fuzzy materials selection

A Fiimy Material Selection Syst,em for Mcchariical Design

Jean-Luc Koning May 28, 1994

CML-R1-TR-94-22

Robotics Institute Carnegie Mellon University

5000 Forbes Avenue Pittsburgh, PA 15213-3890

Copyfight @ I V Y 4 CMI! Robotics Irwlitute

This research was supported by INRIA, Domahe de \~'ololnceau, h c q u c n c u u r t , B.P. 10.5. 78153 Le Chesnay cedex, France.

Abstract

This paper c.oncentratcs on two types of mat,erial-related reasoning that occurs in engineering design: selection and substatution. Selection is based on criteria set out by thc designer and is often associated with the process of designing new art.ifact.s from scratch. Substitution, on the other hand, is done when new designs are created by adapting and re-using parts of previous designs. This involves re-evaluating the reasons for a particular material that was used in thc base design: and determining whether the same, or new criteria are relevant in the target design. Once the new criteria are established, appropriate materials arc selected and substituted. The substitution step, hence, subsumes the selection task. tVe present a material selection/snbstilution system that is used as part of a larger case-based design environment. The material selection syst,em helps the designer adapt a previous designs by suggesting material substitutions t,liat better snit the target application. The system is intended t o provide support, right from the conceptual to the detailed stages of design, and can reason about material properties expressed in both numerical and qualitative terms. The system relies on the material selection charts developed by Prof. Aahby of Cambridge University. To enable qualitative reasoning about the charts, we have developed an cncoding based on fuzzy set theory. This allows access of materials using specifications expressed in terms of fuzzy orders of magnitude.

Contents

1 Introduction 6 1.1 What. is this Report About? . . . . . . . . . . . . . . . . . . . . . 6 1.2 Quick Survey of the Problem . . . . . . . . . . . . . . . . . . . . 6 1.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Select. ion and Substitutio~l of Material . . . . . . . . . . . . . . . 8 1.5 Outline of the Report . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The Material Selection Problem 11 2.1 Overview of the Problem . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Material Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Ashby’s Selection Charts . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Four Possible Representations . . . . . . . . . . . . . . . . . . . . 16 3.1.1 Continuous envelopes . . . . . . . . . . . . . . . . . . . . 16 3.1.2 Discret envelopes . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.3 Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1 .4 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.5 Fuzzy contours . . . . . . . . . . . . . . . . . . . . . . . . 2U

3.2 Polygons for Material Domains . . . . . . . . . . . . . . . . . . . 21 3.2.1 Specifying property boundaries . . . . . . . . . . . . . . . 22 3.2.2 Compiling the data . . . . . . . . . . . . . . . . . . . . . . 23

4 Description of the Materials Selection System 25 4.1 Material Information Attached to a Case . . . . . . . . . . . . . . 26

4.1.1 Justifications in the retrieved case . . . . . . . . . . . . . 26

4.2 Fuzzy Orders of Magnitude . . . . . . . . . . . . . . . . . . . . . 28 4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2.2 Partitioning a set of values . . . . . . . . . . . . . . . . . 29

3 Rcpresentation of the Charts 16

4.1.2 Inferencc of justifications for the adapted case . . . . . . . 27

4.2.3 Combining Matching Degrees for Mat. erials Class Selection . . . .

An int.erface for acquiring new orders of magnitude . . . . 32 34 4.3

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4.3.1 Queries for accessing materials classes . . . . . . . . . . . 34

4.3.3 Example of material selection . . . . . . . . . . . . . . . . 37 4.3.2 Combining matching degree . . . . . . . . . . . . . . . . 36

5 Description of the Program 5 . 1 Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1 Mechanical and engineering properties . . . . . . . . . . . 5.1.2 Materials classes . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Modes of loading . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Compiled data . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 .2 Querying the data base . . . . . . . . . . . . . . . . . . .

5.2.1 Data coherence verification and knowledge base initializa- tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 39 40 41 41 42

42 43

6 Conclusion 44

A Material Selection Charts 47 A.1 Materials Selection Chart for Light Stiff Components . . . . . . . 47 A.2 Materials Selection Chart for Light Strong Components . . . . . 47 A.3 Materials Selection Chart for Light Fracture-Resistant Components 49 A.4 Materials Selection Chart for Cheap, Stiff Components . . . . . . 19 A.5 Materials Selection Chart for Cheap. Strong Components . . . . 49 A.6 Materials Selection Chart for Components Resistant t o Corrosion 50

55 B.l Material Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . DJ

- _ B Materials Classes and Properties

B.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2

List of Figures

1.1 Sccsaw in a playground . . . . . . . . . . . . . . . . . . . . . . . #

1 2 8 1 .3 Sketch of a material selection chart . . . . . . . . . . . . . . . . 9

2 .1 Ashby material selection chart for light stiff components (courtesy) . 13

Hot and cold water faucet using the -saw concept . . . . . . . .

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3 .8 3.9

Discret representation of a region . . . . . . . . . . . . .

Representation by rectangles . . . . . . . . . . . . . . . . Representation by polygons . . . . . . . . . . . . . . . . . Representation by polygons of the worst case . . . . . . . Crisp set . of engineeriug polymers denuities . . . . . . . . Fuzzy set of engineering polymers densities . . . . . . . .

List of threshold values for property P . . . . . . . . . .

Discret represent. ation of a region on a logarithmic grid .

Fuzzy representation of Ashby’s charts . . . . . . . . . .

1.1 4.2 1 ..?

Seesaw concept for a water faucet . . . . . . . . . . . . . Example of non-fuzzy orders of magnitude . . . . . . . . Example of fuzzy orders of magnitude . . . . . . . . . . .

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1.4 Example of wrong fuzzy partition . . . . . . . . . . . . . . . . . 4.5 Example of fuzzy orders of magnitude on a logarithmic scale . . 4.6 4.7 ‘I’hree-clinicnsional representation of fuzzy orders of magnitude . 4 . 8 User defined orders of magnitude . . . . . . . . . . . . . . . . . . 4.9 Matching degree between a material class K and the order of

4.10 Matching degree between a material class K and the order of

Fuzzy compound orders of magnitude . . . . . . . . . . . . . . .

magnitude A uf some property . . . . . . . . . . . . . . . . . . .

rriagnitude A of some other property . . . . . . . . . . . . . . . .

.5 .l Cbhcrence checking on fuzzy orders of magnitude.

-4.1 Material selection chart for light strong components (courtesy) .

17 18 18 19 19 20 20 21 23

25 29 30 30 31 31 32 33

35

35

42

48

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A.2 Matcrial selection chart for light fracture-resistant components (courtesy ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I

4 . 3 Material selection chart for cheap, stiff c0rnponent.s (courtesy). . 52 -4.4 Material select,ion chart for cheap, strong components (courtesy). 53 A.5 ?daterial select,ion chart for components resistant to corrosion

(courtesy). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4

List of Tables

2.1 Sample of propertycombinations which determine performance in design (courtesy) . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Kecesuary data to be acquired . . . . . . . . . . . . . . . . . . . . 2'2 3 2 Compiled data for a particular property P . . . . . . . . . . . . . 24

4.1 Materials satisfying the three behavioral properties . . . . . . . . 38

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Chapter 1

Introduction

1.1 What is this Report About? This report describes a general system for material selection. This system has becn developcd in the framework of the CADET project. It is originally intended for material adaptation when old designs are to be reused in new contexts. The purpose of this system is to help designers pick up the right type of material given some constraints on a mechanical artifact. This system is built on a knowledge base that makes use of materials selection charts from the mechanical engineering field. Queries t o this data base are embedded with imprecision. !!'e handle this aspect by relying on fuzzy set theory.

1.2 Quick Survey of the Problem Mechanical design is the act of devising an artifact that satisfics a useful need, in other words, that performs some function. This requires, in part, a deep understanding of material propert,ies An engineer conceiving a new mechanical artifact can choose from among 50,000 different materials [Ashby: 1989bj.

