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The Tenth International Conference on Machine Design and Production 15 4 - 6 September 2002, Cappadocia, Turkey
MODULAR STRUCTURAL COMPONENT DESIGN
USING THE DECOMPOSITION-BASED ASSEMBLY SYNTHESIS
Onur L. Cetin, University of Michigan, Ann Arbor, MI, 48109, USA
Kazuhiro Saitou, University of Michigan, Ann Arbor, MI, 48109, USA
Shinji Nishiwaki, Kyoto University, Kyoto, 606-8501, Japan
Yasuaki Tsurumi, Toyota Central R&D Labs, Inc., Nagakute, Aichi, 480-1192, Japan
ABSTRACT
The design problem addressed in this study involves simultaneous optimization of two
structures for the locations of joints and joint attributes (assembly synthesis), while
investigating the possibility of sharing some of the components among two products (design
for modularity). A genetic algorithm is employed for the solution of the problem. The objective
function attempts to minimize the reduction of structural strength due to introduction of spot-
weld joints and maximize the manufacturability of the components. The method is applicable
to 3-D beam-based structures, and a case study on automotive bodies is presented to
demonstrate the capabilities of the developed software.
1. INTRODUCTION
Modularity is a tested and proven strategy in product design. One short description would be
having products with identical internal interfaces. The scope of the word ‘interface’ includes
connection between the product components in functional, technology and physical domains.
The interfaces between modules are seen by many as the core issue of modularity and they
must be standardized to allow the ability of full exchange of components [Blackenfelt and
Stake 1998].
Design for modularity is now in widespread use globally. Carmakers prefer to design many
features of a family of cars at the same time, instead of one model at a time. Standardizing
the components and letting several variant products share these parts would save tooling
costs and many related expenses [Kota et al. 2000, Muffatto and Roveda 2000, Sundgren
1999]. Developing of a complex product involves many activities and people over a long
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period of time. Making use of modularity leads to clustering of activities involved in the design
process, so potential group of activities might be scheduled simultaneously, which enables
simplification of project scheduling and management [Blackenfelt and Stake 1998]. As the
identification of modules tremendously affects the entire product development process, the
strategy is usually applicable in the preliminary stages of the design.
Many companies are also actively pursuing to replace the traditional design process, in
which the translation of a conceptual design into a final product to be manufactured has been
accomplished by iterations between design and manufacturing engineers. In particular,
design for manufacturability (DFM) and design for assembly (DFA) methodologies are
utilized to implement early product design measures that can prevent manufacturing and
assembly problems and significantly simplify the production process. Most existing
approaches generate redesign suggestions as changes to individual feature parameters, but
because of interactions among various portions of the design, it is often desirable to propose
a judiciously chosen combination of modifications [Gupta et al. 1997, Yu et al. 1993].
In this paper we introduce a method to combine the DFA and DFM concepts with the design
for modularity process. The objective is to identify the modules in the early development
stage of structural products and optimally design the interfaces. Decomposition based
assembly synthesis [Yetis and Saitou 2001] is applied to search for the basic building blocks
of a product and determine the locations of joints that will result in the minimum decrease in
structural strength. Manufacturability criterion incorporated in the design process helps the
designer quantify the trade-off related to the additional constraint of sharing components. A
2-Phase optimization approach is implemented to identify the shared modules in Phase 1
and design a common interface at the end of Phase 2. The proposed method aims to
constitute an effective tool for the decision-maker at the conceptual design phase.
2. PREVIOUS WORK
In the case of having common elements together with interfaces, the shared subsystem is
usually called the product platform in the literature, and adding different components onto the
platform to end up with the product variants is defined as developing a product family. Note
that conceptually, and also in terms of the methods used, there is no essential difference
between design for modularity and product platform design. In the scope of this paper the
shared parts are to be consistently named as modules instead of platforms.
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There has been considerable increase in the research directed towards modular design
recently, inspecting many different industries and products, and presenting numerous means
of solving the problem efficiently. In this section, we will describe other studies dealing with
relatively complex applications, those which need multi-level optimization processes to
finalize the design of the product family.
