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From: AAAI Technical Report FS-92-03. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.
A Representation for Algorithmic Expansion of Engineering Designs
Vital Aelion1, Jonathan Cagan2, 1and Gary J. Powers
Carnegie Mellon University, Pittsburgh, PA15213
Abstract
A knowledge
representationbasedon first principles and
a library of techniquesfor expandingthe spaceof design
alternatives are reviewed.Theyhavebeendevelopedfor the
algorithmicinnovationof engineeringdesigns.There, suiting
l stpRINCE design methodology uses optimization
information to decide which expansion technique may
produce improved designs and induction to examine limiting
behavior. Starting from an initial design, lstpRINCEhas
algorithmically innovated interesting engineering concepts,
suchastheelectric
bus,thetapered
andhollow
beam,the
wheel,
andtheplugflowreactor.
1
Introduction
Designmaybe describedas a searchfor a goodsolution
in a spaceof possibleconfigurationsthat satisfy the user’s
goals. Design frequently involves three important aspects:
¯ a set of design objectives,
¯ a representation of design alternatives, and
¯ a search technique.
In this paper we review a knowledgerepresentation which is
based on first principles and a library of design space
expansion techniques for innovating designs. Optimization
is used as a search technique but won’t be discussed in any
detail here.
The first principle knowledgeof this paper is represented
as an algebraic objective, a set of algebraic engineering
equations, a design topology, and a set of specifications of
constraint and variable types. The equations are derived from
detailed physical, chemical, manufacturing, safety, and
performance considerations. This knowledgerepresentation
is useful in domainswhich can be described mathematically,
and require additional information on variable characteristics
and constraint applicability, to achieve their reasoninggoals.
The expansion techniques are mathematical heuristics
which innovate designs by introducing new features into a
knowndesign. Optimization information is used to decide
which expansiontechnique is applicable.
The knowledgerepresentation, the library of expansion
techniques, and the optimization-based search have been
incorporated into the lstpRINCE design methodology for
innovation
in engineering
design. The knowledge
representation and the expansion techniques maybe useful to
other research efforts independently.
This paper unifies various pieces of this research effort,
which have been published by Cagan and Agogino [1987,
1991a, 1991b] and by Aelion et al. [1991, 1992]. After a
discussion of each of these research issues, the lstpRINCE
methodologyis applied to the design of a vehicle fueling
strategy.
lDepartment of Chemical Engineering
2Deparlment of Mechanical Engineering
99
Related Literature
Several researchers have published on the issue of
algorithmic design generation. This literature is diverse both
in scope and approach. Research in algorithmic design
generation usually originates from the fields of artificial
intelligence, civil, and mechanicalengineering [Mitchell et
al., 1985; Brownand Chandrasekaran, 1985; Mittal et al.,
1986; Lenat, 1983; Dyer et al., 1986; Faltings, 1991;
Williams, 1991; Murthy and Addanki, 1987; Maher et al.,
1989; Ulrich and Seering, 1988; Joskowicz and Addanki,
1988; Mitchell 1989].
2
3
Motivating
Example
The lstpRINCE design methodology algorithmically
transforms an initial design topology into a new topology
that performs the same function with an improved performance. Both designs are defined by mathematical relations
describing their geometry and their physical /chemical
behavior,i.e. their first principles.
Considerthe initial design of a square, whosefunction is
to roll on a plane. The goal of this design is to produce a
design which performs the same function, better than the
square. The initial design, shownin Figure la, is stated as
follows:
minimize
subject to
resistance to spinning
square area > lower bound
(b)
(c)
(d)
FigureI. Application
of I stpRINCE
to theMinimumSpinning
Resistance
Design
(a)
Afteroptimization,
IstpRINCE
produces
thedesign
of a
square
whoseareais equalto thelowerbound.
An expansion
technique,
whichis partofourdesign
methodology,
changes
thetopology
to thatofFigure
lb.Thenewdesign
isbetter,
because
it hasa lowerresistance
to spinning.
Repeated
application
of thistopology
transformation
produces
the
designof FigureIc, whichis superior
to the previous
designs.