This paper concentrates on twu t,ypes of material-related reasoning that occurs in engineering dmign: selection and substilation. A material is selected based on criteria set out, by the designer. At the early stages of the design process. criteria that define the properties of the required material are usually quite vague. For example! a designer might. know that the material for an artifact component should be of high strength and very high fracture toughness, bul t,he actual values for E (strength) or K (fracture toughness) may not be known. In addition, a designer may have certain gods that require the minimization of weight and cost. Material selection is often associated with the designing of new artifacts horn scratch. As a designer proceeds from the conceptual design stage, through embodiment, and finally to detailed design. the criteria for

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material selecsion arc developed and refined. A large part of cngineeririg design: however! is case-baTed. Yew designs are created by adapting and reusing p a r k of previous designs in a new context, or application. Part of the process of adapting a prior [base) design to a new (target) design is the subst,itntion of materials in the base, to suit the conditions of the target application. This involves re-evaluating the reaSons why a particular material was used in the ha..e. and whether the same or new criteria are relevant in the target. Once the ncw criteria are established, appropriate materials are selected and suhsti- tilted. Thc snbslitution step, hcnce, subsumes the selection task. For examplc, if a honsohold Oven were being adapted for an aerospace application, one would haw t,o reevaluate the material choices based on strict weight considerations, possibly replacing heavy steel components by ceramic coated aluminum ones

1.3 Illustrative Example l,et, UY illustrate this problematic with a detailed example that we will refer t,o later 011 in lhis report. From that example, we willsee (in Chapter 4) how a base case (a see-saw) is reused in a target problem (a faucet). Material justifications in the base will be modified to suit the target context that will lead to the selection of new materials.

Consider the simple see-saw shown in Figure 1.1. The -saw's main component is a beam, hinged at its center. Material selection for the beam is based on a variety of criteria: since it is intended for children: it should not be too heavy; since it is intended for outdoor use it should be water resistant; it should not rnst; it shoiild not cost too much; and it's color should not fade in the sunlight,

t-

Figure 1.1 : See-saw in a playground.

The see-saw ca5e is represented in the database as a frame where the varions attrihules (dimensions, material types, etc.) as well as their intrinsic reason(s) are stored (sce Section 4.1).

Consider the target application for the see-saw: a hot and cold water faucet. Figure 1.2 depicts n faucet that utilizes the see-saw principle. The rotation

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in t,be vertical plane regulates t,he relative quantity of hot and cold water. The vertical nroLionof its hinge increass or decreases the combined flow. The Caucct represents a target, application where the environmental conditions are quite dilTercnt from that of the see-saw in the playground.

n

Figure 1.2: Hot and cold water faucet using the see-saw concept.

Although the (see-saw) principle is identical in both cases the material the faucet, should be made of does not necessarily meet the same rcquirements. On the onc hand some requirements for rhe see-saw may no longer be valid: and on t,he other hand some new requirements may crop up. For example, since the heam is of a much smaller dimension its cost may become less important. Also, the material may have to be resistant to certain environments if used with chemical products.

Clearly, thc problem is to come up with the material(s) that meet, the set of requirements. This calls for typical engineering knowledge and reasoning ability on materials. Both types of informatlion are provided in the materials selection charts settled by M. Ashby.

1.4 Selection and Substitution of Material The selection step is done using special ma,terial selection charts that werc developed by Prof. Ashby of Cambridge University [Ashby, 1989al

Theses chast,s show the relative and absolute positions of various material classes wifh respect to a tradcoff between two different properties. For example the sketch in Figure 1.3 presents the positions of the material classes A, L1: C , D based on their strengt,h and density.

The oblique line L crases material classes with the same strength to densky tradeoff. The higher this line on the chart the better the ratio. Looking Cur materials with a particular combined property (tradeoff between t v o properties) is a problem that, involves two kinds of imprecision. First. the malerial classes plorted on the charts represent the smallest area containing all t.he materials of some class. This means that within a class envelope there may be some regions

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Strength L1

I I b

Density

Figure 1.3: Sketch of a material selection chart

that do not correspond to any actual material. Secondly, one rarely looks for materials with a precise value of the combined property but rather Cor materials within acceptable ranges of values, some of these values being more appropriate t,han others.

This notion is exemplified in Figure 1.3. A designer may look for materials that lie between two oblique lines L1 and L z . The area between L1 and La is the preferred rangc of values for the combined property. The ranges of values next t,o lines L1 and L.1 (shown by the shaded area on Figure 1.3) represent values closc 60 the preferred range. Materials in these areas should not be t,otally rejected although they do not pertain to the set of the preferred materials. This defines three conceptually different regions: (1) above the upper shaded area or under the lower shaded area are the materials one is definitely not interested in, (2) materials between L1 and L2 definitely meet the required combined property. arid ( 3 ) the materials crossed by one of the shaded are= but ontside t,he interval [ L l , L z ] are acceptable to a certain extent (they are not totally rejected). Actually, in this third region, the closer the material t o t,he stripe [L.1: L?] (the preferred threshold) the better.

This type of imprecision can easily be handled by making use of fuzzy set theory. It would enable the ranking of material classes with respect to their degree of preference given some property. For example, one could say that materials class B meets the requirements more adequately than materials class A arid much more than materials class C. Also, the fuzzy set theory provides a means t o aggregate several material rankings relative to different requirements (see Sect,ion 4.3)

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1.5 Outline of the Report The rest of the report is organized as follows. Chapter 2 presents the usage of the materials selection charts. In Chapter 3 several computer representations of the charts are given and discussed. One of them ir selected to be implemented i n the system that will actually perform material modifications. Chapter 4 is lhe key chapter of this document. It introduces the not.ion of justifications attached to a case and highlight its importance. This same chapter explains in detail fuzzy orders of magnitude and also focuses on the system capabilities. Thc queries the syst,ern enables are fully explained and an example of material selection is shown Chapter 5 gives an overview of the program. T h e purpose of that chapter is t o e a e the possible re-use of the code written for our system. Chapter 6 gives some concluding remarks and states the place such a system has in a case-based design application like CADET. Appendix A shows the material selection charts that our system uses so far. Appendix 8 sums up all the materials classes and materials properties taken into account by the system

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Chapter 2

The Material Selection PI: o blem

'l'his section presents the material selection problem in mechanical design. For a more detailed explanation see [Ashby and Jones, 19801, [Ashby, 19911. Only some aspects relevant to an understanding of the system are presented here.

2.1 Overview of the Problem

Mechanical design is usually divided in three stages ([Ashby, 1989L>]):

th.e conceptual dessgn stage; approximate data need for selecting the possible range of materials.

the ernbodmen1 design siage; reducing to a small subset the large material set found previously. This second stage requires data at a greater level of precision and detail.

t k e detailed design. slagr; making use of still higher levels of precision and detail! such as accurate data from handbooks and data sheet,s provided by material suppliers.

This paper is only concerned with the first two levels. The purpose of the system we describe here is to find materials that parts of a device should be made of and t,hat best fit t h e specifications ror t,he whole artifact. Finding the exact best material is not that important sirice this choice may still be altered due to the taking into consideration of parts interactions.

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2.2 Material Classes At Ihc conceptual design stage all niatcrials should be considered as potential candidat,es. They are conventionally c.lassified into nine broad classes: engineering alloys, engineering polymers, engineering ceramics, engineering composites, porous ceramics, glasses, woods. elastomers, polymer roams (cf. Section B.1).

Making a select,ion among these material classes is usually driven by necessary constraints such as: i t must operate a t high temperature, must, be cheap, light, for instance. In other words, materials must satisfy certain individual properties. At the embodiment design stage, the engineering designer seeks to identify the design-limiting combination of material properties. One way of efficiently examining the relationships between the properties of all nine classes is by referring t o Ashby's material selection charts.

2.3 Ashby's Selection Charts The primary advantage of these charts is their simplicity and elegance. They display combinations of properties and allow a. designer to visually compare and pick classes of materials with properties within specified ranges. Moreover, materials with maximum performance can easily be identified.

The idea, illustrated by Figure 2.1 , is to have one property (the young's modulus in this cme: property of elasticity whose symbol is E ) plotted against another (the density: symbol p ) on a logarithmic scale. Each class of rna- terials, occupies a characteristic part of the chart. A class is enclosed in a property-envelope. Within each class, data are plotted for a representative set of materials, chosen both t,o span the full range of behavior for that class and to include t h e mmt widely used members in it. In this way, the envelope for a class encloses dat,a for all members of the class.

The performance, in an engineering sense, of load-bearing components is not limited by a single property but by one or more combinations of them. For example, t he l ights t rotating cylinder that will carry a given tensile load without exceeding a given deflection is that with the greatest value of E / p . The lightest rod that will support a given pressure with minimum deflection is one with the greatest valne of E*/p . The lightest panel which will support. a given compressive load without buckling is one with the greatest value of E f / p . Figurc 2.1 shows how the chart can be used to select materials that maximize any one of these c.omhinations.