Adapting Fujita and Yoshida’s classification [Fujita and Yoshida 2001], we consider that
optimal product design for modularity is to be carried out solving one of the following
problems:
Class I: Given which modules have potential for sharing, impose constraints to keep the
corresponding modules (or some of their variables) equal, then optimize module attributes.
Class II: Given an initial design, solve an optimization problem to find which modules (or
some of their variables) are ideal to share among products.
Class III: Simultaneously make the commonality decision and optimize the module
attributes.
Noting that problems of Class III are considerably more difficult to solve, several researchers
suggested skipping the search for the best platform during the design phase. For instance
Zugasti et al. start with several alternative platforms defined and design variants without
going through an iterative design loop (i.e., re-specifying the platform and completing the
design of variants once again). In this Class I problem the optimal product family can then be
identified based on decision analysis and real options; modeling the risks and delayed
decision benefits present during product development [Zugasti et al. 2001]. Another example
of Class I formulation is reported by Nelson et al., who formulate a multi-criteria optimization
problem to show that with the commonality decision, the performance of the products within
the platform will degrade, and that the amount given up in performance can be quantified.
The optimal platform design should lie in one of the Pareto sets resulting from platform
choices and solution of the multicriteria optimization problem. However, exactly which Pareto
set is ‘‘the best’’ is a question of performance as well as of other business issues and
compromises have to be made based on results of the analysis, taking into account the
specific application and/or the company strategy [Nelson et al. 2001].
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Simpson et al. focus on a slightly different Class I application, introducing the Product
Platform Concept Exploration Method to design and synthesize a scalable product platform
and the resulting product family. The goal is to design a product that can be vertically
leveraged for different market niches. Several variables are chosen to form the platform and
designed optimally to be used in every variant in the family. To instantiate each of the
products within the family to meet the performance requirements, the platform values are
held fixed while scaling variable is varied. A group of individually optimized products are
compared with the product family and it is reported that commonality is achieved without a
considerable loss in performance [Simpson et al. 2001].
There are also some attempts towards the direction of solving optimization problem for both
module combination and module attributes across multiple products, i.e., Class III problems.
Fujita and Yoshida present an optimization method hybridizing a genetic algorithm, a branch-
and-bound method and a constrained nonlinear programming. In their multi-level technique,
they first optimize the combinatorial pattern of module commonality and similarity among
different products, then optimize the directions of similarity on scale-based variety, and finally
optimize the continuous module attributes [Fujita and Yoshida 2001].
Design of modules in structural products is not addressed frequently and this is a difficult
problem as the choice of what needs to be part of the set of modules and what should be
individually designed for each variant creates vast numbers of combinations. Cetin and
Saitou presented a method to solve a Class III structural design problem, formulating a multi-
criteria non-linear program in which the module attributes consist of discrete variables only
[Cetin and Saitou 2001]. The solution is to be searched for using a genetic algorithm (GA).
Even though it is limited to 2-D beam-based products, this previous work on the assembly
synthesis method and its application to modular structural component design establishes the
background for this study.
3. MODULAR STRUCTURAL COMPONENT DESIGN
The design problem we are addressing involves simultaneous optimization of two structures
for the locations of joints and joint attributes (assembly synthesis), while investigating the
possibility of sharing some of the components among two products (design for modularity);
an example problem from earlier work is given in Figure 1. In this article we introduce the
extended method with more realistic joint models, and improvements to handle industrial
applications with 3-D structural models.
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(a)
(b)
Figure 1. A 2-D problem solved in earlier work [Cetin and Saitou 2001]. (a) Two structures to
be decomposed simultaneously (b) The optimal decompositions, together with the weld
angles chosen from a discrete set; shared modules are denoted with ‘s’.
Implementation of the optimization problem in this study is carried out in a different way
compared to the previous application. The first difference is that the joint attributes are no
longer discrete variables; they are now real numbers estimated repeatedly at each iteration.
Though the introduction of this inner optimization loop results in a significant computational
burden, it is now possible to realistically design the joint attributes. The flowchart of the
process is given in Figure 2. The second change is mainly because of the considerable
increase in algorithmic complexity for complicated structures. While before angle similarity
(common interface design) was imposed at the same moment geometrically similar
components were identified, the extended method uses two phases to complete the solution.
