Automating extrusion design: a case study in geometric and topological reasoning for mechanical design

Automating extrusion design: a case study in geometric and topological reasoning for mechanical design M R Duffey and J R Dixon A model for topological and geometric reasoning about mechanical designs is described and demonstrated by a computer program using the 2D domain of extrusion cross-sections as a case study. The design system uses a feature-based representation and has two stages: first, parametric design, in which a numerical, iterative technique is used to vary existing parameters and second, topological design (the central focus of this research), in which the extrusion cross-section is modified by adding new parameters and~or deleting old ones. This paper summarizes recent research related to topological design, then describes representation and reasoning as modelled in the computer program, presents some test cases, and outlines future directions for geometric and topological reasoning in mechanical design domains. computer-aided design, mechanicaldesign, extrusions, topologk: reasoning,geometricreasoning

Mechanical analysis, in particular the structural analysis of beams, is a long-established discipline based on firm, well-understood physical principles. The mechanical design process, however, remains something less than a science. The human designer, especially in the preliminary or conceptual stages of the design, tends to use knowledge in a way that is difficult to articulate and systematically define, let alone transform into a computational model for automatic design. This is particularly true of the topological and geometric reasoning processes used for the design of physical objects. Consider the following simplified model of a design process for an extruded beam cross-section (see Figure l(a)). It begins by an examination of the performance specifications (maximum stress, deflection, etc.) and existing geometric constraints (mating surfaces,

/'°'aOll. t Mat'ing" ~ L surface a

b

c

d

Figure 7. Modelling the extrusion design process points of load application, and 'forbidden area' in Figure l(b)). From this, a loosely-defined topology for the design is chosen based on the geometric constraints, the designer's knowledge of manufacturing constraints, and the relationships between cross-section properties and performance specifications (Figure 1 (c)). Then, with the basic topology more or less fixed in the designer's mind, numerical values for the selected parameters (wall thicknesses, web and flange lengths etc.) are chosen and the design is analysed, using either elementary beam equations or a more sophisticated numerical analysis (Figure l(d)). All or part of this process is iterated, if necessary, until a satisfactory solution is found. Only a limited portion of the above-mentioned design process can be automated using existing computational design tools. What can be automated the routine or parametric stage of this design cycle can be done well, and continues to be the focus of much research; optimization and other iterative techniques that use analysis results are available that can vary the established parameters of a fully dimensioned design to achieve, within a limited portion of the design space, optima for single or multiple criteria objectives. However, very little research has been done on the preliminary or conceptual stages in which the design topology is created and modified.

OVERVIEW OF DESIGN SYSTEM An earlier version of this paper was published in the Proceedings of the American Societyof Mechanical Engineers(ASME)Computers in Engineering Conference, San Francisco, CA, USA (July 31August 4, 1988) Mechanical DesignAutomation Laboratory,Mechanical Engineering Department, University of Massachusetts,Amherst, MA 01003,USA

volume 20 number 10 december 1988

This paper describes an approach to geometric and topological design in the fairly simple 2D domain of extrusion cross-sections as implemented in a working computer program. As seen in Figure 2, redesign of the extrusions is done in two parts. The Parametric design

0010-4485/88/100589-08 $03.00 ~, 1988 Butterworth & Co (Publishers) Ltd

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IMpllCl! (-]()malr/ Krlowleoge ~.e.g. nlovin,~ material away from the neutrai axi~ increases the moment of inertia). A robust system Ior generating topological design~ will probably require a large and carefully defined set of such topologica! operators. Only a small set was constructed fo~ the implementation described here ~,()nlaln

-InuUaldesugr:tOpologlca; [ [Select p a r a m e t r i c 1 _-._..._i~[ design variables Parametric design (Dominic II)

