DTM-1530

Proceedings of The 1996 ASME Design Engineering Technical Conference and Computers in Engineering Conference August 18-22, 1996, Irvine, California DTM-1530

ON A MATHEMATICAL FOUNDATION OF AXIOMATIC DESIGN Stephan Rudolph1 Institute of Statics and Dynamics of Aerospace Structures Stuttgart University, Pfaenwaldring 27, 70569 Stuttgart Germany

ing the early stages of the design process (Pahl and Rieg, 1984; Suh, 1990), it has become a major topic of research to integrate engineering knowledge and manufacturing experience into the design process. Much research eort has been spent on formalizing and systemazing the design process from an engineering viewpoint (Koller, 1994; Pahl and Beitz, 1993; Roth, 1994). This has led to the establishment of guidelines for the design process (VDI 2221, 1986). These techniques intend to support the fast, eective and cost-conscious development of new products (Roozenburg and Eekels, 1995; Ulrich and Eppinger, 1995). Parallel eorts have been made in the eld of arti cial intelligence to support the automatic use, generation and processing of design knowledge by acquiring expert and previous design case experience. This has led to the development of engineering expert systems using rule-based, case-based and qualitative reasoning (Dym, 1994; Sycara and Navinchandra, 1989; ten Hagen and Tomiyama, 1991). However, many of the above stated techniques have only been theoretically demonstrated and there is still a remarkable gap between what seems to be theoretically promising and/or feasible and what is really used in everyday engineering design practice. One engineering method of design practice is the axiomatic design approach developed by Suh (1990). His book covering the topic of axiomatic design has been recently reviewed by Mistree (1992). In industry, axiomatic design is now being applied to real world problems (Suh, 1995) and application examples can be found in engineering design literature (Albano and Suh, 1992; Kumar, 1988). From a purely scienti c viewpoint, any axiomatic foundation of a theory is the permanent object of a scienti c

ABSTRACT The axiomatic design approach and its two underlying postulates, the independence and the information axiom, introduced by Suh (1990) are based on the fundamental claim that \there are generalizable principles that govern the design process and that these in the form of the design axioms are general principles or self-evident truths that cannot be derived or proven to be true except that there are no counter-examples or exceptions" (adapted from (Suh, 1994)). This paper shortly describes a mathematical framework which allows replacing the independence and the information axiom with the so-called evaluation hypothesis (Rudolph, 1995a). This evaluation hypothesis is based on the technique of dimensional analysis known from physics and allows the straightforward derivation of expressions that in special cases match the above stated independence axiom. For the information axiom a dierent mathematical expression is obtained. The new theoretical approach has the advantage of possessing an epistemological foundation in form of four fundamental requirements for the evaluation model of a design object which can be motivated through a rational thought process. The two dierent model assumptions are compared, discussed and illustrated using a published axiomatic design reference example. 1 INTRODUCTION

Because of the growing competition on the worldwide market some of the decisive factors of industrial growth and success are a permanent product innovation, opening up new markets and increasing the productivity. Since approximately 80% of the future product characteristics and the cost for manufacturing a future product are determined dur1 Email: [email protected] and in the World Wide Web http://www.isd.uni-stuttgart.de/rudolph