Eventually,
IstpRINCE
takesthe limitof this
transformation,
andproposes
thedesign
of a wheel(Figure
Id).Thisexampleis described
in detailin [Caganand
Agogino,
1991a].
First
Principle
Knowledge
Representation
First principle knowledgerepresentationsmodelproblemsbaseddirectly on the fundamental
physical andchemical
phenomena,as well as on manufacturing, safety, and
performance
considerations.Thereare manychoicesof first
4
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principle
knowledge
representations.
choice
constraints,
, are the definitions of the corresponding
algebraic relations in symbolic or in numerical form.
assignment variables.
Designs are described by an algebraic objective and a set of
Based on these definitions, the primitive-prototype
algebraic constraints. The representation also models becomes:
fundamentalcharacteristics of each quantity and each relation
primitive-prototype = f boundedby
in the system, by typing the variables and constraints.
Designs can have multiple regions. A region is a section of
{cs} u {cP} u {cb} u {ca} over {xs} u {xr} u {xa}.
a design, which may be independently modeledand mayhave
Withinthis representation, searching for the best design
independent properties [Cagan and Agogino, 1991b]. The involves minimizing the objective, while satisfying the
connectivity amongregions is described by a topology. All
design constraints. The space of solutions is limited by a
of this informationis providedby a designer.
fixed set of features. Design space expansion is one way to
The first principle information is captured by the
remove
this limitation.
primitive-prototype, which is based on an objective function,
f(x), to be minimizedover a set of variables, x, bounded
Design Space Expansion:
5
a set of equality and inequality constraints, h(x) = 0 and g(x)
A Method for Generating New Designs
<0:
Design space expansion introduces new features into a
knowndesign by creating new design regions, each of which
Minimize f(x)
can be independently modeled and assigned independent
Subject to h(x) =
properties (independent intensive variables). Twoexpansion
g(x) _<
techniques are currently available: Dimensional Variable
Wedefine five variable types. The first two, namelythe Expansion, DVE,and Input Variable Expansion, IVE. DVE,
shownin Figure 2, focuses on a region with a dimensional
system and region variables, designate whether these
variables dependon the physical size of the systemor a have variable, in this case Dim1. The initial design, which
a regional effect. The next three variables are subsets of comprises of a single region in this case, is expandedinto
these variable types and are used by lstpRINCEto expand multiple regions along Dim 1. Each new region is
independently modeledand assigned new properties.
the design space.
An example application of DVEinvolving a beam is
s,
Systemvariables, x express quantities that dependon a
shownin Figure 3. The weight of a beamunder flexural load
characteristic size of a system. Examplesof such variables
include weight, reactor volume, capacitor charge and mass is to be minimized, subject to a yield stress constraint.
flowrate. System variables are replicated whennew regions Application of DVEover the length, a dimensional variable,
appear in the design.
Regionvariables, xr, express quantities that describe a
property of a design region. Suchquantities are independent
of all the region characteristic sizes. Regionvariables include
temperature, pressure, density, Young’smodulusand thermal
conductivity. Region variables are replicated when new
regions appear in the design.
Assignment variables, xa, express quantities that have
Figure 2. Conceptual illustration
of DVE
contributions from multiple design regions, or that appear in
specific design locations, such as the first and the last region.
F
These variables facilitate the accounting within a design.
Examples that account for contributions from multiple
regions include total weight of a design, average density,
r
total sales and total cost. Examplesof variables which are
specific to a particular design region include entering and
exiting chemicalreactant concentration.
r I
In addition to the above variable types, we define two
l
ll
/2
morevariable types, whichare subclasses of the types already
Figure 3. Application of DVEon a beam example
defined. Dimensional variables are a subclass of system
variables that denotes the physical dimensions of a design,
BEGIN (*DVE*)
such as length, radius, or volume. Input variables are
1. R = region involving critical dimensionalvariable.
another subclass of systemvariables that describes quantities
RU= connectedregions topologically upstreamof R.
