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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