1

No

Store clesigninStantiated

No

Select best ] • nstantiated design I (end program) J

~,Yes

- - TOPOLOGICAL REDESIGN

Figure 2. Control structure tor extrusion design

module is used to converge on the best numerical values for existing parameters of a given topology. The Topological redesign module, the central focus of this paper, generates redesigns of the essential shape of the cross-section, in which the connectivity of existing primitives is redefined or new primitives are added. Operationally, this involves adding or deleting the 'slots' that define the attributes of objects in the extrusion data structure. Topological redesign in the program is characterized by the following: • Feature representation (~- The extrusion cross-section is composed of both lower-level microfeatures and higher-level features which provide information about the extrusion as a whole. For example, the individual walls which comprise the extrusion are contained within a rectangular envelope, which provides information about the height-to-width ratio and relative locations of the walls. • Functional and spatial context of features o The features are used to relate the shape to the performance specifications and the external geometric constraints. To do this, a set of predicates is required that assess the connectivity and context of the features (e.g. assessing potential lengthening of walls relative to the location of a forbidden-area). • Topological operators o A set of topological operators uses the above predicates to identify specific configurations of extrusion walls, and associates them with specific transformations of the topology. These operators

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Topological redesign is incorporated into the larger design program shown in Figure 2. The control structure of the program can be outlined as follows: • Step 1 : get beam specifications from the user: length, support type, load magnitude and location, locations of external geometric features (mating surfaces, forbidden areas, etc.!. • Step 2: construct an initial topology for the beam cross-section using a modified shortest path algorithm between load and mating surface. • Step 3: select parametric design variables from among the attributes that describe the topology (e.g. lengths and thicknesses of webs and flanges). • Step 4: test the topology for feasibility; i.e. set the chosen parametric design variables to their maximum value (limited by manufacturing) and analyse the design for stress and deflection. If these are within specified limits, then the topology is a feasible one. • Step 5: if the topology is feasible, then perform parametric design to determine optimum or near optimum values for the current parametric design variables. Store the resulting instantiated design. • Step 6: if there are topological changes that can be made which are expected to improve the design performance, then generate a new topology. • Step 7: repeat steps 3-6. • Step 8: select the best instantiated design from the set of stored designs. 1here are a number of design and analysis limitations in the program. The elementary beam analysis is for stress and deflection due to bending only (i.e. load application through the shear centre), and does not consider torsional effects or local shear and buckling. New walls can only be built at right angles to existing walls, and only open sections are allowed (a project to relax these restrictions is in progress). The rest of this paper will summarize related research, describe representation and reasoning in the program, present some test cases, and discuss future directions for geometric and topological reasoning in mechanical design.

BACKGROUND While optimization techniques for parametric design of load-bearing structures have been widely researched, there has been little work in computational modelling for the preliminary design of structural topologies. Shah 1 presented a hybrid system for preliminary design that identifies good candidate structures for numerical optimization. His 'shape algebra' approach represents load-bearing structures as composites of straight line and polygonal elements, and interprets these elements