RD= connectedregions topologically downstreamof R.
to be processed by a design. Examplesof input variables in2. ReplaceR with specified numberof connected
clude material flows in chemicalreactor designs and loads in
expansion regions.
structural designs.
3.
Connectuppermostexpansion region with all regions
Constraints are classified as serial, parallel, boundary,
previously connecting RUand R.
s,
and assignment. Serial constraints, c are functions only of
4. Connectlowermostexpansionregion with all regions
assignment variables and apply across all design regions.
previously connecting R and RD.
Parallel constraints, cP, apply to individual design regions.
Return new topology.
5.
Boundary constraints, c b, are defined for each boundary
END
between neighboring design regions. Finally, assignment
Figure 4. The DVEalgorithm
[ """’U T
1 O0
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produces
expanded
space
of a©stepped
beam.
constraints
Application of DVEalong the radius, another dimensional
variable, would produce a beam with several concentric
regions. The algorithm of DVEis shownin Figure 4.
IVE proposes the parallel processing of an input into
design regions whosetopologies are replicates of the initial
design region, as shownin Figure 5. In doing so, IVE automatically introduces new features in the design, which may
lead to better designs. In Figure 5, Input 1 is distributed to
the multiple design regions which appear in the new design.
The newly created design regions can be independently
modeled. Application of IVE to the beam design over the
load, an input variable, proposes designs of multiple beams,
each of whichcarries a fraction of the original load, as shown
in Figure 6. The algorithm of IVE is shownin Figure 7.
Initial
DGsifn
~
f
the same across several design
generations, then we can induce that this pattern will hold for
an infinite numberof generations, and take the limit of this
expansion activity.
The number of consecutive design
generations, n, required before attempting induction is specified by the user.
Induction is not a required step. Rather, it is an additional analysis step capable of producing certain designs
which would otherwise require infinite design generations.
Induction is an aggressive design policy which risks the possibility that someconstraint be violated at the limit. It is
important to check the result of induction against the design
constraints to ensure that they have not been violated. In
addition, when taking limits a designer should always
determine whether or not the primitive-prototype remains a
valid modelof the design.
NGw Dcflgx
7 lStpRINCE
Design Methodology
The In’st principle knowledgerepresentation, the library
of expansion techniques, the optimization-based search, and
the induction step have been incorporated in the lstpRINCE
design methodology, lstpRINCE is an acronym for FIRST
PR|Nnciple Computational Evaluator. The goal of this
methodology, introduced by Cagan and Agogino [1987,
1991a, 1991b] and extended by Aelion et al. [1991 and
Figure 5. Conceptual illustration of IVE
1992], has been to generate new engineering designs. The
means for generating designs is the introduction of new
FI
F
features in a known design by applying design space
expansion techniques.
In lstpRINCE, expansion techniques operate on a space
of design solutions, defined by first principle knowledge.
This knowledgeconsists of an algebraic objective, algebraic
equality and inequality constraints, a design topology, and
constraint and variable types. An optimum is determined
Figure 6. Application of IVE on a beam example
within the space of solutions, and the space is expanded
BEGIN(*IVE*)
(augmented). This new design space provides an opportunity
1. R = region involvingcritical input variable.
for including better designs.
RU= connectedregions topologically upstreamof R.
1 stpRINCEis described in Figure 8. The initial design
RD= connected regions topologically downstreamof R.
is
optimized.
If the resulting design meets the designer’s
2.
ReplaceR with specified numberof ur~connected
requirements
a
satisfactory
solution is found and the design is
expansion regions.
completed.
Alternatively,
the design space is expandedvia
3. Connectall expansionregions with all regions
the
library
of
design
space
expansion techniques. The
previously connecting RUand R.
resulting
design
is
again
subjected
to optimization and the
4. Connectall expansionregions with all regions
design loop is repeated. Four points are of particular interest:
previously connecting R and RD.
¯ If specific design trends appear after a certain number
5. Return newtopology.
of iterations, then the correspondinglimit is induced
END
to obtain the maximum
benefit of that expansion.