1

c 1996 by ASME Copyright

Stephan Rudolph debate, since axioms mark the currently known epistemological horizon and re ect the present state of scienti c knowledge. The fundamental assumption of axiomatic design that \there are generalizable principles that govern the design process and that these in the form of the design axioms are general principles or self-evident truths that cannot be derived or proven to be true except that there are no counter-examples or exceptions" (adapted from (Suh, 1994)) is therefore not necessarily a nal and timeless fact, but a scienti c argument in the present engineering design discussion based on empirical observation and an inductive conclusion. Purpose of this paper is to show which additional insight into the axiomatic design approach can be gained if the above stated fundamental assumption is replaced by the evaluation hypothesis (Rudolph, 1995a or 1995b) and which conclusions can be drawn from the dierent derivations of the very same design matrix equation. This will allow to extent the epistemological horizon (i.e. the claim of self-evident truth) towards the postulate of four epistemological requirements (i.e. homogeneity, completeness, invariance and minimality) upon which can be reasoned on a rational basis as shown in section 3. As a further advantage, these four requirements are already common concepts in the theory of model building as used in physics (Bridgman, 1922; Gortler, 1975). They allow the linking and the embedding of the idea of axiomatic design into the world of our present physical thought constructs. The evaluation hypothesis is based on dimensional analysis from physics. Dimensional analysis has already been applied to engineering design problems in the past (Dolinskii, 1990; Kloberdanz, 1991, Middendorf, 1986), where it was used to ease conceptual modeling and where it helped to improve understanding the functional behavior of the design object. Besides numerous traditional engineering applications in uid mechanics and heat transfer (Birkho, 1950; Sedov, 1993), dimensional analysis has also been used in the eld of arti cial intelligence as a basis for qualitative reasoning (Bhaskar and Nigam, 1990; Kalagnanam et al, 1994). Dimensional analysis has also been successfully applied to the problem of topology generation and generalization in feed-forward neural networks (Rudolph, 1996a). The outline of the paper is as follows. After the introduction in section 1 a short review of the main principles of axiomatic design is given in section 2. This includes the presentation of a published reference example of the Mit Rim molding machine (Suh, 1990; Tucker, 1978). Section 3 brie y reviews the foundation of the evaluation hypothesis and contains the alternative derivation of the previously shown reference example of the Mit Rim molding machine. The paper closes with sections 4 and 5 containing a brief discussion and summary.

2 AXIOMATIC DESIGN

The axiomatic design approach de nes the design world as a thought construct of mappings between four dierent domains. These four domains are called the customer domain (de ned by the customer attributes, CA), the functional domain (de ned by the functional requirements, FR), the physical domain (de ned by the design parameters, DP) and the process domain (de ned by the process variables, PV) (Suh, 1990; Suh, 1994) as shown in g. 1. Going from

' ' ' C

CA

D

FR

M

DP

PV

' - ' - ' - ?1 C

?1 D

?1 M

Figure 1. Domains and Mappings (according to Suh (1990))

a domain on the right to a domain on the left in g. 1 hereby represents what the designer wants to achieve (i.e. the mappings 'M , 'D and 'C of the design analysis process). On the other hand, going from left to right represents how the designer proposes to achieve these goals (i.e. the mappings '?C 1 , '?D1 and '?M1 of the design synthesis process). The design process with the overall goal of creating a design object can thus be interpreted as the 'zig-zagging' through these four domains by means of these mappings (Suh, 1990). For a linear mapping 'D , the following design matrix A is (Suh, 1990) de ned by

fFRg = [A]fDPg

(1)

For a sensitivity analysis of a proposed design in the general non-linear case, the design matrix equation (1) must be written in dierential form 9 2 8 9 38 > > > > > > > @FR @FR 1 1 > > > d FR1 > d DP1 > 6 > > 7 > > @DP @DP > > > > 1 n > 7 > > = 6 < . > = < 6 7 . . . 6 7 . . .. . = (2) . . . 6 7 > > > > 6 > > 7 > > > > > > > > > 4 @FR 5> > > > > > > m @FRm > ; : d FRm > ; : d DP n @DP 1 @DP n By neglecting second and higher order terms equation (2) can formally be derived from a Taylor-series expansion of a real valued vector function 'D : DP ! FR which maps the n-dimensional space Rn of the design variables 2


On A Mathematical Foundation of Axiomatic Design DP onto the m-dimensional space Rm of functional requirements FR. Such rst order expansions have a long tradition in science and engineering and independent of speci c values for m and n play an important role in dierential error analysis (Stoer, 1989) of numerical algorithms in computer science. In other words equation (2) implies no speci c restrictions on n and m, also allowing for a rectangular form of the matrix A.