As a result, of logarithrnicscales. the set of points where these ratios have for value a particular constant C are straight lines even though the lines describe non-linear relationship between the properties. The three different types of slope are shown on Figure 2.1. They are labeled as "Guide Lines for Material Selection". Translating the appropriate guide line sideways leads to idcntify points with another constant value for the corresponding ratio (i.e., property

Figure 2.1: Ashby material selection chart for light stiff components (courtesy).

combination). It is then easy t o pick off the subset of materials that are optimal for each

loading geometry. For any straight-edge laid parallel to the E ? / p = C line, all the materials which lie on the line will perform equally well as a light plate loaded in compression; those above are better (they can withstand greater loads), those below are worse. If the straight edge is translated towards the top left corner of the diagram while retaining the same slope, the choice narrows. The same procedure, applied to the rotating cylinder ( E / p ) or rod in bending ( E i l p ) , leads t o different equivalences and optimal subsets of materials.

Classes of materials crossed by the same line are equally good with respect t o the criteria the straight edge represents. The criteria for optimal materials selection that we are going to use here are summarized in Table 2.1. They all

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appear on one of the malerial selection charts. The purpose of Table 2.1 is to display the most irriportant criteria given the mode of loading of the artifact.

Among t.he mechanical and thermal properties, there are Len that are of primary irnportance, both in characterizing the material, and in engineering design. In this paper, we have used Ashby’s charts that refer to the following properties: young’s modulus, density, strength, fracture toughness, cost, and corrosion resishnce. Thcse charts are explained in Appendix A.

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Table 2.1: Sample of property-combinations which determine performance in design (courtesy).

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Chapter 3

Represent at ion of the Charts

3.1 Four Possible Representations

As seen in Chapt,er 2 the charts’ goal is t,o display the full range of mat,erial properties a t relatively low precision. Because of this approximation, material classes tend to have a rather rounded shape. In the following sub-sections, four possible ways of representing the material class envelopes are presented in view OC a computer exploitation.

3.1.1 Continuous envelopes

One technique consists in scanning the charts and keeping hack of all the points constituting the envelope. This is similar to what is done in the field of image synthesis where object surfaces are plotted point by point. Needless to say this representation is space consuming.

The chief advantage of Ashby’s charts is that it quickly and efficiently guides n s t,o a subset of materials worth considering, Le., that maximize a combined criterion, and m&e sure not to overlook a promising candidate. The repreuen- tation suggested here generates a great amount of data whose processing con- sequently leads t o slow information retrieval. Evidently the tradeoff between data accuracy and processing time should be in favor of t,he second one.

Another argument for dropping this type of represent,ation relates to the fact, that, the charts themselves do not display precise information. At the conceptual and embodiment design stage approximate data are sufficient to determine lhe classes of potentially interesting materials. Full precision is only required in the detailed design stage.

3.1.2 Discret envelopes

Anot,her technique from image processing consists in representing regions by contiguous squares. The image is applied on a grid. Any square with more than x percents of their surface covered by a specific region is considered to belong to the region. When x equals 0 then the set of contiguous squares represents the srnallest siirface of the grid containing the region. If one imposes thc squares to have hundred percents of their surface covercd by the region then the set of contiguous squares represents the largest discrete surface contained in t,he region. r\ reasonable choice for 3: i s 50%. See Figure 3.1.

.._... C .,__

Figure 3.1: Discret. representation of a region

With t,his method a chart becomes a two dimensional array whose elements contain Lhe name(s) of the classes of materials that would be plotted on the corresponding sqnare. The logarithmicscales used for the charts give rise to Lwo problems. First, if one wants to use the same method with the logarithmic grid one ends up approximating too roughly the regions showing material classes. See Figure 3.2.. The squares do not have the same size. Second, one can avoid t,his drawback by deciding to take into account a linear scale. But this alternative lcads to storing a huge amount of data. For instance the scale for the strength property goes from 0.1 to 10,000 which makes 100,000 echelons of size 0.1 (cf. Figure A.1).

'These two reamus prompted us to abandon lhis representalion.

3.1.3 Rectangles

'I'he t.wo representalions suggested so far require a great deal of data and com- put,ation. Their advantage is that they provide accurate information.

For the sake ol simplicity one c.an represents these regions by rectangles where parallel edges correspond to lower and upper bounds of a class in the chart.

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Figure 3.2: Discret representat,ion of a region on a logarithmic grid.

With this representation one faces now a lack of precision. Figure 3.3 high- lights this drawback. The same representation would be identical for far different classes of materials. The shaded area does not correspond t o any material of the class.

......... ........

......... ........

Figure 3.3: Representation by rectangles.

3.1.4 Polygons The given shape of t,he material class envelopes resemble rounded polygons. One may t,ake advantage of this fact and approximate them by true polygons where each edge corresponds to bounds of the properties shown on the chart. In the case of Figure 2.1 the chart presents five properties: young’s modulus, density, and t,hree types of stiffness ( E l p , E i j p , E f l p ) . Thus, any class of materials on the chart would be represented by an IO-edge polygon. Figurc 3.4 shows

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an example of such a polygonal rrpresentatlon with only one type of stiffness. This is R 6-edge polygon, the four other edges are introduced by the other two guide-lines.

Figure 3.4: Representation by polygons

On rishby's charts material class envelopes are naturally either vertical (parallel t o the ordinate axis): horizontal (parallel t o the abscissa axis) or slopcd to the righl (rather parallel to t,he combined property). Therefore choosing these axis to define the polygons leads to a reasonable approximation of t,he actual regions. The worst c a x (rigure 3.5) would arise when a class envelope iu normal to the slant-wise axis. This case doesn't appear on any material selection chart.

Figure 3.5: Representation by polygons of the worst. case

Y'his ~ y p c of representation does not have t,he drawback of requiring a huge amount of data, and also allows a rather fair approximation of the regions it represents.

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3.1.5 Fuzzy contours

Ashby's charts are not a source of numerical data for precise analysis, they are approximate. The above snggested represen tation does not take into consideration the imprecision of the charts. Fuzzy set theory (Dubois and Prade, 19881 [Zadch, 19651 can help remedy this problem.

Let us first recall some hasic notions. A set is fuzzy if i t s boundaries are not precisely delimited. This concept of fuzzy sets generalizes the concept oC ordinary sets. A fuzzy set A on a universe Cl is characterized by its membership fimct.ion: p~ : Cl + [0,1]. For all w belonging t o Q, pa(*) is t h e extent t,o which w belong lo .A. If pn(w) = 0 it doesn't, belong to it at all, and if ~ A ( u ) = 1: it. belongs fully. If p n ( w ) E ( 0 , l ) : it belongs more or less to A .

Let us illustrate the difference between crisp and fuzzy boundary representat.ion with an example drawn from the material selcction domain. where A is the range of densities for the class of engineering polymers and C2, in this case. is the set of all possible densities. When A is a crisp set, the representation GC ils characteristic function looks like on Figure 3.6 (..e suppose that the class of densities of engineering polymers range from .8Mg/rn3 to 2.0hfg/m3).

R

densities for the class of engineering polymers

" I t - - - - - - - : Figure 3.6: Crisp set of engineering polymers densities

When the boundaries of A are fuzzy the representation of its characteristic fiinction look like on Figure 3.7. Certain densities are typical of engineering polymers (between 1.0Mg/m3 and 1.8A4g/m3), other densities are typical to a smaller extent (at the sides of the and others are not densities of engineering polymers whatsoever.

densaiaes faor the class of engineering polymcrs - - - - - - - - -

8 1.0 1.8 2.0

Figure Xi : Fuzzy set of engineering polymers densities.

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The set {w E n I p~ > 0} is called the support of A; this is the set of va lua for w thaL belong to A at least to a certain extent ((0.8,2) in Figure 3 .5 ) . 'The set {a E 0 I p~ = 1) is called the core of A; this is the set of values for w that. cornplet.ely bclong t o .4 ((1.0,1,8) on Figure 3.7).

In a trvo-dimension universe the core and the support of a fuzzy set become surfaces. This is the case encountered with Ashby's charts since the clnsses of materials are plotted against two properties at the same time. The membership function of fuzzy sets looks then like a trapezoid volume (see Figure 3.8).