Figure 7. The IVE algorithm
¯ If expansion does not improveperformance, the design
DVEand IVE initiate a library of formal expansion
generation is stopped. Performance can be enhanced
techniques. These techniques are the meansof expanding the
either by improving the objective or by satisfying
space of possible design solutions, and provide the powerto
previously violated constraints.
¯ This algorithm uses optimization in two ways: (a)
generate new designs. They have been developed for the
representation presented in section 4, but their principle may
optimizing a given design and (b) suggesting which
be extendedto other representations.
expansion may improve design performance.
¯ The designer is involved in each design generation in
6 Limiting Designs by Induction
two ways: (a) deciding if a design is satisfactory and
Repeated application of design space expansion and
(b) choosing amongcandidate expansion techniques
subsequent optimization of the resulting primitive-prototype
determinedalgorithmically.
frequently reveals patterns of interesting constraint activity.
The algorithm of lstpRINCE is presented in Figure 9.
After each expansion optimization analysis derives sets of
It
consists
of the input specification and the control structure.
active constraints that are evaluated. This sequenceof actions
Step
5
of
the
algorithmis the selection of a candidate critical
is called a design generation. If the activity of the analogous
variable, defined as one which improves the design upon
I
eee
I01
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expansion.
Whenmultiple
candidateCopyright
critical © 1992,
variables
are
x +/3dlv2 _< Q (11 - local constraint)
available,
the designer makes a decision among the
The variable type specifications are given below:
alternatives presented by lstpRINCE.
R = total revenue / week (assignment),
C = total cost / week(assignment),
x = merchandise volume / trip (system, input),
dl = region distance per trip (system, dimensional),
d = total distance per trip (assignment),
Opti~i~n Is[om~n
v -- speed (region).
~
~
Constraints (al) and (a2) define the total revenue/trip
and the total cost/trip respectively. Constraint (a3) specifies
that the trip distance in the design region is equal to the total
trip distance, d, because there is only one design region in
this design. Finally, constraint (ll), the only local constraint
in this design, specifies that the sum of the volumes
occupied by the merchandise and the fuel cannot exceed the
total volumeof the vehicle.
l stpRINCEstarts by optimizing the initial design,
shownin Figure 10. The shaded area represents the fraction
of the total capacity used for storing merchandise, and the
remaining is the fuel volume. Monotonicity analysis
(Papalanabros and Wilde, 1988) indicates that all constraints
are active and constraint (It) is satisfied as an equality.
optimization and backsubstitution of the active constraints
into the objective function we determine that the minimum
net weeklycost is:
Stopwith Solution
of PrcviouJ I.tion
Figure 8. The lstpRINCE design methodology
BEGIN(*INPUTprimitive-prototype*)
1. Specify modeltemplate (objective and constraints).
2. Specify regions of initial design (topology).
3. Specify variable and constraint types.
END
BEGIN(*lStpRINCE*)
4. If f’trst designgeneration,goto step 8.
5. Select critical variable and expansiontype.
6. Choosea numberof expansionregions (default = 2).
7. Updatedesign topology by calling expansion technique.
8. Applyassignment constraints.
9. Create new boundaryconstraints.
10. For each region create parallel constraints.
11. Updateobjective function and serial constraints.
12. Optimize design.
13. If design generations> inductionlimit, and analogous
constraints remainactive (inactive), then inducelimit.
14. If designsatisfactory, returnresult.
15. If designimproved,go to step 5.
16. Returncurrent design.
END
Figure 9. lstpRINCE Algorithm
o
min= 33/2-d3/2 ~1/2 ~P + F!
, ()
L..
[ ~
lStpRINCEis domain independent. Wehave used this
approach to create newdesign alternatives in the domainsof
chemical engineering,
mechanical engineering and
economics.
8 Example: Vehicle Fueling Design
Consider a problem where the net weekly income of a
shipping process is to be maximized. The income depends
on the annual volumeof freight carried and fuel costs. The
vehicle has volume capacity, Q. The volume of the
merchandiseis x. The total numberof trips per weekis v/d,
wherev is the average speed and d is the shipping distance.
