.8943x20-69.368y61/3-2.3606y1215.6724,where xi and z are the independent variables and yi is the output of the ith discipline.This coupled system with 20 disciplines has 21 independent inputs and 20 coupling variables. The DSM of the system is shown as Figure7. The disciplines are coded with ascend digital numbers from 1 to 20. Each diagonal box with a number represents one discipline, and each black dot in the off-diagonal represents a coupling variable, marked on the left or on the right. The system is a strongly connected graph, and the reordered DSM with minimal feedback variables is shown as Figure 8.Figure 7
The initial DSM of the coupled system in test Case  1.Figure 8
The reordered DSM with minimal feedback variables in test Case  1.Table1 shows the solving results starting from the independent variables z=1 and xi=1 and the coupling variables yi=10. Here, each discipline is treated as a black-box simulation. The gradients required by the Newton method and thehybrd solver are computed via forward difference method with a relative step size of 10−6. The Newton method and thehybrd solver would terminate if the 2-norm of discipline residuals is less than 10−10. For the FPI method, the termination criteria are satisfied when the 2-norm of the distance between consecutive solutions is less than 10−10. Starting from other values of the independent variables and other initial values of the coupling variables, similar results are obtained.Table 1
The result data for test Case  1.
Iterative method/solver
r
Evaluation number
D
1
~
D
20
MDSA_AC
FPI
4.95
×
10
-
11
65
Newton method
1.05
×
10
-
12
106
hybrd solver
4.94
×
10
-
11
37
MDSA_IF
FPI
1.79
×
10
-
11
36
Newton method
3.08
×
10
-
14
71
hybrd solver
8.44
×
10
-
11
26
MDSA_MF
FPI
1.07
×
10
-
11
31
Newton method
2.22
×
10
-
13
29
hybrd solver
2.80
×
10
-
11
16
r represents the 2-norm of discipline residuals and Di represents the ith discipline. The evaluation counts of different disciplines are the same.The data in Table1 indicate that the coupled system is successfully solved using all the MDSA methods. For a given iterative method, for example, thehybrd solver, the MDSA_MF uses the fewest function calls to disciplines, followed by the MDSA_IF. And for the MDSA method based on the same selected couplings, for example, MDSA_MF, thehybrd solver uses the fewest discipline evaluations. The coupled system has 20 coupling variables. As shown in Figures 7 and 8, 13 feedback variables, y3, y4, y6, y8, y9, y10, y3, y4, y6, y8, y9, y10, and y9, are selected for MDSA_IF, and six coupling variables, y2, y5, y6, y9, y12, and y14, are selected for MDSA_MF. Sequentially, MDSA_MF uses less function calls to compute the derivative information. As the Jacobi matrix of the discipline residuals to coupling variables at each iterate is computed by the finite difference method, the Newton method uses more function calls to the disciplines.Figure9 shows the convergence histories of the 2-norm of discipline residuals. Similarly, for a given iterative method, for example,hybrd solver, MDSA_MF uses the fewest iterates, followed by MDSA_IF. The Newton methods converge to the final solution by the fastest speed, for derivative information is used at each iterate. However, the discipline evaluations increase when the finite difference method is used to compute derivative information.Figure 9
Convergence histories of the 2-norm of discipline residuals on test case  1.