The complex process of automatically matching the corresponding joint locations of shared
modules is thus avoided. The procedure is summarized below:
PHASE 1
Given: Two beam-based structures.
Find: Components to share between two structures (modules); optimal joint locations and
attributes.
Criteria: Minimize the manufacturing costs for the resulting subassemblies; minimize the
reduction in structural strength.
Constraints: All the decomposed components have to be manufacturable.
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PHASE 2
Given: Two beam-based structures and modules to share.
Find: Optimal joint locations and attributes.
Criteria: Minimize the reduction in structural strength; minimize the manufacturing costs for
the resulting subassemblies.
Constraints: Further decomposition of the modules is not allowed; corresponding joints of
modules have to; all the decomposed components have to be manufacturable.
The user examines the optimal solution at the end of Phase 1 and decides if the suggested
component sharing is feasible. If the common module is confirmed, the matching joint
attributes between two components are marked and Phase 2 starts to repeat the optimization
process for common module interface. The advantage of utilizing a two phase approach is
that the user can examine the effects of sharing different modules by holding different
components common at the end of Phase 1 and running Phase 2 repeatedly. Phase 2 can
also be used by itself if the designer does not need any sharing suggestion from Phase 1
and wants to specify modules by intuition only. Note that both phases of the process
essentially use the same program. The only difference in Phase 2 is that, the objective
function no longer includes the modularity terms and two constraints are added to design a
single interface for each module specified in Phase 1. For convenience, only the Phase 1
formulation will be given in the following sections.
Initialize
Joint locations
Optimizer
Optimal joint attributes
Convergence?
Optimal design
Specify newjoint locationsN
Y
Outer opt. loop
Inner opt. loop
Figure 2: Flowchart of the optimization process.
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Assembly Synthesis via Product Topology Graph Decomposition
In the assembly synthesis, given beam-based structures are optimally decomposed into an
assembly consisting of multiple members with simpler geometries. There are two main steps
in the process:
1. The topology of the problem is examined and a product topology graph is developed.
2. The topology graph is split into subgraphs to generate a decomposition of the actual
product.
In the topology graph generation, the members (beams) of the structure are mapped to
nodes and the intersections are mapped to multiple edges as they can be joining more than
two members. The decomposition of the topology graph is customarily called graph
partitioning. So the problem can be defined as: given the topology graph of the structure,
obtain the partition representing the optimal decomposition and the mating feature for each
joint, subject to a cost function evaluating the decomposition quality. Figure 3 illustrates a
simple case.
(a) A beam-based structure (b) Corresponding graph
(c) A sample partitioning (d) The resulting decomposition
Figure 3. Decomposition of a simple beam-based structure.
Definition of the design variables
Let the members of the structure be mapped to the nodes of the product topology graph and
the intersections be mapped to the edges. So the whole structure can be represented as
G=(V, E) with a node set V and an edge set E. The problem of optimal decomposition
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becomes one of finding a partition P of the node set V such that the objective function, c(P),
is maximized. For the ease of formulation, a partitioning of G can be represented by a vector
x=(xi) of a binary variable xi representing the presence of edge ei in the decomposition
defined by the partitioning P. It is obvious that i=1,…,|E| since there are |E| edges in the
topology graph. Let y=(yi) denote the vector of joint attributes calculated by the inner loop
optimizer if there is a weld at edge ei.
Definition of the constraints
It is assumed that when the product is decomposed, the resulting substructures are to be
manufactured with stamping processes. An important restriction then would be the
manufacturability of that component, as the building blocks of a structure are expected to be
easy to develop. A classification adapted from (Gupta et al. 1997) on manufacturability
measures is as follows: a) Binary measures: It is simply reported whether or not a given set
of design attributes is manufacturable, b) Quantitative measures: Here designs are either
given qualitative grades based on their manufacturability or assigned numerical ratings along
some scale. Time and cost of an operation can be also used for quantification. In this paper
both classes of measures are used. The binary measure consisting of imposing a constraint
that does not allow a 3-D substructure, as it is practically impossible to manufacture with
stamping. So all the resulting components are constrained to be on a 2-D plane, as
illustrated in Figure 4. This evaluation is done by the function called MFG(x), which returns
TRUE when all the decomposed components are manufacturable. The quantitative measure
pertaining to the cost of manufacturing each component, essentially die cost estimation is
included as a part of the objective function.