computer-aided design

as symbols for algebraic manipulation. An algorithmic technique can be used to generate a large number of configurations of these elements, which are then culled using heuristics for load-support paths, stress patterns and other structural principles. Individual elements can be designed for shape with a separate set of heuristics. Nevill and Paul2'3 have reported their work with Mosaic, a program for the preliminary design of 2D load-bearing structures composed of rods, columns and beams. Using features to represent supports, forces and forbidden areas, the physical design space is decomposed into subregions of 'feature clusters'. A partial solution is arrived at by using heuristics to locate supports within these feature clusters. The use of high-level 'macro' features (e.g. regions containing clusters of lower-level features such as beams and columns) is considered important to spatial reasoning in this domain. Fenves and Baker4 have proposed adapting the 'shape grammars' developed by Gips and Stiny s for spatial and functional reasoning in preliminary structural design. Shape grammars are context-free grammars that provide a language for the recursive generation of spatial elements in a design. It has been suggested that design 'semantics' might be used to describe shapes in terms of function or some other characterization, and provide rules for the generation of new designs in architectural and engineering applications 6. The approach of Fenves and Baker incorporates a 'syntax' for both structural and architectural attributes in a designgenerating mechanism similar to a production system. Rules are used to generate complex objects from simple objects using either structural or architectural transformations based on their respective grammars. As part of their formalism, they have defined some terminology relating to physical representation that is useful for our research. Topology defines the connectivity of design primitives. Geometry defines the numeric properties that describe a design primitive in space: dimensions, location, orientation, etc. (The common nomenclature, however, applies the term 'geometric reasoning' to both geometry and topology.) Fenves and Baker have also emphasized a distinction between 'preliminary' and 'detailed' design which has implications for representation: It is the purpose of preliminary design to determine the topology and geometry, and to develop estimates of member properties. For detailed design, when the topology and geometry of the structure have been determined, analysis tools must be used iteratively so that the preliminary values of member properties may converge to acceptable values satisfying strength and serviceability requirements. Features research at the University of Massachusetts has focused on the representation of geometric primitives for interactive, menu-driven interfaces in design-formanufacturing applications. Libardi 7 developed such a system for extrusion design (named Pedro) which provided a foundation for the work described in this paper. In Pedro, the user constructs the extrusion cross-section by a series of add, modify and delete

volume 20 number 10 december 1988

operations on primitive intersection, fillet and wall features. By specifying load and support locations along the beam, as well as requisite physical properties, Pedro can then generate a finite element mesh for analysis of beam stresses and deflections. Other similar programs for casting8and injection-moulding 9 have been developed which use features representations for manufacturing evaluation. Dixon et a/. have worked on a program called Dominic 1°, for domain-independent, parametric (i.e. 'detailed') design. Dominic does not use symbolic representations of geometry and reasons only with numerical values for design variables and problem specifications. It has been successfully used in a number of domains involving parametric variation of design geometry, including extruded heat sinks, solar heating systems and post and beam structures. The types of problems that can be addressed are similar to those for structural optimization programs. A recent version ~1, Dominic II, was adapted for the Parametric design module of the program described in this paper.

REPRESENTATION AND REASONING ABOUT EXTRUSION TOPOLOGIES The overall design goal for the simplified extrusion domain described here is to meet specifications for stress o" and ~ due to bending and manufacturing constraints with the minimum amount of material (i.e. minimum cross-sectional area). Basic physical principles were used to derive the heuristic domain knowledge used by topological operators. From elementary beam theory, the stress and deflection can be related to the distribution of material in the crosssection via the moment of inertia about the neutral axis Io such that

6h~o~,~goc1 ~In

and

o'u~d,ngoc c/In

where c = outermost fibre distance from neutral axis. To achieve a minimum-area design, one should locate as much material as possible away from the neutral axis to increase the moment of inertia. This heuristic relates functional and spatial attributes of the extrusion and is implicitly contained in the set of topological operators used for redesign in the program.

Representation The design problem is represented by: • microfeatures (same as those used by Pedro)(see Figure 3) • specifications-and-constraints • macrofeatures • external-geometry Microfeatures, as adapted from Pedro 7, consist of walls and intersections. For the C-channel shown in Figure 3, the representations for a wall and intersection would be:

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\ 0 . [?

Intersection 4 Wall 3

Wall 2

\

Intersection 1 Wall 1

Figure 3. Wall and intersection microfeatures trom Pedro

Intersection3: meet-at (wall2 w a l l 3 ) location (1 4)