I =?

n X i=1

ln( xxi )

(3)

i

The information axiom therefore de nes the best design as the one with the minimum information content among all acceptable designs.

Axiomatic design as seen by Suh (1990) is concerned with a special case where m = n. This means the design matrix is only allowed a square form. Depending on the speci c form of the functional requirements of a design object, certain elements aji = @ FRj =@ DPi of the dierential design matrix A may or may not be zero. For m = n three qualitatively distinct cases are classi ed by Suh (1990) as:

2.1 Mit Rim Machine (Part I)

To illustrate the axiomatic design approach the example of the MIT RIM molding machine (Suh, 1990; Tucker, 1978) is brie y repeated here. This way it is possible to compare both derivations of a well documented design problem as contained in parts I and II of the Mit Rim machine sections by only focussing on the theoretical issues. A scheme of the MIT RIM molding machine is shown in gure 2. This machine consists of two pumps that charge two accumulators A1 and A2 with the pressure P of liquid chemicals from the reservoirs R1 and R2 . A metering pump turning with an angular velocity ! and an ori ce diameter D is then used for mixing and metering both chemicals in the mixing head M . For the two design performance char-

Coupled design. In general, the matrix A is a full matrix (the aji = 6 0 for all i; j ). Such a design is called coupled

design, since a small change in one design variable may aect all other functional requirements. Decoupled design. If the matrix A is either a upper or a lower triangular matrix (the aji 6= 0 for i j , and aji = 0 for i > j , or vice versa), the design is called a decoupled design. The design variables can then be independently changed, if a certain order in these changes is observed. Uncoupled design. If the matrix A is a diagonal matrix (the aii 6= 0 and aji = 0 for i 6= j ), the design is called an uncoupled design. Any change in one design variable will only aect one functional requirement at a time.

R A j A j P P ! j j D D R1

2

1

For axiomatic design this concludes that any acceptable design should be uncoupled or at least decoupled (Suh, 1990). Therefore, the so-called independence axiom in its declarative form (Suh, 1990) states:

2

M

Q ?Z

Maintain the independence of FRs.

Figure 2. Mit Rim Machine Design (Suh, 1990)

To select the best design among all acceptable designs which ful ll and are conform to the independence axiom, the socalled information axiom in its declarative form (Suh, 1990) is used:

acteristics of delivering a certain mold ow rate Q while maintaining a certain mix quality Z , the following relationships have been determined experimentally (Tucker, 1978) to be approximately equal to

Minimize the information content of the design.

Q = K2P 1=2 D2 Z = K3P ?3=8 D1=4

The information content I of the design in equation (3) is hereby de ned as the relation between the designer speci cations (the set of design parameters DPi , also denoted as xi ) and the manufacturing capabilities (expressed in tolerances of the design parameters (xi =2))

(4) (5)

where K2 and K3 are numerical constants determined in the experiments. In terms of axiomatic design, Q and Z 3


Stephan Rudolph object

evaluation

*H YH HH

evaluation 1

evaluation 2

1Pi 6 PPPP

mapping 'D

object 1

object 2

3Qk QQ

object 4

object 3

object 5

Figure 3. Description, Mapping and Evaluation (Rudolph, 1995a)

constitute two of the functional requirements, while P and D represent the design variables. This leads to the following fully coupled design matrix (Suh, 1990): 8 9 2 38 9 > > ? 1 1 > > > > = < d Q = 6 21 K2 P 2 D2 2K2P 2 D 7 > 6 7 = (6) 4 3 > > ?11 1 1 ?3 ?3 5 > > > > > ; ; :dZ > : 8 4 8 4 dD ? K3P D K3 P D 8