/ Figure 3.8: Fuzzy representation of Ashby's charts,

This type of representation is not in opposition to the previous ones. Ac- tually it extends them by providing a way to take into accounl the imprecise nature of the charts. Any of the previous crisp representations can be extended to a fuzzy one. This only requires two contours instead of one: one delimiting the support, the other one delimiting the core of the domain.

3.2 Polygons for Material Domains As stated earlier the representation to he adopted should preserve data accuracy and allow efficient computation. The best tradeoff seems to be at,tained by the polygon represen tation since it doesn't require a, t,rrmendous amount of data to code the niaterial domains, and it also enables quick retrieval of information which is the primary purpose of the charts.

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Here, only a crisp representation by polygons h s been envisaged. Let us uoticc that the fuzzy counterpart would not have led t o much more complex computations. In that case material domain would have been represent,ed by t,rvo polygons. \t‘e didn’t choose the frizzy representation because Ashby’s chart,s only give information concerning the contour delimiting the support (there‘s no way to infer the core of such sets given the charts). Indeed, no material of a class may be found outside the envelope of the class.

In the pdygon-type representation, edges of material domains corrcspond to upper and lower bounds of the properties shown on the chart. Encoding the charts t,hen conies down to a two step process. First. the various bounds must be defined for any property of a material class. Second, these data must he compiled to ease information retrieval.

3.2.1 Specifying property boundaries Property boundaries are represented in a two-dimensional array where colurrins correspond to properties and rows to material classes. Each cell contains the lower and the upper bound for the related property and mat.erials cl,lss. ‘The combined properties shown in the charts are also included in that array. Some cells may stay empty when the informationis not available. Table 3.1 shows the necessary data. Ifif(&’) and z 1 ~ ( P ) are respectively the lower and upper bound of property P for materials clavs A [ .

Table 3.1: Necessary data to be acquired.

This way of storing data is not suited for fast access to the ‘best’ material c l a s . One has to look up every possible class before retricving the right one. On the other hand with Ashby’s charts one directly looks at the correct, region of the chart. If one wants a high (resp. low) value of the property shown on the horizontal axis then one focuses on the right-hand (resp. left-hand) side of the chart without even taking into consideration materials on the left.-hand (resp. right-hand) side. If one wants a high (resp. low) value of the property shown on the vertical axis then one focuses on t,he top (resp. bottom) half of the chart u-ithout even taking into considcration materials on the bottom (resp. top) half. If one wants a high (rcsp. low) value for the combined property one focuscs on the top left (resp. hott,om right) corner of the chart. Sce Figures 2.1, A . l , A.2, A.3, A.4.

22

'Ihe tabular data is thcn compiled to speed access.

I

3.2.2 Compiling the data In the two examples just qnot,ed the materials are retrieved via their properties. A c.onvenient way to store the data is thus by properties. For each property we rank the materials according to their range of values. Since tN.0 different ranges may overlap onc needs t o keep t.rack of bhreshold values, Le., values corrmpond- ing to lower or upper bound of properties of materials classes. Figure 3.9 gives an exarriple of such a list of thresholds for a particular property P . Materials classes A ; B , (7 and D ate plotted according to their range Cor property P.

Figure 3.9: List of threshold values for property I'

In ordcr to improve the retrieval of the right material given some characteristics ( e & low stiffness, high strength! etc.) pieces of solution may be pre- compiled and attached to the threshold values. By pieces of solution we mean information such as (1) the material classes that appear beforr the threshold, (2) t,hr ones that appear afler. and (3) the ones that overlap i t . These threc lists are called less, greater and equal respectively. See Table 3.2.

A compilation of this type is made for each property. Looking for mat,erials with a particular characteristic (for instance: density less than . S M g / m 3 , strength c q u d to ~ O O M P Q or stiffness gnater t,han 3 x lo3) becomes straightforward. If the value being looked for is a threshold value for that property the answer is given by the corresponding slot (either less, e p 0 l or gmaler). Othcr- wise, the answer is derived from the three lists of material classes att,ached to the threshold values just prcceding and just following the value looked for. The iriterscction between the list, greaterlt) and ltss(t + 1), where t and t + 1 are two adjaceiit t,hresholds, givcs the list of all the material classes whose property value belongs to ( t , i + I ) .

Two default threshold values are added to the table of compiled inCormat,ion for each property. 'The first threshold gets the smallest possible value, its slot

23

Table 3.2: Compiled data for a particular property P

named ymnter contains all the material classes like the slot named less attached to the last threshold. The obher threshold (the last one) gets the maximum possible value for the particular property.

One could divide the lists of threshold values in several parts corresponding to different orders of magnitudes in order to facilitate the access t,o malerials whose property is in a particular range like low density, high stiffness, medium strength, etc. This idea is exploited in a slightly different way. A more detailed discussion iu found in Chapter 4.

24

Chapter 4

Description of the Materials Selection System

In chapter 1 we have succinctly seen an example showing the reusing of the see-saw coucept for a water faucet (see Figure 4.1).

Figure 4.1: See-saw concept for a water faucet.

The process of finding the right material Cor the water faucet involves the following steps:

1 . 1,ook up thc material and material justifications of the base case,

2. J,ook up the features of the target artifact,

3. Deduce (possibly new) mat,erial properties the base component should

4 Infer t,he fuzzy values these propcrties should have.

5, Suggcst the best materials for substitution

In this chapter we shall detail these various steps

possess in the target. context.

25

4.1 Material Information Attached to a Case

4.1.1 In the see-saw cme Cor instance, various pieces of information are provided regarding the material it is made of.

Justifications in the retrieved case

See-saw in a playground Material: wood parallel to grain Inherited specifications: -1- mode of loading: bending of rods and t u b a -2- normal load: Behaviora l justif ications: -1- bending resistance: high (or better)

-2- torsion resistance: medium (or better)

-3- fracture resistance: high (or better)

support a weight of at least 1OU kg

mechanical reason - to avoid the see-saw t o bend under normal load

mechanical reason - to avoid t,he see-saw t o bsvist under normal load

mechanical and safety r e s o n - to avoid the see-saw to break under normal load

h n c t ional justif ications: -1- UV radiation resistance: at least good

aesthetic reason - to avoid color fading -2- t,emperature of use: very cold to yery hot

safety reason - t,o avoid painful skin contact

The inhxrided specifications are deduced from the general specifications given by the designer. In the example shown in Section 1.3 concerning the see-saw the description of the artifact mentions that a beam, hinged at its center, should Yupport, the weight of two people sitting on both extremit,ies. Knowing t,his the mode of loading is looked up in the table of modes of loading. Such a table can be found in [Ashby! 19911

There are two types of justifications used in the system: behavioral and functional. A reason is attached t o each justification. nere only three types of reason are mentioned: mechanical, safety and axsthetic reasons. A degree ol importance can also be added t,o t,he justifications that would convey the extent f o which the justification must be t d e n into account in determining the right rnatcrials class. For instance, one may consider aesthetic reasons less important and thercfore focus first on satisfying mechanical and safety constraints. Aesthetic reasons would become secondary. The introduction of a degree of importance would lead to a ranking ofjustifications and then lead fo enhancing the choice between classes of materials.

The beharioraf jrasdificotiorrs constitute the intrinsic reasons that led to the rriatcrial choice. They relate t o the function of the artifact itself and thus always hold. A behavioral characteristic of a see-saw found on a playground is to

26

support the weight of two people sit,ting at Loth ends. The mode of loading of t,liis ar t i fxt pertains to the class of “bending of rods and tubes”. Such information can be found in [Ashby, 19911 where a table (see Table 2.1) of the ten modes of loading relative t,o any mechanical design are given. Each mode of loading delermines t,he relevant mechanical property/properties the material should possess. In the see-saw case, bending resistance, torsion resistancc and fracture resistauce are t,he three important properties. Chapter 2 explains how their Inathemat,ical expressions are exploited.

The values given t o the properties are conveyed by means of orders of magnitude that can he given by the designer either direct,ly or through a user interface. A user interface could help the designer specify the range of weight the see-saw should support for instance and the qualitative values for the three types of resistance could t,hen be derived. An example of how a user interface could he useful in dekrmining fuzzy qriantities is given in Section 4.2.3.

The funciioaal justij5cations take into consideration the environment of the artifact. Let us say, we are talking about an outdoor location for the see saw. If the see-saw is intended to he used outside in very cold weather, one wants to avoid metal because of possible skin contact. The other functional justificatio~i relates to an aest,het,ic point of view. One doesn’t want the color of the see-saw to fade t,oo quickly when exposed t o the sun’s radiation. These two types of juslification are directly specified by the designer.