Wall2 : flanked-by:(intersection2 intersection3) thickness: 0.4

Walls and intersections also have perim-points properties which together define the perimeter of the crosssection. Using this features database, cross-section properties such as moments of inertia, area, outermost fibre distances, etc. are computed automatically. (See Libardi et al.7 for an in-depth description of the extrusion microfeatures.) Other parts of the representation are similarly defined in record structures using Lisp's defstruct function. The specifications-and-constraints structure contains information about the design not tied to a particular geometric configuration. This includes beam specifications (maximum stress and deflection, modulus of elasticity, beam length, etc.) and manufacturing constraints (minimum and maximum wall thicknesses, allowed wall thickness ratio, etc.). Macrofeatures are attributes of the topology which relate to the extrusion as a whole. For example, the envelope describes the dimensions of the smallest rectangle enclosing the extrusion walls. The external-geometry consists of simple primitives for the load-points, mating-surfaces (these are not supports, but simply adjacent surfaces), forbidden-areas and outer-boundary (an exterior forbidden-area that defines the perimeter of the physical design space (see Figure l(b)). These contain simple points and lines data and provide the external geometric constraints in the physical design space. Functional and spatial context of features The functional and spatial context of the features in the design space must be assessed to provide a basis for reasoning about topology. To do this, a set of predicates was created that returns properties of the

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features relevant to a specihc context, lhe~e propertie.s change continually as the design is nlodified. The predicates are based on simple geometm operations for walls, intersections and external-geometry, and include the following: • Wall-sight o The distance between an existing or proposed wall and external objects in the direction of potential movement is a consideration in topological modification. Two simple operators, side-wallsight and end-wall-sight, are used to identify external objects and their distance from the side and end of a given wall respectively. This facilitates both the choice of location for adding new webs and flanges, and the maximum allowed values for parametric design variables. • Envelope-position c The position of a wall with respect to other walls within the envelope is useful: walls which are in the 'interior' of the envelope are candidates for repositioning to the outer edge of the envelope. The aspect-ratio (envelope height/envelope width) is one indicator of how and where to add material: for example, a low aspect-ratio suggests possible improvement by extension of an existing web. • Attach-lines © Lines of contact between walls and mating surfaces or loads. Walls which directly support a load or provide a mating surface have constricted motion, and cannot be deleted or moved without providing a suitable replacement support. • Web-or-Flange © Identifying a wall as a web or a flange (i.e. the angle relative to the load vector) aids in reasoning about changes in length, position, etc. For instance, it is desirable to move a flange (i.e. horizontal wall) from the interior to the outside of the envelope when possible.

Topological o p e r a t o r s An analogy with rule-based production systems is useful to understand the utility of topological operators: by using the predicates described in the previous section, 'left-hand side' conditions are formed which provide a mechanism for the identification of subsets of the extrusion topology, and associated transformations. The topological operators have two purposes: • Generating new topologies from existing designs (e.g. selecting location to add new wall(s), moving walls from interior to exterior of the envelope etc./in the Topological redesign module. • Identifying parametric design variables for a given topology. Topological operators assess the connectivity of features and their context for a specific configuration. The context is critical; the 'shape' of the extrusion alone is insufficient to guide the transformation. Figure 4 shows

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Initial

topology

a

Context

Transformation

Unconstrained web and flange extension above and to left

Parametric :

no change in topology

Constrained short flange and web

Minor-top©logical: add new web and flange

b

Flange at outer

d

add new flange

lengthened

Constrained long flange and web, but free space above flange to move material

Parametric design variables

Lengths o f e x i s t i n g web and flange

Lengths of new web and flange

Minor-top©logical:

boundary limit; web cannot be

C

Type of change

Lengths of new and existing flanges

Major-top©logical: add new web and

Height of outer flange, change flange (simultaneous connectivity between changes in length of existing web and the two webs} flange

E

Figure 4. Different transformations for web-flange portion of extrusion topology

four cases where similar portions of topologies are transformed differently as a result of context. The example topology is an L-shaped flange and web. The remainder of the extrusion is not shown. Figure 4(a) shows a section of extrusion unconstrained by mating surface, outer-boundary etc. There is no topological change associated with this transformation, but two parametric design variables are identified by the topological operator (wall thickness variables for the cross-section are assigned once the topology is complete). In Figures 4(b) and 4(c) there are minortopological changes identified by the topological operator, and in Figure 4(d) a major-top©logical change is identified. Minor-top©logical changes are defined as those in which two or fewer walls are added or deleted to the existing topology. Major-top©logical changes are those in which two or more walls are added or deleted and/or the connectivity of existing walls is altered. As mentioned, topological operators are formed by interrogating the features representation with predicates such as those previously described. For example, the major topological change from Figure 5(a) to 5(c) is effected by the following: • As part of the implicit goal to move material to the outside of the envelope, candidate flanges in the interior of the envelope are identified with envelopeposition.

