Management of Dependencies of Analyses in
Multidisciplinary Aerospace Systems
A S Shaja1, Amitay Isaacs2 and Prof. K Sudhakar3
Center for Aerospace Systems Design and Engineering, Aerospace Engineering Department
Indian Institute of Technology Bombay, Mumbai - 400076, India.
Abstract
“How to decide what to change, and to what extent to change it, when
everything influences everything else?”
Aerospace systems are multidisciplinary in nature. Design of such complex systems
through formal optimization techniques is of great interest. Before design begins it is
necessary to identify various disciplinary analyses that will be used in the design process. Use
of high fidelity analysis tools like CFD, FEM are presently gaining grounds. Disciplines
participating in the design process of Multidisciplinary systems have dependencies on each
other that are referred to as coupling between analysis modules. This paper focuses on finding
a preferred or best sequence of executing the analysis modules, with dependencies.
This paper briefly surveys methods like PERT, CPM etc. that have been developed
over the years for planning, scheduling of tasks in engineering project management. It is found
that iterative tasks (which are part of any design process) are not captured in these methods. So
Design Structure Matrix (DSM), which captures iterations, is introduced. We try to utilize this
framework in engineering management to multidisciplinary complex aerospace design by an
analogy of tasks to analysis modules. We take a standard synthetic problem from literature
offering wide flexibility in defining a multidisciplinary problem to test proposed strategies for
re-sequencing. In short this paper focuses on methods, which will help to sequence the tasks
(analysis modules) in a better way in a coupled multidisciplinary environment.
Keywords: Multidisciplinary Design Optimization (MDO), Design Structure Matrix (DSM),
Analysis Modules, DSM strategies and Framework
1. Introduction
Engineering design is the process of devising a system, component, or process to meet desired
needs. It is a decision-making process (often iterative), in which the basic science and
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3 Professor
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mathematics and engineering sciences are applied to convert resources optimally to meet a
stated objective (ABET Definition of Design)[1]. Multidisciplinary engineering systems are
complex to design. Complexity scales with the number of objects as well as the type and
number of interconnections in a system. This increasing complexity has sparked increasing
interest in Multi-disciplinary Design Optimization (MDO). Multidisciplinary design cycle
begins by identifying various analyses to be performed, which will together generate all the
information required to take design decisions. These analyses may have dependence
(coupling) on each other. Our basic aim is to build a methodology for ordering the sequence of
the various analyses in an efficient way in a multidisciplinary optimization framework.
2. Planning and Scheduling Techniques
Within the framework of any system all elements must be properly organized and executed
and therefore planning and scheduling of elements is essential. Over the years impressive
collections of methods have been developed. Gantt Charts is a graphical way of showing task
durations, task schedule etc. Network techniques like PERT, CPM represents a project (set of
tasks) as a network using graph theory by capturing task durations and capturing task logic
(dependencies)[2]. PERT is a network model that allows for randomness in activity
completion times. It attempts to eliminate some of the shortcomings of CPM by assigning
probability distribution to completion times.
Fig.1 Drawback of PERT/CPM
However, iterations (which are part of any design) are not captured using these network
techniques (fig.1), which leads to the motivation for DSM.
3. Design Structure Matrix (DSM)
The design structure matrix (DSM) layout is as follows (fig2): the analyses modules are placed
as row headings and also as column headings [3]. If analysis i requires as input what is output
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by analysis j then the value of entry ij (column i, row j in the matrix) is unity (or marked with
an X). Otherwise, the value of the entry is zero (or left empty). The diagonal elements of the
matrix do not have any interpretation in describing the system.
Fig. 2. A sample DSM
As per figure 2,

Analysis D requires information from analyses E, F, and L and
 Analysis B transfers information to analyses C, F, G, J, and K.