4.1.2

In water faucet: Figure 4.1, the behavioral properties previously found for the seesaw in t,he play ground hold although their values need t o he adapted. The modo of loading remains identical to the see-saw’s mode of loading, as well as the three iinportant mechanic.al properties. However, the forces on the see- saw niechanism are much lower. The designer recognizes this and inputs the following to the system:

Inference of justifications for the adapted case

bending resistance: medium (or better) torsion resistance: medium (or better) fracturc resistance: low (or better)

Orders of magnitude are used here rather than actual values This facility allows designers to work with incomplete or inexact information about an evolu- ing design concept. They usually don’t have a precise knowledge of the value a property must get but know roughly the suited range of value. Thesc quanti- lies can easily be represented as fuzzy intervals [Dubois and Prade, 19881. We can divide the whole range of possible values in five categories: very low, lorn. medium. high and very high (see Section 4.2 for a more detailed discussion).

Fnnct,ionaljustifications can also be modified. In case of a water faucet, resistance to ultra-violet radiation is not required, and t,emperatures below freezing

27

are unlikely t o be encountered. Therefore, t,he corresponding justifications attached to the retrieved case can hc discarded. In addition, one may require the artifact to be resistant. to salt corrosion if. for instance, the faucet is going to he placed in a marine environment.

4.2 Fuzzy Orders of Magnitude

4.2.1 Definitions

Referring to aspecific value for amaterial property can be performed in diffcrent ways.

h precise value can be given for the material properties. For Example the statement: “the material density is 5Mg/rn3”. In this statement the density is precisely known. Houever, sometimes one doesn’t have the knowledge t o specify such a precise information. In that Case it is more realistic t,o provide a range of possible values.

e The statement becomes then i m p w i s e : ‘;The material density lies between 4.5Mg/m3 and 5.5A.fg/rn3”. Now one is given a range of value where thc honndaries are precise. This allows to find materials with roughly the same property. For instance, materials with a density ranging from 4.5Mg/rn3 to 5.5Mg/m3 would be retrieved. but a material with a slightly greater (or smaller) denvity (e.g.! 5.6My/m3) would not even be considered. In order t,o avoid this drastic discontinuity, crisp boundaries may be changed into imprecise ones.

The range of values then becomes B fuzzy interval, i.e., an interval with imprecise boundaries, whose definition is given in Section 3.1.5.

To facilitate the retrieval of materials we partition the set of possible values for each property in several intervals each referring to a specific order of magnitude. W e have defined five different orders of magnitude that apply to all properties hut the o n e concerning resisbance to corrosion: v e r y lorn. lorn, medium. hiyh, and w r y high. The choice of fiye different orders of magnitude stems from t,he way materials selection charts are presented (see appendix A). They feature grids with five columns and five rows. This feature has heen re- tained.

‘I‘he scales used to measure t,he various propcrties are not ident,ical. The five orders of magnitude thus need to be adapted for each of them. For instancc a low density has nothing in common with a low shength. Section 1.2.2 explains the process to define the fuzzy orders of magnitude for each property given that t,heir corresponding crisp thresholds are known.

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4.2.2

The scales used in Ashby’s material selection charts are logarithmic. but this does not really pose a problem when one wants to define a non-fuzzy partit,ion over a universe O C possible values. Figure 4.2 shows the characteristic functions of the five orders of magnitude for the set of possible material densities in Ihe case of non-fnzzy orders of magnilude on a logarilhmic scale.

Partitioning a set of values

Charact,eristic function: p

very very high

Densit.y

.I .3 1 3 10 30

figure 4.2: Example of non-fumy orders of magnitnde,

Although the materials densities range from 0.1 to 30,tfg/nr3 the ends are cxt.ended to infinity. The lower (resp. upper) bound of interval very-low (resp. very-high) is --x (resp. +m). One has the general property that any value only pertains to one interval at a time:

PA(.) = 1 * J L B ( Z ) = O.V/R # A E 8

0 is the set of possible values, 8 the set of the five orders of magnitude. p~ is the Characteristic function of the set described by order of magnitude A .

One intuitive way to gct to a fumy partit,ioning of a domain from a non- fumy one is to choose supports and cores boundaries of fuzzy subsets slightly greater and lower than the thresholds taken for the non-fuzzy partition. In order to derive these two new thresholds from the crisp one, one may apply a mult,iplicative coefficient. A reasonable coefficient to find the fuzzy boundaries was selected at 20% above or below the crisp threshold, see Figure 4.3. Rote that the scale shown is linear.

In the crisp case the upper hound of Lhe set of values called high is 10. In a fuzzy setting t.his t,ranvforms to tx.0 thresholds. The core’s upper bound becomes 8 (10 - .2 x 10) and the support’s upper bound becomes 12 ( lo+ .2 x 10). The same coniputat,ion applies for the lower bounds.

29

high

( 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3

Figure 4.3: Example of fuzzy orders of magnitude.

One should be careful in choosing Ihe multiplicative coefficient to obtain the fuzzy boundaries. If it becomes too large this may lead to fumy intervals where the support’s upper bound of fuzzy set n is greater than the support’s lower hound of fuzzy set R + 2, see Figure 4.4. This leads to fuzzy sets wit.11 no core.

Figure 4.4: Example of wrong fuzzy partition.

Matching this fuzzy partition on a logarithmic scale leads to noli-trapezoid

The general property of such a fuzzy partition ensures that: fuzzy sets, see Figure 4.5

With a crisp partition one has the relation p ~ ( z ) > U, (i.e., the value z is A ) for only one order of magnitude A. Thus one is able to state Lhat “1: is A” or ‘‘z is not. A ” . On thc other hand, the advantage of having a fuzzy partition over a crisp partition is that it allows to determine that a value z is more A t,han B , and also t,o quantify to what extent x is A or z is more A than B

~ “ ~ ( ~ ) > ~ B ( z ) , ~ : E R : A , B E ~ ~ E is more A than B

30

Charperistic function: r

very high

Density t

1 3 10

Figure 4.5: Example of fuzzy orders of magnitude on a logarithmic scale

p,,(r) is interpreted as the degree to which the value 2: is A, and pa(2:) - p ~ ( z ) as the degree t o which z is more A than B. Both pa(.) and pa(.) - ps(r) belong to [0,1]. pa(t) = 0 means that z is not at, all A . p a ( z ) = 1 means that s is fully A. p n ( z ) E ( 0 , l ) means that I is .4 t o some extent.

Fuzzy orders of magnitude can easily be combined. For instance, the compound order of magnitude found in Section 4.1.2 a t feast medium refers t o the values that are considered as medium or above medium. The compound luzzy order of magnitude would look like on Figure 4.6.

Pat-kart-medium 4

Figure 4.6: Fuzzy compound orders of magnitude.

This can be stated as:

where .4 is a fuzzy order of magnitude. In this section we have been considering fuzzy orders of magnitudes on a

twiAimensional representation. Applied on materials selection charts the representation hecomes threedimensional as shown on Figure 4.7.

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/ Figure 4.7: 'Three-dimensional representation of fuzzy orders of magnitude.

4.2.3 An interface for acquiring new orders of magnitude A iiser interface is no1 critical to this project but could certainly help better specifying ranges or values looked for about material properties. This aspecl is not actually implemented but in this section insights are given about how t o define such an interface and in which way it ma? be useful.

'The issue of acquiring vague knowledge reprsented by means of fuzzy scts has already been taken up: see [Zemankova, 19891 in the data base setting and [Koning, 19901 for decision systems.

The orders of magnitude defined above are somewhat rigid. In a numbcr of cases this way of splitting a set of possible values into five fuzzy intervals may be too rough whereas sometimes it may lead to over-specification. The purpose of the interface is to let the user choose hisfher proper mode to define thc order of magnitude hefshe really needs. For instance, if some materials classes are concentrated on a tiny area of the scale measuring a property, then it is wise to split this area in more int,ervals than the rest of the scale. Orders of magnitude lou' and medium shown on Figure 4.5 cover a rather small space of valucs (compared to high). This discriminates among materials more efficiently if a lot of them are gathered in that same area.

Evjdently, sometimes i t is more relevant t o deal wit,h at most three different orders of magnitude when referring to a certain scale, sometimes at lcast t,en different orders of magnitude could be necessary.