This is represented by an increasingly growing description graph (Rudolph, 1995a or 1995b) on the right hand side of g. 3. During design analysis, the designer analyses the current state of his design by mapping the design object's description onto the design evaluation hierarchy. This evaluation hierarchy is represented as an evaluation graph (Rudolph, 1995a or 1995b) on the left hand side of g. 3. The procedure of design evaluation therefore formally consists of identifying the mapping 'D of the design description onto the design evaluation by means of transfer functions (Rudolph, 1995a or 1995b). In order to obtain objective and reproducible evaluation results it can be stated that the transfer functions 'D will be formally independent of the designer`s belief and cultural background etc., if the parameters of the transfer functions depend on the design description in form of the design parameters x1 ; : : : ; xn only. This means that an objective and reproducible evaluation is an unique property of the design object description. The motivation of creating an objective design evaluation method is based on the following four epistemological postulates (Rudolph, 1995a or 1995b): a reproducible and objective evaluation can only exist if it is based on and derived from some type of law which has to be dimensionally homogeneous, under the assumption that such an objective evaluation procedure exists, it may not depend on the arbitrarily chosen de nitions of physical units and therefore has to possess a dimensionless form, an evaluation method should turn into exact physics and be consistent over all hierarchical evaluation levels in the case of complete physical knowledge about a certain design object, and an evaluation method should consists of a minimal set of evaluation criteria only. If not, some partial evaluation criteria will distort the overall evaluation result when taken multiply into account. Once these four epistemological requirements are accepted

4

Rewriting and normalizing equation (6) by dividing with

Q or Z and using the equalities d Q^ = d Q=Q = d Q=K2P 1=2 D2 and d Z^ = d Z=Z = d Z=K3P ?3=8 D1=4 leads to

8 9 2 38 9 > > > > > > 1 2 < d Q^ = 6 2P D 7 > = 6 7 = 4 3 1 5> > > > > ; : d Z^ > ; :dD> ? 8P 4D >

(7)

Setting P = 1 and D = 1 and rewriting equation (7) one obtains the dimensionless design matrix as stated in Suh's book (Suh, 1990) ([P ] and [D] denote the dimensions of P and D) 8 9 2 38 9 > > > > > < d[PP] > = < d Q^ > = 6 21 2 7 > 7 6 (8) = 4 3 1 5> > > > > > > d D : ; : d Z^ > ; ? 8 4 [D] In the following section, the evaluation hypothesis is brie y reviewed and then applied in section Mit Rim machine part II for the derivation of the very same example. 3 EVALUATION HYPOTHESIS

During the design synthesis, the designer creates a more and more detailed description of the design object. 4


On A Mathematical Foundation of Axiomatic Design 2 ?11 1 ?1r 1 8 9 6 ?x211 2 ?x2rr 2 > > > x1 xr < d . 1 > = 666 . .. .. .. = 66 . > > > > : d m ; 64 ?m?1;1 m?1 ?m?1;r m?1 x1 x1

?m1 m

xr xr

?mr m

as meaningful assumptions, it can then be shown that a universal method exists for constructing the required dimensionless design evaluation quantities from the design descriptions in form of dimensionally homogeneous function equations by means of the Pi-theorem. The Pitheorem (Bridgman, 1922) states that from the existence of a dimensionally homogeneous and complete equation f of n physical quantities xi , the existence of an equation F of only m < n dimensionless quantities j can be shown

f (x1 ; :::; xn ) = 0 F (1 ; :::; m ) = 0

r Y x?ji

i=1

i

.. . 0 0

...

0 0

(9)

d xr+m

1995b). An application example of the evaluation hypothesis is shown in section 3.2. 3.1 Replacing the Epistemological Basis

According to the evaluation hypothesis, the evaluation problem consists of identifying the description space X = fx1 ; : : : ; xn gT for a physical design object, identifying the design goals = f1 ; : : : ; m gT in the evaluation space, and establishing an appropriate mapping 'D . Comparing the properties of both the evaluation hypothesis and the axiomatic design approach, the space of the design parameters DP is matched by the space X of the physical variables, while the space of the functional requirements FR is matched by the space of dimensionless groups associated with these physical variables. This can be written as

