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Artificial Intelligence for Engineering Design, Analysis and Manufacturing (1997), //, 379-394. Printed in the USA. Copyright © 1997 Cambridge University Press 0890-0604/97 $11.00 + .10 Innovative dome design: Applying geodesic patterns with shape annealing KRISHNA SHEA AND JONATHAN CAGAN Computational Design Lab, Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. (RECEIVED November 27, 1996; ACCEPTED April 21, 1997) Abstract Shape annealing, a computational design method applied to structural design, has been extended to the design of tra- ditional and innovative three-dimensional domes that incorporate the design goals of efficiency, economy, utility, and elegance. In contrast to deterministic structural optimization methods, shape annealing, a stochastic method, uses lat- eral exploration to generate multiple designs of similar quality that form a structural language of solutions. Structural languages can serve to enhance designer creativity by presenting multiple, spatially innovative, yet functional design solutions while also providing insight into the interaction between structural form and the trade-offs involved in multi- objective design. The style of the structures within a language is a product of the shape grammar that defines the allowable structural forms and the optimization model that provides a functional measure of the generated forms to determine the desirable designs. This paper presents an application of geodesic dome patterns that have been embodied in a shape grammar to define a structural language of domes. Within this language of domes, different dome styles are generated by changing the optimization model for dome design to include the design goals of maximum enclosure space, minimum surface area, minimum number of distinct cross-sectional areas, and visual uniformity. The strengths of the method that will be shown are 1) the generation of both conventional domes similar to shape optimization results and spatially innovative domes, 2) the generation of design alternatives within a defined design style, and 3) the gen- eration of different design styles by modifying the language semantics provided by the optimization model. Keywords: Topological Layout; Structural Optimization; Dome Design; Simulated Annealing; Shape Grammars 1. INTRODUCTION Structural design can be described as the ability to create innovative and highly appropriate structural solutions to both familiar and new problems that balance efficiency, econ- omy, elegance, 1 and utility (Billington, 1983; Addis, 1990). The motivation for this work is to create the foundation for Reprint requests to: Dr. Jonathan Cagan, Department of Mechanical En- gineering, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. Tel: (412) 268-3713; Fax: (412) 268-3348; E-mail: [email protected] 1 For the present work elegance is used in the sense of formal elegance or elegance derived from formal geometric properties such as rhythm, sym- metry, or ratios. A discussion of formalist critiscism and the distinctions between formal, material, and associative aesthetic values can be found in Mitchell (1990). This interpretation of elegance only uses a portion of what was meant by Billington (1983) who used the word elegance to mean the expressive power of a form that is a combination of all parts of aesthetic value. Because material and associative aesthetic values are dependent on personal interpretation, this portion of aesthetic value is left to be evalu- ated by the designer when judging the resulting designs within a structural language. a computational tool for structural design that embodies this definition of structural design. The development of compu- tational structural design tools has largely focused on de- terministic structural optimization methods that provide one design solution for a specified problem that maximizes func- tional efficiency subject to behavioral and geometric con- straints. This limited scope of structural design goals does not consider utility or appropriateness of the design, eco- nomic factors in constructing the design, and visual impact of the structural form. Because structural design is more than iterative analysis, an effective computational tool to aid the designer that models the practical design goals of effi- ciency, economy, elegance, and utility is presented. In ad- dition, the method presented could enhance designer creativity by providing alternative innovative solutions of similar quality and insight into the relationship between form and function. The discrete topology layout of three-dimensional struc- tures is a complex problem that has been approached by using 379 https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0890060400003310 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 30 Jul 2017 at 10:22:22, subject to the Cambridge Core terms of use, available at

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  • Artificial Intelligence for Engineering Design, Analysis and Manufacturing (1997), / / , 379-394. Printed in the USA.Copyright © 1997 Cambridge University Press 0890-0604/97 $11.00 + .10

    Innovative dome design: Applying geodesic patterns withshape annealing

    KRISHNA SHEA AND JONATHAN CAGANComputational Design Lab, Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

    (RECEIVED November 27, 1996; ACCEPTED April 21, 1997)