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• Each candidate flange is then tested with the subset of topological operators that effect the repositioning of existing flange material. In this case, the flange configuration is matched and transformed by the following operator: if

o The aspect-ratio of the envelope is greater than 0.4 (an arbitrary value). o There is only one web attached at each end. © One of the two webs is attached to a mating surface (mate-web). © The other end of the mate-web (mate-end) is at the top or bottom of the envelope. o The remaining web (free-web) is extended in the opposite direction from the mate-web. © There are no conflicts with external geometry at the outer edge of the envelope where the flange would be moved. then

© Reconnect the flange at the opposite end of the mate-web. © Move the intersection connecting the flange and free-web to the appropriate location at the outside of the envelope. © Propagate these changes to the rest of the features

593

a

b

c

before trying maff)r-topological changes. Operators are recursively applied only for the initial topology and subsequent major topological variations. The simple heuristic used to guide the transformation~ t i.e. moving material to the outside of the envelope to increase moment of inertia) does not necessarily guarantee a 'better' design. At this point, the authors have no generally valid way to compare different topologies except by comparing instantiations after the parameters have been assigned. TEST C A S E S

Area [in21

0.83

~.157

"~bending (psi) (in} °bending

355.4

320.1

0.0~

0.0a

d

e

Area(in 2)

0.695

0.691

c bending (psi) g (in) bending

327.9 0.04

327.5 0.0a

Figure 5. Recursive generation of topologies for Case C. Performance specs are given for instantiated designs representation (e.g. construct a new set ot perimeter-points for the extrusion walls). o Assign a parametric design variable for changing the height of the flange. Note that, in all cases, the topological operators identify variables for parametric design. These design variables are either individual or combinations of microfeature properties (e.g. varying the lengths of flanges in the outer zones, simultaneously lengthening two webs connected by a flange, varying the thickness of a set of flanges etc.). Once all the variables are selected, a dependency order list is created (by preset priorities for variable type) to provide a general guide for the Parametric design module about the relationships between design variables and performance specifications (see Orelup et al. 1~for a discussion of dependency order lists in Dominic II).

By varying the positions of mating surface, forbidden area and applied load, the program is capable of generating designs for different test cases in its simple domain. The designs produced are generally acceptable, but not necessarily optimal. Three test cases are discussed here. Each extrusion design was analysed as a simply-supported beam using material properties, beam specs and some simple manufacturing constraints suitable for polycarbonate resins 12 (Table 1). Figure 6(a) shows the external geometry and initial topology for Case A. This topology was generated by the Initial design module which connects the matingsurface to the load-point. As described in the flow chart in Figure ,2, .the topology is then tested for feasibility. This is done by setting selected parametric design variables to their maximum allowable values consistent with manufacturing constraints, and then performing simple beam analysis for stress and deflection. (The load is a point load midway along a simply-supported beam. Torsion is not considered in the analysis.) In this case, three parametric design variables are chosen by the program: • thickness of flange • thickness of web • length of flange Table 1. Beam specifications for test cases

P L E ~ ~m~x t .... t ...... t~,,'tm,o

Applied load Beam length Flexural modulus Yield strength Deflection Maximum wall thickness Minimum wall thickness Allowable a

b

10 Ibs 48 in 340 000 psi 6 000 psi 0.04 in 0.2 in 0.04 in 2.0

c

d

1.549

0.491

G e n e r a t i o n of topologies

Developing a set of topological operators for a robust design system - particularly one with multiple design goals - will involve (at least) the same complexities as any large scale production system (e.g. consistency, redundancy and completeness of the rule set). For the program described in this paper, a very simple control scheme was used to generate acceptable design topologies. Only one topological change is 'fired' at a time, and minor-topological changes are attempted first