(10) (11)

where r = n ? m is the rank of the dimensional matrix constructed by the xi and with dimensionless quantities (dimensionless products or parameters) j of the form

j = x j

1 xr+1 0 0 2 0 0 xr+2

8 9 > 3 > d x 1 > > > > 0 > > > > . > > . 7 > > . > > > 0 77 > > > < = d x 7 r .. 7 . > d xr+1 > > > m?1 0 7 7 > > > > > > 5 > . xr+m?1 > . > > . > > m > > 0 xr+m > > : ;

DP fx1 ; : : : ; xn gT FR f1 ; : : : ; m gT

(12)

(13) (14)

The fact that the evaluation hypothesis is applicable (i.e. that the assumptions of equations (13) and (14) are correct) can be veri ed later by inspecting both alternative derivations yielding identical results. Since the 's are de ned by equation (12), writing the derivatives with respect to all variables leads to a form similar to the dierential design matrix as shown in equation (2) and results directly in equation (9). This general equation results in interesting conclusions when several variables xj are held constant (d xi = 0). This leads to the elimination of the corresponding columns in the above matrix in equation (9). This matrix can thus exhibit all behaviors, from a purely diagonal form to a fully coupled matrix, as stated by the axiomatic design approach. It can also be stated without loss of generality that the dimensionality of both spaces m and n is principally dictated by the number of dimensionless products associated with any dimensionally homogeneous physical equation, for which the relation m = n ? r is generally valid. These facts

with j = 1; : : : ; m 2 N and the ji 2 R as constants. An example of the dimensional matrix is shown in table 1 in section 3.2. (The term dimensional homogeneity only means that an equation of physical variables applies to the dimensions of the variables as well. An equation F = m a does not only result in 12 = 3 4 or 6 = 2 3, but also in [N ] = [kg] [m=s2 ]). As a consequence of the ful llment of the four stated epistemological requirements by the Pi-theorem the socalled evaluation hypothesis (Rudolph, 1995a or 1995b) can thus be expressed as follows: \any minimal description in the sense of the Pi-theorem is an evaluation." This hypothesis is based on the fact that for any engineering design object, for which a description in the form of equation (10) exists, an evaluation in the form of equation (11) can automatically be generated with the Pi-theorem. The evaluation hypothesis is discussed in great detail in Rudolph (1995a or

5


Stephan Rudolph are guaranteed with no special restrictions on f , as long as f is a dimensional homogeneous function equation to which the Pi-theorem can be applied. This variety in behaviors can thus be used to establish a mathematical framework in which upper and lower limits for the principles of design (Rudolph, 1996b) can be proven.

the remaining exponents in each matrix column needs to add up to zero to obtain a dimensionless product, the signs of the elements in the row of Q in the modi ed dimensional matrix just need to be inverted to obtain the appropriate exponents ji in equation (12). This leads to the following dimensionless quantity

1 = QP ?1=2 D?2 K2?1 Q = K2 P 1=2 D2

3.2 Mit Rim Machine (Part II)

In this section, the evaluation hypothesis based on dimensional analysis is applied to the very same example as in section Mit Rim machine part I. Using equation (4) of the reaction injection machine with

Q = K2P 1=2 D2

Applying this procedure to the second design equation (5) results in a second dimensionless product

(15)

2 = K P ?Z3=8 D1=4 3

as an example, the implicit notation of the function f (P; D; K2 ; Q) = 0 leads to the dimensional matrix shown in table 1. The elements eij of this dimensional matrix represent the dimension exponent of the i-th physical variable in the j -th physical dimension of the SI-unit system. Using

P D K2 Q

[L] [T ] -1 -2 1 -1/2 3/2 3 -1

SI-units kg m?1 s?2 m kg?1=2 m3=2 m3 s?1

(17)

According to the general context of the evaluation hypothesis as shown in equations (13) and (14), this leads to a dimensionless expression of the design goals, i.e. the functional requirements, and according to equation (9) is equal to 9 8 d K2 > 9 2 8 > > 3> > > > > > > 6 ? 1 0 ? 1 ? 21 1 0 7 > > d 1 > > > d K3 > = < < K2 2P D Q 7 d P = 6 = (18) 4 5> dD > > > > > > 3 2 2 2 2 ; : d 2 > > > 0 ? K3 8P ? 4D 0 Z > > > > > ; : dQ >