Fuel consumptionvaries with the square of the speed and the
distance, fldv 2, where fl is a constant. The weekly fuel
consumption, flv 3, is equal to the fuel consumptionin each
trip,/~dv 2, times the numberof trips, v/d. P is the revenue
per unit volumeof delivered merchandise and F is the cost
per unit volumeof fuel. R and C are the total revenue per
weekand the total cost per weekrespectively. The input to
lstpRINCEincludes variable and constraint typing:
Minimize C - R
Subject to R = Pxv/d
3C = flFv
dl -- d
3/2( e__e_ll/2
eQ
fl,
()
I
d
"-I
Figure 10. First Generation of the Fueling Design
The designer may either accept the above design, or
search for a better one. Application of DVE.targets the only
dimensional variable which is the distance, dl. The region
which contains dl is expanded to the default value of two
regions. The new design is expressed with the following
primitive-prototype:
Minimize C - R
(J2
Subject to R = Pxv/d
(al
3C = flFv
(a2
dl + d2 = d
(a3
x + fldlv 2 < Q (11
x + fld2v 2 < Q (12
(Jl - objective)
(al - assignmentcon.)
(a2 - assignmentcon.)
(a3 - assignment con.)
l !il flair2 I
- objective)
- assignment con.)
- assignmentcon.)
- assignmentcon.)
- local constraint)
- local constraint)
refueling stop
vI
Figure 11. Second Generation of the Fueling Design
102
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thisTechnical
two-region
design, Copyright
shown ©in1992,
Figure
constraint (a3) specifies that the sumof the distances in each
region be equal to the total distance of the trip. Again, all
constraints are active and the local constraints, (11) and (12),
are satisfied as equalities. Backsubstitution and symbolic
optimization indicate that dl is equal to d2 and the minimum
weeklycost of this design is:
_2pQ3/2 [ p ~1/2.
!f2’ min-33/2 d3/2 fll/2~+ F
In this formulation several simplifying assumptions
have been made: the average speed of the vehicle, v, remains
unaffected with the creation of refueling stops and there is are
no costs associated with stopping to refuel, building
refueling stops, or building an energy source. The problem
can be solved without these assumptions and perhaps
innovate still other designs. The maininterest of conceptual
design is to observe trends of improvementby expanding the
design alternatives.
Ultimately, the designer must recognize the meaningand
implement the design ideas produced by lstpRINCE. The
application of this design methodologyhelped in exploring
alternative designs and provided the stimulus of a new design
idea. lstpRINCEworks interactively with the designer who
critiques the emerging designs and provides guidance on
which expansion types to be applied. The automation of this
methodologysupports rapid exploration of a certain class of
innovative designs.
lstpRINCEhas also been applied to the design of beams
and chemicalreactors.
This objective, f2,min, is smaller than the previous
objective, fl,min, indicating an improved performance.
lstpRINCEhas been able to innovate by introducing a new
feature into the original design. The newfeature, stopping to
refuel, provides a larger fraction of the vehicle capacity for
storing merchandise, as shownby the increase of the shaded
area in Figure 11. The stop was created by recognizing the
existence of a dimensional variable and applying DVEto the
9
Discussion
design space.
This
paper describes a knowledgerepresentation which is
The same procedure can be carried through for another
based
on
first principles and a library of mathematical
design iteration to determine that the four-region design
heuristics
for expanding the space of design alternatives
produces further improvement, l stpRINCEnotes that the
These
have
been combined in the lstpRINCE methodology
analogous conswaints remain active across design generations
for
innovation
in engineering design.
and induces the limit of infinitely manyand differentially
The
first
principle
knowledgerepresentation is based on
small design regions which sum up to the total distance of
the
engineering
paradigm
of modeling systems with
the trip, d. The objective becomes:
mathematical relations. The representation also includes
variable and constraint typing. Variable typing is domain
dependent. Constraint typing describes the region(s)
which each constraint is applicable in a design. Typing
provides additional system information necessary for
min- "2pQ3/2 [ l’ 11/2
fn.
reasoning
within lstpRINCE.