32

very very low low medium high high

Figure 4.8: User defined orders of magnitude

A simple algorithm t,o define t,he various orders of magnitude could be:

1. Give the sequence of names of the desired orders of magnitude

2. Givc the two houndaries of the set of possible

3. Give the boundaries of each fuzzy orders of magnitude.

This last step could be performed fairly simply by answering questions like:

Abooe whrch value u muterial mould be considered as having definitely a h g h s t i f i e s s ?

h d e r which d u e a material .wou.ld be considered as nol hauiny a high. stiflness at all?

The answer to these two questions, for instance, would provide t h e lower bound of the core and the support respectively for the order of magnitude high attached to the materials property stiffness. These two questions are generic when the t,erms ‘high’ and ‘stiffness‘ are replaced by any order of magnitude and material property. As a matter of fact these two questions are sufficient to define all the fuzzy intervals (See Figure 4.3) since:

lorvcr bound (support, (n)) = upper bound (core (n-1)) lower bound (core (n)) = upper bound (support (n-1))

Only the two exlreme intervals are not regular t,rapezoids. The luwer (resp. upper) hound of the core and the support of the first (resp. lart) int,erval equal the minimum (resp. m;utimum) possible value.

The other way to perform step 3 of the algorithm is to specify what would be thc crisp thresholds of the orders of magnitude and let the interface compute their fuzzy houndaries via a multiplicative coefficient as explain in Section 4.2.2. That’s what is actually performed by our system.

33

4.3 Combining Matching Degrees for Materi- als Class Selection

4.3.1 The fuss? sets based representation in the system, support t,he following types of queries t,o the materials knowledge base:

Queries for accessing materials classes

1. Gnuen a materials class, what es the range of R possible properly?

The purpose of this query is simply t o ret,rieve the data associated with each material, e.g.: the range of densities associated with the class of engineering alloys. It is used to guess the intent behind the selection of some material in a given base case.

2. Gioen an order of rnagniinde of a property, whai are ihe corresponding material classes?

For example, one may ark for material classes with a very high streugt,h. This query can bc used to look for materials that are similar to a given property. The answer to this type of query is a set, possibly empty, of material classes with associated matching degree. The matching degree captures the extent to which the property of the materials class matches the specified order of magnitude. It is given by:

\.!'here A is t h e order of magnitude and R the property range for the marerials class IC. If the property range for the materials class overlaps the c.ore of the fucay set that represents the order of magnitude then the matching degree equals 1. If the property range is outside the support then the matching degree is 0: otherwise its value belongs to ( O ! 1) (see Figure 4.9)

Iu the answer to the query, the materials classes are ranked according to their matching degree. h,laterials close to the magnitude sought appear first. A matching degree of 0 implies the materials class is not suited at all and thus this class is not part of the answer.

3. Gioen a materials class, what is/are the urder/oniers of magnitude of R

particular property?

For instance, one may want to rctrieve the order of magnitude of thc fracture toughness for the class of elastomers. This query does not always give a single answer. Indeed, because of the fuzzy definition of the orders oC magnitudel the property of a materials class can be of several orders

34

p7, property . property * scale scale

)f,,(I<) = 1 pa(1i) = ;

Figure 4.9: Matching degree hetween a material clms I< and the order of rrrag- nit,ude A of some property.

of magnitude at the same time. Figure 4.10 shows in detail this aspect, where 4 . B and C are three different orders of magnitude and R is the property range for the materials class K .

Figure 4.10: Matching degree between a material class I< and the order of maguilude A of some other property.

Thc answer to this query is a list of orders of magnitude where each of them i s ranked according to its computed degree of matching. 'The example given on Figure 4.1U would lead to t,he answer { ( B , l ) , ( A , $),(C, +)}.

4. Gzven a design code: what am the relevant criteria one should take into cunsideration? The answer to such a query is a set of criteria that are optimal Cor materials selection of an artifact whose mode of loading is known. For example, for a light cylinder with internal pressure the t,hree important criteria are stiffness of type f , ductile strength of type and brit,tle strenglh of type y . This type of d a h is sine qua non to determine which materials selection charts t o use. The answer to this hype of query depends on the mode of loading of the artifact. It is given by Table 2.1

4.3.2 Combining matching degrees Once the important criteria that should be taken into considerat,ion have been found, the system looks for the materials classes that best fit each of t,hem. For instance, in the example given in Section 4.1.2 three crireria t,urn out to be important: bending resist,ance, torsion resistance and fracture resist,ance. In that caSe Ihe system retrieves (1 ) materials classes that possess at le& a medinm bending resistance: (2 ) materials classes that possess at least a medium Lorsion resistance and (3) materials classes thal possess at least a low fracture resistance. As seen in Section 4.3.1 this gives rise to three lists where materials c.lasses are ranked according to t,he extent to which they match the property requirement.

Aggregating these three lists can be performed in various ways. The st,rateg? should however meet intuitive characteristics. Some of them are dpscribed here after. See [Dubois and Prade, 19851 for a more general analysis on aggregation fnnctions.

1. A materials class that oompletely matches all t h e requirements should get the maximum aggregated matching degree. This statement can be conveyed as:

q ( K ) = l : , .. , Z " ( K ) = 1 3 z ( K ) = f(Zl(K), , . . , .c , ,(K)) = I! VI<.

\%'here q ( K ) is the matching degree of the materials class I< with the requirement, d (an example of requirement could be a very high stiflriess), z ( K ) is the aggregated matching degree, f is the aggregation function.

2. The antonym characteristic should also be true. The materials class that completely mismatches all the requirements should get 0 as the aggregated matching degree:

zs(1i) = 0: V i E 11.. . n) + z ( K ) = f ( ~ l ( X ) . . . . ,z,,(K)r)l = 0, VK

3. f does not depend upon the class of materials:

4. The aggregation function f should be defined for any possible combination of matching degrees:

VZI(K) , . . . , t , (K) E [ O , 11% 3 z ( K )

such that z ( K ) = f(rl(K), .... zn((K))

36

5. II‘ at lcast one single matching degree increases then lhe global (aggregated) matching degree should not decrease.

6. f should he continuous:

f(Zl(K), , . . , z j ( K ) : , . . , tn(l<)) The operation nain is a straightforward aggregation function meeting all

t,hese characterist,ics if one considers all the materials properties equally relevant. On the other hand, one may want to weight the material properties differenlly. The weight wd assigned to a material property i could be a value bclonging to [ O , 11 and conveying tu what extent the associated property is relevant, 1 meaning that the property is definitely relevant and 0 that it is not relevant at all.

Some material propert,ies may be crucial whereas others may have a srnallcr importance. The weighted minimum function reads [Dubok and Prade, 19861:

f(zl(Ii) . . . . , z n ( K ) , w l : . _ _ , wn) = ,min niax(wi,zi(I<)), where,min uii = 0

In this cme f becomes a weighted aggregation function.

,=I ... n *=I... n

In the example detailed in Section 4.3.3 we assume the set of properties is homogeneons. the aggregation function used is the operator min.

4.3.3 Example of material selection Tlie materials of mechanical and structural engineering fall into nine broad classes, cf. Chapter 2 . Each of t.hese classes is represented on the select,iun charts. In the example of the see-saw fur the water faucet. three charts are necessary to find ant the malerials that match the behavioral properties on bending, torsion and fracture resistance. Table 4.1 summarizes the results drawn from these t,liree charts.

Chart 1 guides selection for light stiff component. It allows finding that fix materials fully possess the required bending rcsistance. Engineering alloys, engineering ceramics, porous ceramics, engineering composites and woods all trdve a1 least a medium bending resistance. These materials get degree 1 in the first colnmn of ‘lahle 4.1. From the secund chart, that guides for light strong components, six classes of materials are retrieved as satisfying the torsion resistance property. see lhe second column. The class of engineering polymers doesn’t,

37

E. Alloys E. Polymers E. Ceramics P. Ceramics E. Composites Woods Elastoniers

1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

Table 4.1: Mat,erials satisfying the three behavioral properties.

completely meet the requirement of a t least a medium torsion resistance; it,s degree (1/3) therefore belongs to (0, l). The third chart guides for light fracture- resistant c.omponent,s. The degrees found in each column are aggregated via the operator min. This conveys an intersection between the classes of materials found from each chart. The classes that best meet the requirements get the degree 1. Engineering composites and woods are the t,wo preferred classes. 0 means the corresponding class is definitely rejected. One can see that t he clavv or engineering alloys is not totally rejected (degree 4 / 5 > 0).