4. A Synthetic Problem
Research in the area of MDO has looked into several single level and bi-level architectures for
realization of Systems Design. Many of the architectures have been proposed based on
heuristic and without much mathematical basis. This has also brought in several test suits to
compare various architectures. Balling and Wilkinson [4] have discussed a scheme to generate
a set of analyses of user prescribable extent and strength of coupling. Present study proposes
to use this to create a set of synthetic analyses. As part of this work a ‘test suit’ generator is
developed that can create analysis modules as per [4], and has following features:
(i)
number of analyses can be specified by user
(ii)
dependencies are user specified through DSM representation.
(iii)
coupling strengths between analyses can be controlled
Let us first create a system containing 4 analyses (say 1,2,3,4). Dependencies between each
analysis are as shown in fig.3, and uniform strength of coupling between the dependent
analyses. Also each analysis takes constant execution time.
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Fig. 3. DSM showing dependencies
After creating the analyses modules they need to be ordered in some way to be executed. If we
execute these 4 analyses sequentially there are 4! (four factorial) ways of firing these analyses
modules (dependencies being preserved).
Fig.4.a
fig.4.b
fig.4.c
fig. 4.d
Fig. 4 some orders of executing the same 4 analyses modules problem
If an analysis that is fired earlier in the sequence requires output from an analysis fired later in
the sequence, an iteration till convergence is required. An initial guess will also be required.
Number of iterations will depend on strength of coupling and initial guess. The analyses
modules are allowed to iterate, till multidisciplinary feasibility [5] is achieved. Execution
times for 4 orders shown in Fig 4 is given in Table 1.
Execution Time for convergence (in sec)
Sequence Problem A Problem B Problem C Problem D
1-2-3-4
1-3-4-2
4-3-2-1
2-3-4-1
Time
6.480
4.860
6.460
3.220
Table.1. Execution times for 4 analyses problem
It is interesting to observe that the same problem when executed in different order results in
different execution times. In Problem A, analyses 2 & 3, which are between analyses 1 & 4
are re-executed unnecessarily till 1 & 4 converges which is reflected in the table1. In Problem
B, analysis 3 in between 1 & 4 is re-executed unnecessarily and is then added by the execution
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of 2 which cause re-execution of 3 leading to change in 4, causing another un-necessary reexecution of 3 till 1 & 4 converges. Likewise in Problem C, analyses 3 & 2 are re-executed
unnecessarily between 1 & 4. It is interesting to note Problem D, where the execution time for
convergence is minimal (hence the best sequence out of the four). This is because no unnecessary re-execution of analyses modules is done if this sequence is chosen. The outcome
of simple example in figure 2 may seem intuitive. As the problem size increases (which is
highly likely in complex systems like aerospace) it becomes difficult to infer intuitively the
best sequence and hence a formal method for finding the sequence of analyses modules
becomes necessary.
5. DSM Operators
Let us extend the DSM terminology explained in section 3 to also imply the sequence of
analyses modules. Let us assume, the order of row headings and/or column headings (analyses
names) represent the order in which the analyses is to be fired. So, over this DSM, the marks
below the diagonal represents feed forward coupling and marks above the diagonal represents
feedback coupling. Since feedback implies cycle (iterations), any sequence that has least
entries above the diagonal are preferred; and if a cycle is present smaller its span (number of
modules within the cycle) the better. Some operators to be applied over such a DSM to get the
best way of executing analyses modules.
(i) DSM Clustering
Clustering finds the subsets of DSM analyses (i.e. clusters or modules) that are mutually
exclusive or minimally interacting subsets [6]. Clusters absorb most, if not all, of the
interactions (i.e. DSM marks) internally and are useful in finding set of those analyses, which
are closely related.
(ii) DSM Partitioning
Partitioning is the process of manipulating (i.e. reordering) the DSM rows and columns such
that the new DSM arrangement does not contain any feedback marks, thus transforming the
DSM into a lower triangular form. For complex engineering systems, it is highly unlikely that
simple row and column manipulation will result in a lower triangular form. Therefore, the
objective change from eliminating the feedback marks to moving them as close as possible to
the diagonal [7]. In doing so, fewer system analyses will be involved in the iteration cycle
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resulting in a faster development process. Please refer Appendix A for steps used to partition
DSM.