33/2 d3/2 ill/2 [~ + F]
The expansion techniques introduce new independently
modeledregions in designs and produce new features in these
lira [fn, min] = 3/2
"2p3/2Q
designs. Currently the library of expansion techniques has
two members: DVEand IVE. These techniques are the
n --,**
33/2 d312,fit/2 F 112
grammar for modifying design topology, so the more
This objective represents the maximumimprovement membersin this library, the stronger the powerto create new
that DVEcan produce with this system description.
designs. DVEand IVE have been defined as mathematical
l stpRINCE has been able to innovate the concept of a operators. The concept of design space expansion, however,
vehicle which refuels continuously along a trip, shownin
is useful independently of knowledgerepresentation. In their
Figure 12, and resembles the operation of an electric bus. In current form, design space expansion techniques target
this design all the volumecapacity of the vehicle is used for specific variable types. As we identify variables with other
storing merchandise.
interesting roles, we can define new variable types and
The designer also uses lstpRINCE to try IVE, which develop corresponding expansion techniques.
proposes the design of two vehicles making this trip in
Withthe exception of designs producedby induction, the
parallel. Given the initial system description, this design results of lstpRINCEare sound, because the newly created
does not improvethe objective.
regions are modeled by physically valid equality and
inequality constraints (assuminga correct initial primitiveprototype). Induction, on the other hand, is an aggressive
energy source
design policy which poses two risks: (a) some constraints
maybe violated at the limit and (b) the primitive-prototype
may or may not be a valid model of the substantially
different topology. Wheninduction has been used, the user
must act as a critic of the resulting designs. With this
I
additional check, lstpRINCEbecomesa sound algorithm.
I_..,
vl
The generation of designs by lstpRINCE is combinaFigure 12. Final fueling design
toric. The system offers a choice amongexpansion types,
103
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Intelligence
in Engineering Design, Analysis,
and
target
regions
for Report
expansion,
and Copyright
number of© 1992,
newlyAAAI
created
Manufacturing,
1(3), 169-189.
regions. The combinatorial
explosion can be managed in
Cagan, J. and A. M. Agogino [1991a]. Inducing Constraint
two ways: (a) automatically prune out certain expansions
Activity in Innovative Design. Artificial Intelligence in
based on domain knowledge and Co) interactively include the
Engineering Design, Analysis, and Manufacturing, 5(1), 47user in the design loop to help make decisions when the
61.
preferred course of action is ambiguous. In limited cases,
Cagan,
J. and A. M. Agogino [1991b]. Dimensional Variable
currently we can assess a priori which expansion type or
Expansion - A Formal Approach to Innovative Design.
region will produce better designs.
Research in Engineering Design, 3, 75-85.
l stpRINCE could be a part of a larger system for
Dyer, M. G., M. Flowers, and J. Hodges [1986]. EDISON:An
engineering design, where a front-end methodology would
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could then be incorporated
into a
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Williams [1990] presents a methodology for discovering
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by introducing new features into known designs. Presently
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In Preprints of Modeling Creativity
and
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are available:
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Mittal,
S. et al. [1986]. PRIDE:An Expert System for the Design
The knowledge representation and the library of design
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Murthy, S. S. and S. Addanki [1987]. PROMPT:An Innovative
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Papalarnbros, P., and D.J. Wilde [1988]. Principles of Optimal
examine limiting behavior. Starting from an initial design,
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lstpRINCE has innovated interesting
engineering concepts,
Ulrich, K. and W. P. Seering [1988]. Function Sharing in
such as the electric bus, the tapered and hollow beam, the
Mechanical Design. Proceedings of AAAI-88, St. Paul,
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1 1 Acknowledgments
Overview. P. Winston and S. Shellard (eds.). Artificial
The authors would like to thank Leo Joskowicz, Tom
Intelligence
at MIT: Expanding Frontiers.
MIT Press,
Mitchell,
David Steier,
and Brian Williams for their
Cambridge, MA.
comments on this manuscript.
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