Let us siippose one intends to manufacture the artifact in great number. One may want to add a property concerning its cost while designing it. From the chart that guides the selection for cheap stiff components (cf. Section A.4), it. appears that the two best classes of materials are poroiis ceramics and engineering alloys and t o a lesser extent woods (degree 1/2), If this property is equally important t,o the three previous on=: the class of woods gets a grade of 1/2 (mini 1/2,1)) while engineering alloys category gets 4/5 (min{ 1.4/5}) and thns becomes the preferred class of material. If all the properties are not equally important, one may assign t o them some weight and even some fuzzy weight conveying their degree of importance. The aggregation operat,ion then would become a weighted min [Dubois and h a d e , 19881.

38

Chapter 5

Description of the Program

The whole code has been written in Common Lisp [Steel Jr., 1990] on a VAX station.

The program is divided in two parts, One containirrg all the data definitions that will constitute the knowledge base and the other part contains all the functions that create and access the knowledge base.

5.1 Data Structures A nahural way to program our type of application and in particular. to set up our kind of knowledge base is to rely heavily on object-oriented programming. I could have used Knowledge Craft, but this choice would not have been in favor of easy portability since Knowledge Craft is a very large/expensive system that is not so widespread. I could have used Common Lisp Object SysWrn that emerges as the standard for object-oriented programming in Common Lisp, hut again not. every version of Common Lisp integrate this package yet. Therefore, I chose to work only with structures and defsfructs functions [see Chapter 19 of [Steel J r . , 19901) which is not really penalizing for the application.

5.1.1 Mechanical and engineering properties Materials properties definitions are stored under a global variable called *property-descriptorst that is actually a vector ofstructures of type property-descriptor The slots for such a structure are:

name: name of the material property (e&, density)

description: a string describing t,he property.

p-symbol: the propert,y symbol (e.g., p )

39

unit: the appropriate unil for measuring the property (e.g., M g / m ” ) .

thresholds: the crisp thresholds t,hat, lead to dcfine the fuzzy ones? cf. Scc- tion 4.2.2

’I‘he value this last slot gets is an instance of structure of type thresholds chat contains the four crisp thresholds t-1, t-2, t -3 and t-4 t,hat. separates between the five fuzzy orders of magnitude, plus the two boundaries for the properly called t-min and t-max. For instance, the property density would gct the thresholds:

t-min 0.1 t-1 0.3 t-2 1 t-3 3 t-4 10 t-max 30

Once all the materials properties have been defined a special structure called properties is created with the name of the defined properties as its slot names. This structure is used to define the materials classes.

5.1.2 Materials classes The various types of materials classes are represented as a hierarchy of structures The root of this hierarchy is the structure materials. Under this root I have defined five structures corresponding to the five general classes of materials metals, polymers, ceramics and composites. The leaves of this hierarchy are the subclasses:

engineering alloys that belongs to the class of metals,

s engineering polymerss! polymer foams and elustoners that belong t o the class of polymers.

engin.eering cemmics and porous ceramics that belong t o the class of ceramics,

engineering composites and moods that belong to the class of composites. The woods subclass is itself divided into three sub-subclasses wood parnilel l o grazn, mood perpendicular l o grain and wood products.

The root structure is a generic structure with only one slot that will capture an instance of the structurc properties where its slot names are the names of the materials propert,ies defined (see Section 5.1.1). Each of these slots are instanLiated with a pair (structure range) of values representing the lower and upper bound of that property for the corresponding materials class. Fix example, the struclure engineering-polymers will look like:

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Engineering Polymers Density

lower bound: - upper bound: -

lower bound: - upper bound: -

Fracture Toughness

A l l tile materials classes are stored under a global variable. This variable IS called materials-sequence*. It is a vector of the structures defining t h e various materials classes.

5.1.3 Modes of loading The information displayed in Table 2.1 are stored in a global variable called *modes-of -loading* that is actually a structure of nine slots corresponding t,o the nine different modes of loading. The value attached l o each slot is an instance of structure loading-mode-properties that has itself three dots:

4 stiffness: the type of stiffness that should be Laken into account given the mode of loading like E/p, E * / p , etc.

0 ductile strength: the 1.ype of ductile strength like u y l p , u:/pl etc.

a brittle drength: the type of brittle strength, like U ; c / p

"I'he different kinds of stiffness, ductile strength and brittle strength h a w heen defined as regular properties with generic names like st'fnessl for E / p : sfifness.9 for Et/,, etc.

5.1.4 Compiled data The data stored in the global variable *materials-sequence+ are not directly used by the system. As explained in Scction 3.2.2 they are first compiled into sequence3 of structures containing four slots:

change-value: a value of t,he property,

8 m-less: list of materials classes whose property range is situated at least part.ially before "changevalue",

e m-equal: list of materials cl,zsses whwe property range overlaps "changevalue''

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m-greater: list of materials classes whose property range is situated at least partially after “changcvalue”.

Such a structure is called change-point because i t refers to a point where at least one of the lists changes from a previous point,. All the sequences of change points are stored in the global variable *change-points-sequences+. This global variable is in fact an instance of the structure properties where each d o t (called after a property name) is bound to the corrcsponding sequence of change points of that propcrty.

5.2 Program

5.2.1 Data coherence verification and knowledge base ini- tialization

A lot of information tnay be stored in the data base. Currently we take into consideration 9 materials classes and 24 properties (see Appendix D ) . For each property of a materials class two values are provided (the lower and upper bounds). When these two values are defined one wants t o make sure that the ripper bound is greater than the lower bound. This type of checking is performed because inistakes are easy to do when typing in the data.

The second type of coherence checking deals with the dcfinition of the orders of magnitude. One wants to avoid the situation shown on Figure 4.4. As said in Section 4.2.2 the boundaries of the core and the support of the fiizzy orders of magnitude are 20% greater or lower than the crisp bounds. If a and b are two successive crisp thresholds one checks that b - p is always greater than a + $ so that the interval [a + g , t - k] representing the core of the fuzzy order of magnirude is non empty (see Figure 5.1).

t

Figure 5.1: Coherence checking on fuzzy orders of magnitude

Initializing the data base ronsists in filling thc structure which contains the lists of change points sequences (Le., *change-points-sequences+), onc per material property. In the beginning the sequences are empty vcctors. For each

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lower bound or upper bound of a property an instance of the change-point structure is rreated with this value as the change point value. The slot m-equal is hound to a list containing the material the value refers to. Once all the change point,s have been generated the sequence is sorted in increasing order of thc \ d u e ? then Ihe slots m-less, m-equal and m-greater are updated according t,o the malerials preceding, overlapping or following the change point value. Two change-point structures are added at bot,h ends of each sequence. The first one gets only its slot la-greater updated and the last one only its slot m-less. If two different change points appear t o have the same change-point value in a scquence than they arc merged into one.

5.2.2 Querying the data base 'The different types of queries are given in Section 4.3.1.

The first type of query is coded in a straightforward way. The two boundaries are looked up in the global variable *materials-sequence+.

The answer to the second type of query is built from three partial solutions: (1) the list of materials that overlap the core of the order of magnitude, (2) the list, of materials whose property range begins between the core's upper bound or the order of magnitude and its support's upper-bound, and (3) the list of materials whose propert,y range ends between the support's lower bound ol the order of magnitude and the corc's lower hound. These three lists are derived form the sequence of change points for t h e property one is interested in. More specifically, they are derived from the change points whose values are between the lower and upper bound of the support of the order of magnitude. These change points are easy to locate since they are sorted in increasing order of their values.

A matching degree is attached t,o each material. In the first caSe (in the first l i d ) the matching degree is always 1. In the second chse it is calculated via the formula where ia is the value of the core's upper bound, b the value of the support's upper hound and u the lower bound of the property range (the value of t,he corresponding change point). In the third case the degree of matching is calculated via the same formnla where a is the support's lower bound, b the core's upper-bound and v the upper bound of the property range. The three lists are merged into one that is sorted according to bhe matching degree attached to each material.

'The answer to t,he third type of query is a list of orders of magnitude where Lhe attached matching degrees have been calculated in exactly the same way as explained ahovc.

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Chapter 6

Conclusion

The approach presented here lies on the concept of materials sclection charts created by Ashby [Ashby, 19911. AII implementation of these charts has been suggested as well as a way t o exploit them via queries based on fuzzy orders of magnit.ude. This work has led to the development of an autonomous system that can be hooked up t o more general systems. The initial purpose of this syslem was to be implemented in bhe framework of the casebased design project CADET [Sycara et al., 19921.