(a) (b)
Figure 4: a) An acceptable component resulting from the decomposition b) A substructure
to be rejected as not manufacturable.
Definition of the objective function
Objective function will evaluate a decomposition according to the following criteria:
• Structural strength of the products.
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• Assemblability of the structures.
• Manufacturability of the components.
• Modularity of the structures.
To evaluate the decomposition according to the structural strength criteria, projected force on
the weld planes are calculated. Considering that spot welds are much weaker against tensile
forces, no term is added to the cost function when the resultant force is compressive, thus
punishing tensile forces only. In the below expression Nwelds is the total number of welds in
the decomposed structure and Fit is the tensile force at joint i, calculated using the projection
Fi.nweld, i.e., dot product of the reaction force at the joint and the normal of the weld plane.
( )11
( ) Nwelds t
ii
f w Fs=
= ∑x y, (1)
Assemblability criterion consists of the number of welds in the structures, taking into account
the fact that the assembly time and cost increase as the structure is decomposed into many
small components:
(2) 2( ) weldsf w Nw =x
As the third term in the objective function, the manufacturing cost for the decomposed
substructures is taken into account. Note that instead of the exact cost of the die to develop a
certain component, some part of the criteria used to estimate the cost would give a
quantification of manufacturability. Following Boothroyd et al.’s formulation (Boothroyd et al
1994), two major components of the die cost estimation procedure are used in our method:
a. Usable area (Au), b. Basic manufacturing points (Mp). The usable area Au is related to the
cost associated with the size of the die, and easily computed as the bounding box that
covers the substructure. Basic manufacturing points Mp actually measures the die
complexity (Figure 5), first calculating the complexity index Xp:
Xp = P2/(LW) (3)
where, P = perimeter length for the die,
LW = length and width of the smallest rectangle that would surround the punch.
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So the third term that brings about a punishment for complex, big and thus costly to
manufacture parts is given as:
*( ) wu3 4f w A Mc p= +x (4)*
where Au* and Mp
* can be taken as the maximum values encountered while examining all
decomposed components. A summation of production costs of all substructures can also be
used instead.
0 50 100 150 200 25028
30
32
34
36
38
40
42
44
Complexity Xp
Mp
Figure 5: Basic manufacturing points vs. complexity index.
The cost function term for modularity is incorporated to evaluate the following two attributes
of the components to be shared between the structures:
1. Similarity in interfaces: this condition is simply implemented by maintaining that joint
angles of the components should be close to each other.
2. Similarity in shapes of the components in a given (user-specified) tolerance: this attribute
is checked by comparing the components with respect to their areas.
Note that this procedure to check geometric similarity requires that all components that come
out of the decomposition process of one structure be compared with the components in the
second one. However, probably only a few of the components at each iteration will be similar
enough to be considered potentially sharable. Therefore there is no need to try matching joint
attributes of dissimilar substructures, but only the components passing several tests for
geometric closeness can be considered.
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First test for similarity consists of evaluating the most basic property, areas of the geometric
entities. This comparison is conveniently carried out using the convex hulls of the
components. A convex hull of a set of points is the smallest convex set to cover all the given
points. In practice, this is equivalent to wrapping a rubber band around the ‘outside’ points.
Using this boundary, which can be quickly obtained, this similarity measure is realized by the
calculation of first moments of component areas with respect to the centroids; so the
similarity is assessed in a rotationally invariant way. Again using the convex hull, it is easy to
get the number of vertices on the boundary of each substructure. Since each vertex is also a
joint location, this property has to be shared among two components to be similar, and to
share the same interface. A last test can be quickly performed to vaguely evaluate the
isomorphism between the subgraphs decomposed at each iteration. Except for some special
cases two similar components have to possess the same number of nodes and edges in
their topology graphs.
