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.
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 [
Several methods can be used to perform MDSA for coupled systems. Fixed point iteration (FPI) is a common-used method for MDSA [
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 [
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
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 (
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.
Fixed point iteration (FPI) method uses the original equations of the system as the iterative functions from a starting point of coupling variables [
Newton-like methods convert the original coupled system into its residual form as formulation (
The Newton-Raphson iterative equations [
Nonlinear least-squares (NLS) methods [
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.
Design Structure Matrix (DSM) [
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 [
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.
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
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 [
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
The DSM representations of an example system.
The initial DSM
The reordered DSM
Figure
The Boolean DSM representations of an example system.
The initial Boolean DSM
The reordered Boolean DSM
To minimize the number of feedback variables in the lower triangle, an optimal sequence problem is stated as formulation (
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.
As the feedback is removed, the disciplines are executed sequentially. The outputs with respect to the selected feedback are determined by formulation (
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 [
Figure
An example DSM of a coupled system.
The initial DSM
The reordered DSM
The procedure of MDSA_MF is presented as follows.
Search strongly connected components, and divide the system into a series of strongly connected components. Tarjan depth first search algorithm [
Topologically order the strongly connected components to generate the solving sequence.
For each strongly connected component, reorder the discipline sequence to obtain a sub-DSM with minimal number of feedback variables by formulation (
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.
Complete solving, and present outputs.
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.
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, like
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, including
In addition,
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.
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 Section
The main user interface of FlowComputer, shown as Figure
The main user interface of FlowComputer.
Dependence relationships between any two components could be constructed visually. Mapping viewer shown as Figure
The mapping viewer for dependence relationships.
The DSM representations of a coupled system in FlowComputer are shown as Figure
The screenshot of the DSM representation of a coupled system.
The initial DSM
The reordered DSM
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 the
The test case is modified from the scalable problem [
This coupled system with 20 disciplines has 21 independent inputs and 20 coupling variables. The DSM of the system is shown as Figure
The initial DSM of the coupled system in test Case 1.
The reordered DSM with minimal feedback variables in test Case 1.
Table
The result data for test Case 1.
Iterative method/solver |
|
Evaluation number | |
---|---|---|---|
|
|||
MDSA_AC | FPI |
|
65 |
Newton method |
|
106 | |
|
|
37 | |
|
|||
MDSA_IF | FPI |
|
36 |
Newton method |
|
71 | |
|
|
26 | |
|
|||
MDSA_MF | FPI |
|
31 |
Newton method |
|
29 | |
|
|
16 |
The data in Table
Figure
Convergence histories of the 2-norm of discipline residuals on test case 1.
The second test case, from [
The system has 21 disciplines and 21 coupling variables. The DSM of the system is shown as Figure
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 Figure
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 Figure
Tables
The result data for test case 2 solving as a whole coupled system.
Iterative method/solver |
|
Evaluation number | |
---|---|---|---|
|
|||
MDSA_AC | FPI |
|
33 |
Newton method |
|
111 | |
|
|
33 | |
|
|||
MDSA_IF | FPI |
|
17 |
Newton method |
|
57 | |
|
|
21 | |
|
|||
MDSA_MF | FPI |
|
17 |
Newton method |
|
45 | |
|
|
18 |
The result data for test case 2 solving the three strong components sequentially.
Iterative method/solver |
|
Evaluation number | |||
---|---|---|---|---|---|
Group A | Group B | Group C | |||
MDSA_AC | FPI |
|
33 | 32 | 20 |
Newton method |
|
37 | 37 | 19 | |
|
|
20 | 15 | 11 | |
|
|||||
MDSA_IF | FPI |
|
17 | 16 | 12 |
Newton method |
|
25 | 19 | 13 | |
|
|
13 | 11 | 8 | |
|
|||||
MDSA_MF | FPI |
|
17 | 16 | 7 |
Newton method |
|
21 | 16 | 10 | |
|
|
11 | 9 | 6 |
The data listed in Tables
Figure
Convergence histories of the Group A in test case 2.
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.
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work is supported by the National Science Foundation of China [Grant no. 51575205] and the Manufacturing Industry Product Innovation Knowledge Service Platform’s Development and Application [Grant no. 2013AA040603].