Casebased design stems from the general theme of case-based problem solving whose philosophy is l o make use of pas1 experiences in solving new prohlems, exploiting its memories instead of relying solely on a base of proredures or rules. I n other words, a casebased systcm is not only an expert or multi-expert system but also a system that takes into account similar cases already encount,ered.

The CADET system has been conceived for the field of mechanical design, its intent is to play the role of a designer’s assistant by retrieving and re-using previous succasful designs while avoiding previous failures such as poor material8 or high cost.

The process of case-based design consists of several steps that are not ner- essarily strictly applied in t,he order given here; some may overlap.

1. Development of u Functional Description. At the simplest level, the desired artifact. c a l ~ be viewed as a black box which takes cert,ain inputs and produces desired outputs. The functiori of the black box is described by qualitalive relations explainiug how the inputs and outputs are related. ‘The system’s job is to realize an artifact which will convert the inputs into the desired outputs.

2. Retrieval of Cases. A set of design cases (or case parts) bearing similarity l o a given collection of features are accessed and retrieved. Retrieval is performed using not only the existing features of t.he input specification and also behavior description.

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3. Dcl;elopment of a Synthesis Strategy. A synthesk strategy is a description of how the various c a e s and case pieces will fit together to yield a working design.

4. Physzcal Synthesis. Realization of the synthesis strategy at a physical Icvcl. This is a difficult problem since undesirable inleractions among case parts may occur. In addition, since it k very rare to retricve cases that m a d l y match the design specifications, cases and case pieces must he physically modified before actual synthesis.

5. Verification. Advcrse interactions could lead t o nou-conformance of the desigo to the desired specifications. This is verified through quantitative and qual i thve simulation. If the simulation is correct, and if all the const,raints are satisfied! then the dmign is successful. If not, repair (next step) is attempted.

6 . Debugging. Debugging involves a process of asking relevant questions and modifying them based on causal explanation of the bug.

So far. CADET implements the first four steps indicated above. Verification and debugging is currently left to the human designer.

Once a cave has been retrieved, it, may need some adaptation to become part of the whole artifact. This adaptat,iou phase can be seen as twofold. First, the case itself may need to be modified to suit the features of the output specifications. ‘lhis s k p only brings into p h y the intrinsic characteristics of the case. Secondly, due to the interact,ions between the (possibly already modified) case and its new environment, other adaptations may be needed. This second step intervenes at a lower level of abstraction right before the actual synthesis.

The tcrm adaptation has a broad meaning. In this report we have focused on material adaptation and more precisely on the one that first takes place. Mate- rial modification due t o interaction with ot,her devices has not been considered here.

CADET’S architecture [Sycara et al., 19911 is centered around a design blackboard. This blackboard is used to maintain information about the various design alternatives that are being actually considered at any given time. The materials selection system is consulted during case selection to check material requirements and during case adaptation.

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Acknowledgements

1 wish to acknowledge:

Katia Sycara and D. Kavinchandra who have contributed many ideas and substantial time to the creation of the material selection system prescnt,ed here.

Mark Fox who accepted me as Visiting Scientist at the Center for Inte- grated Manufacturing Decision Systems.

Michael hshby who authorized the reproduction of his charts

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Appendix A

Material Selection Charts

This appendix gives all the material selection charts we are using in our system. The figures and text presented below are taken from [Ashby, 19911.

A . l Materials Selection Chart for Light Stiff Components

This chart is shown on Figure 2.1. Its three guide lines are given on Table 2.1:

1. b,'/p = C: criterion for axial tension of t ies

2 . E i / p = C: criterion for bending, torsion, or buckling of beams, shack and c.oliirnns;

3. E f j p = C: criterion for bending of plates

h,lat,erials offering t,he greatest stiffness-to-weight, ratio lie towards t,he upper left corner.

A.2 Materials Selection Chart for Light Strong Components

On Figure h.1 the property of strength, whose symbol is uv , is plotted against density. Let us n0t.e that for metals and polymers the strength referred to is called yield strength, Cor ceramics and glass- i t is called compressive strength. for e la tomers it is called t,ensile tear strength, and for composita tensile failure:.

'Ihe three guide lines are given by:

1. u Y / p = C: criterion for plastic failure of ties,

47

1 2. uy’ J p = C : criterion for plastic bending, torsion of beams and shafts,

1 3. u$ J p = C: criterion for plastic bending of plates

Figure A. l : Material selection chart for light strong components (courtesy).

Materials offering the greatest strength-to-weight ratio lie towards the upper left comer.

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A.3 Materials Selection Chart for Light Fkacture- Resistant Components

On Figure A.2 the property of fracture toughness, whose symbol is lirc. is plotkd against density. The three guide lines are given by:

1. I i rClp = C: criterion for brittle failure of ties,

2 . Iir+Jp = C: criterion for brittle failure of beams and shafts,

3. Ii;c/p= C: critcrion for brittle failure of plate8 I

Materials offering the greatest, t.oughness-to-weight ratio lie towards the upper left corner.

A.4 Materials Selection Chart for Cheap, Stiff Components

On Figure A.3 the property of young’s modulus is plot,ted against, relative cost. This lat,ter property, whose symbol is CR, is calculated by:

cost per unit weight of the material cost per unit weight of mild steel

Cn =

The three guide lines are given by:

1. E/Cjrp = C: criterion for axial tension of ties,

2. Ef/C,p = C: criterion for bending, torsion or buckling of beams, shafts and columns,

3. E k l C ~ p = C: criterion for bending of plates

Mat,erials offering the greatest stiffness per unit cost lie towards the upper left corner.

A.5 Materials Selection Chart for Cheap, Strong Components

On Figure A.4 the property of strength is plotted against relative cost. See the different meanings of strength in Section A.2.

The three guide lines arc given by:

1. u y / C ~ p = C : criterion for axial tension of ties,

49

2. o;/C,p = C: criterion for bending, torsion or buckling of beams: shafts and columns,

3. ui /Cnp = C: criteriou for bending OC plates,

Matcrials offering the greatest strength per unit cost lie towards thc upper lcft corner.

A.6 Materials Selection Chart for Components Resistant to Corrosion

The chart given on Figure A.5 indicates where environmental problems may exist. I1 shows the response of each group of materials to prolongcd exposure, undcr normal working loads? to six environments: clean but aerated water, salt materl acids alkalis, organic solvents and direct sunlight.

'Chose in the inner D-zone are rapidly attacked. The radius can be seen ay a scale conveying the time of exposure to the environment before corrosion. The outer circle represents time 0 and the cent,er represents an infinke time.

Materials which lie in the outer A-zone have excellent resistance.

50

Figure 11.2: Material selection chart for light fracture-resistant components (courtesy).

51

Figure A.3: Material selection chart for cheap, stiff components (courtesy).

52

RELATIVE COST PER UNIT VOLUME C,p (Mg/rn’)

Figure A.4: Material selection chart for cheap, strong components (courtesy)

53

Figure A.5: Material selection chart for components resistant to corrosion (courtesy).

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Appendix B

Materials Classes and Properties

This appendix summarizes the materials classes and the mechanical and en- gincering properties that arc currently exploited by the syst,em (taken from [Ashby. 19911).

B.l Material Classes

Enginerring Allows : The met,als and alloys of engineering.

Engineering Polymers ; The thermoplastics and thermosets of engineering.

Engineering Ceramics : Fine ceramics capable of load-bearing applicalions.

Engineering Composites : The composites of engineering pract,ice

Porous Ceramics : Traditional ceramics cements, rocks and minerals.

Glasses ~ Silicatc glass and silica itself.

Woods : Separate envelope3 dexribe propdies parallel to t,he grain arid normal to it, and wood products.

E l a s t o m e r s : Natural and artificial rubbers.

P o l y m e r Foams : Foamed polymers of engineering.

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B.2 Material Properties Derivity P Young’s Modulus E Strength “f Fracture Toughness I ~ I C Stiffness Elp , E i l p , E: J p

Ductile Strength “ J l P ! “ : / P , u: /P

Brittle Strengt,h I i r c l p , x , ~ c l p > ii,&Jp Relative Cost CR Resistance Lo Aerated Water Resistance to Salt, Water Resistance to Strong Acids Resistance to Strong Alkalis Resistance to Organic Solvent Resistance to U-V Radiation

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