(iii) DSM Tearing
Tearing is the process of choosing the set of feedback marks that if removed from the matrix
(and then the matrix is re-partitioned) will render the matrix lower triangular. The marks that
we remove from the matrix are called "tears". Tearing aids in deciding if certain couplings
could be removed (or temporarily suspended) from consideration to achieve computational
savings without a significant loss of system accuracy [8]. Tearing uses coupling strengths to
tear the analyses (using sensitivity information).
6. Synthetic Test Problems using 12 analyses
Using the ‘test suit’ generator that is developed, 3 synthetic test problems containing 12
analyses (say A, B, C…L) are created. Each of the 12 analyses takes same execution time.
Whenever dependency is introduced between any two analyses the strength of coupling is
same.
Problem 1
There is no dependency between any analyses modules and when executed in any sequence
takes 5.410 sec (table 2).
Problem 2
Fig. 5. DSM representing 12 analyses
Fig.6.DSM after partitioning
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Dependencies between analyses for this problem is shown in figure 5. DSM operators when
applied (partitioning) reveals a sequencing that can remove feedback altogether
(fig.6).
Execution as per this sequence takes time of 5.390 sec (table 2). This timing is similar to the
timings for Problem-1, since both require to execute each analysis just once.
Problem 3
This problem has larger number of dependencies (fig. 7). Execution as per sequence in
figure 7 takes 6.2 sec. Application of DSM partioning over yields DSM as in figure. 8.
Fig. 7. DSM representing 12 analyses
Fig.8.DSM after partitioning
Execution time for the sequence in figure. 8 is 7.410 sec. New sequence effectively brings the
coupled analyses nearby (together).
Problem
Sequence
Total Time
Number
(in sec)
1
B-C-A-D-F-E-L-K-I-J-G-H
(arbitrary)
5.410
2
E-B-K-L-A-H-F-C-G-J-I-D
(figure 6)
5.390
3
A-B-C-D-E-F-G-H-I-J-K-L
(figure 7)
10.620
3
B-C-A-K-L-J-F-I-E-D-H-G
(figure 8)
7.410
Table.2. Execution times for test problems
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So, we observe that DSM strategies define a formal way to find the best way of sequencing the
coupled analyses modules.
7. Numerical DSM
In binary DSM notation (used so far) a single attribute is used to convey relationships between
different analyses; namely, the "existence" attribute, which signifies the existence or absence
of a dependency between any two analyses. Compared to binary DSMs, Numerical DSMs
[9,10] could contain a multitude of attributes that provide more detailed information on the
relationships between two analyses. Such an improved description/capture of the relationships
provide a better understanding of the system and allows for the development of more complex
and practical partitioning and tearing algorithms, which will form future work.
8. Conclusion.
A ‘test suit’ generator that can create multidisciplinary analyses has been implemented with
controlling features like number of analysis, dependency and coupling strength. DSM
operators are implemented and shown to sequence analysis so as to reduce execution timings.
Synthetic test problems have been created, DSM operators applied and superior sequencing
shown in a coupled multidisciplinary environment.
Appendix
Steps used for DSM partitioning
All partitioning algorithms proceed as follows, with a difference of how they identify loops.
(1) Identify analysis modules that can be executed without input from the rest of the analyses
in the matrix.
(2) Identify analysis modules that deliver no information to other analyses in the matrix
(3) If after steps (1) and (2), if there are no remaining analyses in the DSM, then the matrix
is completely partitioned; otherwise, the remaining elements contain cycles (fig.9)
(4) Determine the cycles by one of the following methods:
(i)
Powers of the Adjacency Matrix Method [11] (ii) Path Searching Method [12]
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Fig. 9. Steps used in DSM partitioning
(5) Collapse the analyses involved in a cycle into one analysis and go to step 1.
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