Thus the modularity component of the objective function is defined as:
function fm(x1, x2)
1. for each pair of subgraphs ( g1k,, g2
l )
2. if Area_Moment( g1k) - Area_Moment(g2
l ) < Tol
3. if Nb_of_Vertices(g1k) = Nb_of_Vertices(g2
l )
4. if Topology(g1k) = Topology(g2
l )
5. modules = modules + 1
6. if modules = 0
7. return a large number
8. else 9. return 1 / modules
where Area_Moment(g), Nb_of_Vertices(g) and Topology(g) are functions that return the
area moment, number of vertices and node/edge numbers respectively, given a subgraph
(substructure) g.
Optimization problem
Eventually the constraints and objective function combine to give the following optimization
problem:
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minimize f (x1,y1, x2,y2) = fs(x1,y1) + fs(x2,y2) + fw(x1) + fw(x2) + fc(x1) + fc(x2) + fm(x1,x2)
subject to
MFG(x1) = TRUE
MFG(x2) = TRUE
(x1)i ∈ {0,1}, i = 1, …. , |E1|
(x2)i ∈ {0,1}, i = 1, …. , |E2|
Next section presents a case study on automotive bodies to demonstrate the capabilities of
the method.
4. CASE STUDY
Figure 6 shows two beam based automobile bodies, slightly different in topology but similar
enough to conveniently act as candidates for modularity analysis. As shown in the figure,
displacements of two nodes at the rear are constrained, while loads of same magnitude and
opposite directions are applied on the front part, leading to torsion in both structures. Note
that to decrease the complexity of the problem the underbody members of the models will
not be decomposed, however they are still in use to calculate the forces at joints.
This problem is to be solved by using a steady-state GA. Empirical advantages of steady-
state GA are that it prevents premature convergence of population and reaches an optimal
solution with fewer number of fitness evaluations (Saitou and Yetis, 2000). Using GA to solve
the optimal partitioning problem requires that the variables, i.e. the decisions of which edges
to cut, be defined using a chromosome representation. A binary genome including all edges
is sufficient for this purpose, where ‘0’ in a gene would mean the edge is cut, while ‘1’
denotes the corresponding edge is in place. In this specific application of simultaneous
optimization of two structures, a single chromosome of length |E1| + |E2| or a composite
genome can be conveniently used for the representation of edges (Figure 7).
In the previous work, the joints were defined by their weld angles, which consisted of a
discrete set, namely 4 different possible angles (Cetin and Saitou 2001). This joint
representation is now replaced by a more realistic model, by defining the weld interface as a
plane, and defining the weld angles to be orientation of that plane with respect to local
coordinate axes as given in Figure 8. One point on the beam and the direction vectors
(rotations along axes y and z) uniquely define the weld plane and the resultant joint force on
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that joint can then be easily obtained by estimating the projection of the reaction force on the
plane.
Figure 6: Loading and boundary conditions for two auto body structures: Sedan and Wagon.
Figure 7. Chromosome representation for the edges (variable x).
(a) (b)
Figure 8. Joint angle modeled as a plane rotating around a local coordinate system. (a) and
(b) illustrates the weld angles θyy and θzz respectively.
The optimization results are given in Figure 9 and Figure 10, respectively presenting the
welds and the shared modules, together with the common interfaces.
Figure 9: Optimal locations of welds. The symbol ‘s’ denotes the shared components.
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Figure 10: The modules that result from the optimization.
5. DISCUSSION AND FUTURE WORK
The optimization process in this study involves simultaneous synthesis of two structures to
determine the locations of joints and joint attributes, while investigating the possibility of
sharing some of the components among two products. The method is tested by using a real
life problem, and the 3-D extension of the ideas in previous studies is verified.
Future work includes the development of an advanced user interface that will allow the
designer to quickly compare the relative effects of the separate design criteria (objective
function terms). Another add-on to the software will be the quantification of the profit
achieved by sharing the components suggested by the method. This cost model will serve
the user to help understand the trade-offs when the module sharing decision is made.
ACKNOWLEDGMENTS
The first author has been partially supported by National Science Foundation under
CAREER Award (DMI-9984606), the Horace H. Rackham School of Graduate Studies at the
University of Michigan and General Motors Corporation through General Motors
Collaborative Research Laboratory at the University of Michigan. These sources of support
are gratefully acknowledged.
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