Table 1. Dimensional Matrix RIM Machine

[M ] 1

(16)

meaning pressure diameter constant

ow rate

dZ

rank preserving operations on the columns of that dimensional matrix, the following modi ed dimensional matrix with an upper diagonal form is obtained, see table 2 right. The rank of the dimensional matrix is equal to r = 3, thus

This equation is the dierential form of the design matrix as derived from the evaluation hypothesis. Now considering the special case of all designs with constant evaluation, i.e. j = const and thus d j = 0, and considering K2 and K3 as true constants results in

Table 2. Dimensional Matrix Calculations

8 9 2 38 9 > > > > > > d Q 1 2 = < Q = 6 2P D 7 > 7 6 = 4 3 1 5> > > > > ; :dD> ; : dZZ > ? 8P 4D >

P D K2 Q

[M ] 1

[L] [T ] -1 -2 1 =) -1/2 3/2 3 -1

[M ] 1 1/2

[L]

1 2

[T ]

(19)

Rewriting this once more leads to a dimensionless design matrix 8 9 2 38 9 > > > > > < dPP > < dQQ > = 6 21 2 7 > = 6 7 = (20) 4 3 1 5> > > > > > > d D : : dZ > ; ; ?

1 1

with n = 4, m = n ? r = 1 dimensionless product can be constructed according to equation (12). Since the sum of

Z

6

8 4

D


On A Mathematical Foundation of Axiomatic Design Comparing both derived equations (8) and (20) veri es that both results are identical. Equations (16), (17) and (20) are already contained in the book on axiomatic design (Suh, 1990), but no further statement about the use of dimensionless groups outside their traditional applications in engineering, such as in terms of the evaluation hypothesis, is made therein. The fact that in the reference example the identical equations can be derived based on a dierent epistemological basis however suggests that both approaches match eortlessly.

ter. For certain dimensionless groups, such as eciency coecients or Reynolds numbers, much experience on both qualitative and quantitative interpretation of form and value of these dimensionless parameters has already been gained during the past (Birkho, 1950; Gortler, 1975). The general concept of dimensionless groups serving as evaluation parameters oers the possibility of implementing the formalism of dimensional analysis into a future software module called Symbolic Front-End for CAD (Rudolph, 1996c). This software module will be capable of monitoring and tracking the current evaluation status of a design object during the early design stages.

4 DISCUSSION

Regarding the evaluation hypothesis in its dierential form, the design matrix A as based on dimensional analysis in equation (9) generates the following four major consequences: The size of the design matrix A is (m n), i.e. the design matrix A is generally rectangular. In physics, there exists a xed relationship between m (the number of dimensionless groups) and n (the number of dimensional variables) with m = n ? r. Here r is the rank of the dimensional matrix. The focus and purpose of the paper is therefore on the mathematical understanding of the origin of the axiomatic design approach: to show that there is no need to de ne the independence and the information axiom as self-evident truths that cannot be proven to be true (Suh, 1994) but that they can rather be derived as a special case from the theoretical framework of the evaluation hypothesis. The fact that the mechanism of axiomatic design is embedded as a special case in this theoretical framework based on dimensional analysis results from the general validity of equation (9). However no justi cation is provided nor any claim is made or even intended to support any of the commonly made interpretations in axiomatic design concerning its general and unlimited validity outside the area of physics, such as the design of company or organizational structures (Suh, 1990). In order to reach an objective discussion and to help the future scienti c understanding of axiomatic design it is crucial to clearly distinguish between the facts given in form of equation (9) and the interpretation and conclusions drawn from this equation. In terms of the evaluation hypothesis, the goal hierarchy graph in g. 3 represents the dimensionless function m = F (1 ; : : : ; m?1 ) related to the function xn = f (x1 ; : : : ; xn?1 ) of the design object description graph. According to whether xn needs to be maximized or minimized, a high(er) or low(er) Q value of the dimensionless coecient m = xn ri=1 x?i mi is sought af-