594

Area (in 2 ) ¢ b e n d i n g (psi) 5 bending (in)

337.7

199.6

0.04

0.04

figure 6. Topological and parametric designs t'or Case A

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As shown in Figure 6(b), the wall thicknesses are then set to their maximum (as defined by a manufacturing constraint), the flange is extended to its maximum at the outer-boundary, and the design is analysed. Since the deflection is within specification, this topology is feasible and a parametric design is performed that results in the design shown in Figure 6(c). After this, there is only one topological operator that matches this topology: the addition of a flange at the unattached end of the web. This topology is also feasible and a second parametric design is produced (Figure 6(d)). Since costs of extrusion are directly related to weight, the area is a good indicator of design quality: the lower the area the better. (Other than meeting wall thickness constraints, manufacturability evaluations are not considered.) Thus Figure 6(d) shows the best design found by the program. Figure 7 shows a slightly more complicated example. The initial topology (Figure 7(a)) fails the feasibility test, as does a redesign in which one flange is added (Figure 7(b)). A flange and a web are then added to the initial topology to produce a feasible one (Figure 7(c)). Note that Figures 7(b) and 7(c) are both minor-topological changes to the initial topology. The following major-topological change to the initial topology is also infeasible (Figure 7(d)), but the final two minor-topological variations (Figures 7(e) and 7(f)) allow for parametric design. It may be noted that there is a design not found by the program which is better than Figure 7(f) (Figure 8). The reason the program missed this improved design is that it cannot, at present,

a

t b

C

d

Area ( in 2 ) o bending (psi)

331.6

6bending {in}

0.04

e

f

0.710

Area ( in 2 )

2.130

') bending (psi)

139.0

446.6

6 bending (in)

0.04

O.Oq

Figure 7. Recursive generation oi topologies for Case B. Performance specs are given for instantiated designs

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mf Figure 8. Design for Case B not found by program

handle coupled constraints like those represented by the configuration of the web, lower flange and the forbidden area (i.e. the length of the lower flange is constrained by the forbidden area even when the web is extended below the forbidden area). Figure 5 shows another problem with a similar development of topologies. Note that Figure 5(c) is a major-topological variation of the initial topology, and therefore subject to Topological redesign.

IMPLEMENTATION This program (as well as Pedro and Dominic II) was written in Common Lisp on a VAX 11/785 running VMS. A Tektronics 4105 was used for terminal graphics. The Initial topological design module was written in OPS5.

CONCLUSIONS AND FUTURE DIRECTIONS A model for the automated design of load-bearing extrusion cross-sections was developed using a featurebased representation. Design activity was divided into two parts: parametric design, in which cross-section parameters are varied with a numerical, iterative technique, and topological design, in which crosssections are modified by adding new parameters and/or deleting old ones. Although the design domain and program implementation were relatively simple, the results suggest that, in general, design-with-features representations can provide a valuable schema for automated geometric and topological reasoning about mechanical designs. It is useful to employ a hierarchy of features that relates lower-level geometric entities to higher-level abstractions of the connectivity of features and their context in the design space. Evaluating and comparing topologies - critical for efficient search of the design space - is very difficult. To instantiate each feasible topology with a parametric design (as was done for this program) is certainly inefficient, perhaps even prohibitive. Instead, comparison must be based on some more abstract or qualitative assessment of the essential shape of the object. This is particularly important in problems where there are multiple performance criteria (e.g. bending, torsion and manufacturing factors for extrusion). The ability to generate topologies is limited by the appropriate selection and quantity of 'topological operators'. One way to enhance this capability might be to provide a mechanism for learning new topological operators by extracting and generalizing topological transformations performed by human designers in an