## 5.2. Test Case  2

The second test case, from [39], is stated as formulation (14).(14)t1=23.521x112+45.122x121/2+0.9352y11/2-0.2835v2,t2=23.175x12+30.645x13-0.5392v1+0.0024z12,t3=12.944x11+43.846x12+18.345x132-0.0113v22-0.0021z22-0.1635u2,t4=12.347x112/3+21.478x12+7.745x132+0.563w21/2-0.7256v4+0.9476y3,u1=8.435x212+12.563x22+0.1577z1-0.9388w31/2,u2=24.547x21+24.356x22-1.457t12+0.3846t3+0.9862w21/2-0.7377z31/2,u3=9.367x222+0.0285t11/4-0.6758y11/2,v1=16.846x31+10.360x322-0.3367t2+0.157u21/2+0.0054z1-0.4783w21/2,v2=23.644x321/2+14.466x33-0.0064t12+0.5347y2,v3=8.846x312+44.467x331/2+0.3748u31/2-0.1094w1,v4=14.536x322-0.0949y1+0.4503t4+0.1003w31/2+0.0083u1-0.0776y31/2,w1=14.896x413+13.746x42+0.4567y1-0.0144y22+0.0944z2,w2=16.436x422-0.0113z12-0.0673z2+0.0356z3+0.0275u2,w3=8.638x41+26.693x42+6.536x432-0.6685t4-0.8467v4,y1=27.783x51+35.552x52+10.377x532-0.0059t12+0.0983u3+0.0633v31/2,y2=15.367x512+18.653x53+0.1079u3+0.2763w1-0.831v2,y3=22.561x52+35.649x53+0.6463z3-0.7366t4-0.0667u1,y4=9.882x512+0.0011t32-0.0359u2+0.2033z3,z1=8.629x61+7.932x622-0.0274t2-0.0235t3+0.0021v1,z2=14.367x612+48.837x62+0.0009v22-0.0328w11/2-0.0825v3,z3=29.663x62-0.0846w1+0.0024t21/2+0.0436u2+0.0875w2-0.0289y41/2,where xij represents the independent input and ti, ui, vi, wi, yi, and zi are the coupling variables.The system has 21 disciplines and 21 coupling variables. The DSM of the system is shown as Figure10. The disciplines are coded with ascend digital numbers from 1 to 21. The boxes on the diagonal and the black dots on the off-diagonal denote the same as Figure 7.Figure 10
The initial DSM of the coupled system in case 2.The coupled system could be divided into three strongly connected components, and the reordered DSM is shown as Figure11. The strongly connected components are denoted byGroup A,Group B, andGroup C, respectively. Within each strongly connected component, the discipline sequence with minimal number of feedback variables is presented.Figure 11
The reordered DSM of the system in case 2 with minimal feedback variables.Within FlowComputer, the coupled system is successfully solved using several methods starting from some given values of the independent variables and different initial values of the coupling variables. This coupled system could be solved by constructing an MDSA problem with 21 disciplines and could also be solved by constructing three sequential subproblems corresponding to the three strongly connected components in Figure11. Both of the two solving methods are investigated. The settings for algorithms, the step size of finite differences, and the termination criteria are the same as the first test case.Tables2 and 3 show the results starting from the independent variables xij=1 and the coupling variables (ti,uj,vk,wl,ym,zn)=(10,10,10,10,10,10). The evaluation number of the three groups of disciplines and the 2-norm of discipline residuals at the final solution are presented. Solving from other initial values of coupling variables presents similar data. However, solving from some given values of the independent variables far from the point xij=1 might fail in finding a multidisciplinary solution.Table 2
The result data for test case 2 solving as a whole coupled system.
Iterative method/solver
r
Evaluation number
D
1
~
D
21
MDSA_AC
FPI
3.00
×
10
-
11
33
Newton method
2.70
×
10
-
14
111
hybrd solver
1.03
×
10
-
11
33
MDSA_IF
FPI
5.61
×
10
-
12
17
Newton method
2.02
×
10
-
14
57
hybrd solver
1.57
×
10
-
12
21
MDSA_MF
FPI
4.01
×
10
-
12
17
Newton method
2.32
×
10
-
14
45
hybrd solver
1.56
×
10
-
12
18
r represents the 2-norm of discipline residuals and Di represents the ith discipline. The evaluation counts of different disciplines are the same.Table 3
The result data for test case 2 solving the three strong components sequentially.
Iterative method/solver
r
Evaluation number
Group A
Group B
Group C
MDSA_AC
FPI
5.01
×
10
-
11
33
32
20
Newton method
7.24
×
10
-
11
37
37
19
hybrd solver
3.94
×
10
-
11
20
15
11
MDSA_IF
FPI
6.35
×
10
-
12
17
16
12
Newton method
3.65
×
10
-
14
25
19
13
hybrd solver
1.04
×
10
-
11
13
11
8
MDSA_MF
FPI
6.51
×
10
-
12
17
16
7
Newton method
8.33
×
10
-
15
21
16
10
hybrd solver
8.31
×
10
-
12
11
9
6
r represents the 2-norm of discipline residuals.The data listed in Tables2 and 3 show that the coupled system is successfully solved with good accuracy using all of the methods. For each iterative method, MDSA_MF uses the fewest discipline evaluations, followed by the MDSA_IF method. If the system is solved as a whole coupled unit as in Table 2, the FPI method uses fewer function calls when the initial feedback, or the minimal feedback, is selected to be broken. However, thehybrd solver uses the fewest function calls when the three strongly connected components are solved sequentially. The possible reason is that the strategy solving the strongly connected components sequentially reduces the MDSA size. The strategy, however, does not influence the efficiency of Gauss-Seidel iteration significantly. Hence, MDSA_MF could use the fewest function calls, and solving the strongly connected components sequentially could further enhance the MDSA efficiency.Figure12 shows the convergence histories of the 2-norm of discipline residuals for solving the subsystem,Group A. The histories of solving the other two subsystems present the similar behavior. The convergence data indicate that MDSA_MF could use the fewest iterates for a given iterative method. The FPI iteration shows slow convergence. As derivative information is updated at each iterate, the Newton iterations converge to the final solution by fastest speed. However, computing the derivative information using the finite difference method increases the function calls. When the derivative information is easy to be obtained, the Newton iteration could generally converge to the final solution as the fastest speed.Figure 12
Convergence histories of the Group A in test case 2.


## 6. Conclusions

The paper proposes a two-phase MDSA method based on minimal number of feedback variables, called MDSA_MF, to enhance the solving efficiency. In phase 1, DSM is introduced to represent a coupled system, and each off-diagonal element of the DSM is denoted by a coupling variable set mapping from one discipline into another. An optimal discipline sequence problem is constructed to obtain a reordered DSM with minimal number of feedback variables in the lower triangle. In phase 2, the feedback in the lower triangle is broken, and the coupled system is transformed into a directed acyclic graph. Then, regarding the inputs depending on the broken feedback as independent variables, a least-squares problem with respect to these new independent variables is constructed to minimize the sum of residuals of the broken feedback to zero, and to further achieve a multidisciplinary feasible solution. Searching strongly connected components is also used to aid in the discipline reordering. Besides, the MDSA_MF method is implemented in a multidisciplinary design platform, called FlowComputer. The platform also provides the capacity of discipline integration based on COTS wrapping and modeling and solving GUI for coupled systems.Two test cases of coupled systems are modeled in FlowComputer, and several MDSA methods using different iterative method are investigated. The results demonstrate that MDSA_MF could use the fewest function calls. And the strategy dividing the system into several strongly connected components could further enhance the efficiency.The MDSA_MF selects the minimal number of feedback variables as unknown variables to solve coupled systems. Thus, the disciplines are executed sequentially, and parallel computing is not considered in the present work. Besides, the paper is focused on deterministic multidisciplinary analysis and does not include the uncertainty factors in engineering problems. The ongoing and future work includes (a) employing the available parallel computer resources to improve the MDSA efficiency and (b) implementing the MDSA_MF method on engineering problems under uncertainty.



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*Source: 1526041-2017-12-21.xml*