5 SUMMARY

In this work the axiomatic design approach has been derived as a special case of the evaluation hypothesis. It has been shown that the postulate of the independence and the information axiom can be replaced by the evaluation hypothesis. A direct consequence of the evaluation hypothesis is the fact that the form, the content and the interpretation of the evaluation graph are a result of a mapping of the description graph. Due to the consistency of the basic physical model formation, the units of measurement pertaining to the single evaluations (the criteria), and also the aggregation functions (the weighting) of the single criteria are de ned in relation to one another and are consistent throughout all hierarchic levels. The evaluation hypothesis is restricted to the validity of the method of dimensional analysis and relies on a deductive approach based on epistemological considerations. The technique of dimensional analysis is de ned in physics and therefore all statements are limited to relationships of physical variables and no attempt is made to extend what is said outside this original range of validity. The fact that identical equations can be derived from dierent epistemological assumptions shows that the evaluation hypothesis is a substitute for the postulates of the independence and the information axiom (Rudolph, 1996b). ACKNOWLEDGMENT

Special thanks go to Sean Hetterich for improving and proofreading several english draft versions of this paper. The nancial support of this work by the Deutsche Forschungsgemeinschaft (DFG) in form of the interdisciplinary research group Forschergruppe im Bauwesen (FOGIB) \Ingenieurbauten | Wege zu einer ganzheitlichen Betrachtung" (Structural Engineering | Ways Towards a Holistic Approach), is acknowledged. 7


Stephan Rudolph copy of this translated PhD thesis is available on request by email from: [email protected], or by writing to the Institute of Statics and Dynamics of Aerospace Structures, Stuttgart University, Pfaenwaldring 27, D-70569 Stuttgart, mentioning reference number 02-94. Rudolph, S. (1996a), \On a genetic algorithm for the selection of optimally generalizing neural network topologies", in: C. Parmee, ed., Proceedings of the 2nd International Conference on Adaptive Computing in Engineering Design and Control'96, University of Plymouth, United Kingdom, March 26-28th, 1996, 79{86. Rudolph, S. (1996b), \Upper and lower limits for \The Principles of Design"", to appear in Research in Engineering Design, 1{14. Rudolph, S. (1996c), \On a symbolic CAD-Front-End for design evaluation based on the Pi-Theorem", in: Gero, J. and Sudweeks, F. (eds.): Proceedings of the IFIP WG5.2 Workshop on Formal Design Methods for Computer-Aided Design, June 13-16th, 1995, Mexico City, Mexico, 165-179, Chapman and Hall, London, 1996. Sedov, L. (1993), Dimensional Analysis, CRC Press, Boca Raton. Stoer, J. (1989), Numerische Mathematik. Band 1, Springer, Berlin. Suh, N. (1990), The Principles of Design, Oxford Press, New York. Suh, N. (1994), \Axiomatic design of mechanical systems", Preprint, Invited Paper on the 50th Anniversary of the Machine Design Division of ASME. Suh, N. (1995), Personal Communications, Department of Mechanical Engineering, Massachusetts Institute of Technology. Sycara K. and Navinchandra, D. (1989), \Integrating case-based reasoning and qualitative reasoning in engineering design", Proceedings Applications of AI in Engineering, 231{250. ten Hagen P. and Tomiyama, T. (1991), Intelligent CAD Systems I, Springer Verlag, New York. Tucker, C. (1978), Reaction Injection Molding of Reinforced Polymer Parts, PhD thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering. Ulrich K. and Eppinger, S. (1995), Product Design and Development, McGraw-Hill, London. VDI-Richtlinie 2221 (1986), Methodik zum Entwickeln und Konstruieren technischer Systeme und Produkte, VDIVerlag, Dusseldorf.

REFERENCES

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