    Abstract

    Shape annealing, a computational design method applied to structural design, has been extended to the design of tra-ditional and innovative three-dimensional domes that incorporate the design goals of efficiency, economy, utility, andelegance. In contrast to deterministic structural optimization methods, shape annealing, a stochastic method, uses lat-eral exploration to generate multiple designs of similar quality that form a structural language of solutions. Structurallanguages can serve to enhance designer creativity by presenting multiple, spatially innovative, yet functional designsolutions while also providing insight into the interaction between structural form and the trade-offs involved in multi-objective design. The style of the structures within a language is a product of the shape grammar that defines theallowable structural forms and the optimization model that provides a functional measure of the generated forms todetermine the desirable designs. This paper presents an application of geodesic dome patterns that have been embodiedin a shape grammar to define a structural language of domes. Within this language of domes, different dome styles aregenerated by changing the optimization model for dome design to include the design goals of maximum enclosurespace, minimum surface area, minimum number of distinct cross-sectional areas, and visual uniformity. The strengthsof the method that will be shown are 1) the generation of both conventional domes similar to shape optimization resultsand spatially innovative domes, 2) the generation of design alternatives within a defined design style, and 3) the gen-eration of different design styles by modifying the language semantics provided by the optimization model.

    Keywords: Topological Layout; Structural Optimization; Dome Design; Simulated Annealing; Shape Grammars

    1. INTRODUCTION

    Structural design can be described as the ability to createinnovative and highly appropriate structural solutions to bothfamiliar and new problems that balance efficiency, econ-omy, elegance,1 and utility (Billington, 1983; Addis, 1990).The motivation for this work is to create the foundation for

    Reprint requests to: Dr. Jonathan Cagan, Department of Mechanical En-gineering, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. Tel:(412) 268-3713; Fax: (412) 268-3348; E-mail: [email protected]

    1 For the present work elegance is used in the sense of formal eleganceor elegance derived from formal geometric properties such as rhythm, sym-metry, or ratios. A discussion of formalist critiscism and the distinctionsbetween formal, material, and associative aesthetic values can be found inMitchell (1990). This interpretation of elegance only uses a portion of whatwas meant by Billington (1983) who used the word elegance to mean theexpressive power of a form that is a combination of all parts of aestheticvalue. Because material and associative aesthetic values are dependent onpersonal interpretation, this portion of aesthetic value is left to be evalu-ated by the designer when judging the resulting designs within a structurallanguage.

    a computational tool for structural design that embodies thisdefinition of structural design. The development of compu-tational structural design tools has largely focused on de-terministic structural optimization methods that provide onedesign solution for a specified problem that maximizes func-tional efficiency subject to behavioral and geometric con-straints. This limited scope of structural design goals doesnot consider utility or appropriateness of the design, eco-nomic factors in constructing the design, and visual impactof the structural form. Because structural design is more thaniterative analysis, an effective computational tool to aid thedesigner that models the practical design goals of effi-ciency, economy, elegance, and utility is presented. In ad-dition, the method presented could enhance designercreativity by providing alternative innovative solutions ofsimilar quality and insight into the relationship between formand function.

    The discrete topology layout of three-dimensional struc-tures is a complex problem that has been approached by using

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  • 380 K. Shea and J. Cagan

    deterministic methods for both continuous and discrete lay-outs. Shape annealing has been presented as a stochasticmethod that combines discrete topology changes with con-tinuous shape and sizing of topologies to generate opti-mally directed planar trusses (Reddy & Cagan, 1995). Thedome design problem presents an intermediate step be-tween planar layout and three-dimensional layout because adome can be designed as a two-dimensional truss that is pro-jected onto a curved surface, thereby requiring the third di-mension to be a dependent variable of the planar layout.The shape annealing method will be shown to be capable ofgenerating two-dimensional topologies in three-dimensionalspace to form dome designs.

    The shape annealing method is a stochastic method thatallows for lateral exploration of the design space to gener-ate a variety of design styles to a given structural designproblem. As Simon (1975) wrote,

    A style is one way of doing things, chosen from a numberof alternative ways. Since design problems generally donot have unique optimal solutions style may enter inchoosing any one of many satisfactory solutions.

    In shape annealing, style is incorporated in the design pro-cess through multiobjective optimization of design goals.As a stochastic method, shape annealing provides a meansof discovering alternative design styles that satisfy the glo-bal design goals and are functionally feasible, that is, a validtruss structure that meets the required behavioral and geo-metric constraints. While stochastic optimization sacrificesfinding exact global optima, it benifits from design explo-ration that can generate good quality but suboptimal de-signs with interesting qualities and side effects not explicitlyoptimized but that are beneficial in satisfying the designer'sneeds. In addition, stochastic optimization is suited to search-ing an ill-defined design space and, in combination with adiscrete representation, can be used to incorporate practicaldesign goals in addition to structural efficiency.