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intera(tive design-with-features environment. That is, the system would inspect the change in the teature representation made by the human designer, construct a set of 'left-hand side' conditions that generalize the change according to basic physical principles (e.g. moving material from the shear (:entre!., then assign the appropriate transformation. For most real-world design problems, multiple functional criteria must be considered, and conflicting or parallel design requirements must be resolved. This requires a different functional 'viewpoint' for each design specification that has bearing on the form. We are currently developing a new program that incorporates multiple viewpoints for bending, torsion and manufacturing. This includes: • Calculation of torsional constant for torsion analysis of the extrusion. Also heuristics about torsion for the topological operators. • Extended use of manufacturing knowledge related to die shapes (e.g. minimizing perimeter/crosssection ratio, reducing void areas for less pressure on die tongue etc.). • Coordination of multiple functional criteria using a 'manager' module that arbitrates proposed design changes with respect to each functional viewpoint. • Better reasoning for the initial design topology. • Development of a formalism for shape classification and topological operators. The automation of topological and geometric reasoning processes for mechanical design is an area of research that has only just begun to be explored. The development of exploratory programs in testbed design domains such as the one described helps to build a foundation for computational models of these processes. ACKNOWLEDGEMENTS This research was sponsored by a grant to the University of Massachusetts at Amherst from the National Science Foundation (DMC 8603076). REFERENCES 1 Shah, I I 'Synthesis of initial form for structural shape optimization' ASME Trans.Journalof Vibrations, Acoustics, Stress and Reliability (January 1988) 2 Nevill, G E and Paul Ir, G H 'Knowledge-based spatial reasoning for designing structural configurations' Pro(:. ASME Computers in Engineering Conf. New York, USA (July 1987)

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3 Nevill, G E and Paul Jr, G H 'Heuristic evaluation of partial structural designs' Proc. ASMf Computers in Engineering ConL New York, USA ~July 1987) 4 Fenves, S J and Baker, N C 'Spatial and functional representation language for structural design' Preprints of Working Group 5.2, IFIP Conk, rence on I-xpert Systems in Computer-Aided Design Sydney, Australia (February 1987! 5 Gips, J and Stiny, G 'Production systems and grammars: a uniform characterization' Environment and Planning B Vol 7 (1980) pp 399-408 6 Sliny, G 'A note on the description of designs' Environment and Planning B Vol 8 (1981) pp 257.-267 7 Libardi Jr, E C, Dixon, J R and Simmons, M K 'Designing with features: design and analysis of extrusions as an example' ASME Paper No. 86-DE-4 Proc. Design Engineering Conf. Chicago, USA (March 1986) 8 Luby, S C, Dixon, J R and Simmons, M K 'Designing with features: creating and using a features database for evaluation of the manufacturability of castings' Pro(:. ASME Computers in Engineering Conf. Chicago, USA (July 1986) 9 Vaghul, M V, Dixon, J R and Simmons, M K 'Expert systems in a CAD environment: injection molding part design as an example' Proc. ASME Computers in Engineering Conf. Boston, USA (August 1985) 10 Dixon, I R, Howe, A, Cohen, P R and Simmons, M K 'Dominic I: progress towards domain independence in design by iterative redesign' Proc. Amerkan Society of Mechanical Engineers ( ASME) Computers in I-ngineering Conf. Chicago, USA (July 1986) 11 Orelup, M F, Dixon, J R, Cohen, P R and Simmons, M K 'Dominic I1: meta-level control in iterative redesign' Proc. 7th Nat. Conf. Artificial Intelligence (AAAI-88), St. Paul, MN, USA (August 23-26 1988) 12 Lexan Products Division Designing with lexan General Electric, Pittsfield, MA, USA (1983) BIBLIOGRAPHY Dixon, J R, Libardi Jr, E C, Luby, S C, Vaghul, M and Simmons, M K 'Expert systems for mechanical design: examples of symbolic representations of design geometries' Eng. with Computers Vol 2 (1987) pp 1-10

computer-aided design