    2. BACKGROUND

    Related work in structural topology optimization focuses ondeterministic methods for finding one functionally efficientsolution, that is the minimum amount of material requiredfor a specified performance. A presentation of deterministicmethods for structural topology optimization can be foundin Bendsoe (1995). The general structural layout problemhas been investigated as early as Michell's (1904) analyti-cal work. According to Kirsch (1989), structural topologyoptimization methods can be broken down into two catego-ries: distributed material optimization and discrete topol-ogy optimization. Distributed material optimization treatsthe layout problem as a continuum of material broken downinto a grid of elements. Among the distributed parametermethods, much work has been done with the homogeniza-

    tion method that discretizes a specified space into finite el-ements. The optimal density of each element is thendetermined from the stress limit resulting on the principalstresses (Bendsoe & Kikuchi, 1988; Bremicker et al., 1991;Diaz & Belding, 1993). The homogenization method hasbeen adapted for practical structural design (Chirehdastet al., 1994) through the development of a three-phasemethod that combines homogenization, vision algorithms,and shape optimization to generate parameterized manufac-turable objects. Two continuous material methods that varyfrom homogenization are a simulation of adaptive bone min-eralization used to generate structural topologies (Baum-gartner et al., 1992) and a skeleton-based method (Stal &Turkiyyah, 1996).

    Discrete topology optimization methods use either aground structure on which the design is based or heuristicsto introduce new members. Discrete methods have been for-mulated as a sizing optimization problem by using a highlyconnected ground structure where topology changes onlyoccur as the removal of members with a minimum area basedon stress limits (Hemp, 1973; Rozvany & Zhou. 1991; Bend-soe et al., 1994). The discrete layout problem also has beenformulated as a grid of allowable nodes to lay out members(Dorn et al., 1964; Pederson, 1992). The primary disadvan-tage of these methods is the strong dependence of the re-sulting design on the ground structure on which it was basedbecause often the only design variables are the size of mem-bers with no allowance for changes in the grid point loca-tions (Bendsoe, 1995). A method that utilizes exhaustivetopological search based on a heuristic to introduce newmembers was presented by Spillers (1975). A review of dis-crete topology optimization problems can be found in Kir-sch (1989). Additional approaches to both discrete andcontinuous topology design can be found in Bendsoe andSoares (1993). Multiobjective approaches to structural to-pology design also have been developed that trade off be-havioral attributes including stress, buckling, displacement,and frequency while searching for the best compromise de-sign (Eschenaueretal., 1990; El-Sayed& Jang, 1991;Gran-dhietal., 1993).

    The methods described previously are all deterministicmethods that focus on the optimization of functional effi-ciency based on behavioral and geometric constraints. Sto-chastic methods that focus on design exploration rather thanon optimization have been investigated by using an ap-proach similar to homogenization to generate optimal to-pologies from a genetic algorithm (Chapman et al., 1994)and a simulated annealing-based method for part design thatconsiders performance, manufacturing, and material cost(Anagnostou et al., 1992).

    Structural optimization methods can be categorized ac-cording to their representation, either discrete or continuousmaterial, and their optimization technique, either determin-istic or stochastic. Homogenization, a deterministic contin-uous method, has been applied successfully to the generationof monolithic parts in design domains such as the automo-

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  • Innovative dome design shape annealing 381

    no

    \

    accept based onMetropolisalgorithm

    V JX no

    revert to old design

    analyze structure

    inew design cost < old design cost

    yes 1

    yes ^ f 1accept new design 1

    1

    design frozen or^ schedule completed ?

    yes 1

    design completed

    1 create groupsSTART

    n o T yes y

    —| group members?

    4-1 generate initial1 structure

    \( n 0 -

    )

    • apply grammar rule

    FINISH

    Fig. 1. Shape annealing algorithm for structures.

    tive and aerospace industries, where structural efficiency isthe primary goal. In the design of structures that have bothfunctional and visual interactions with humans, the designgoals are broader and cannot be modeled solely by physicallaws, making discrete methods advantageous for addingproblem-specific design knowledge to the representation andoptimization. Using a stochastic optimization technique incombination with discrete design parameters, such as thosein shape annealing, allows for optimally directed explora-tion of a design space consisting of multiple discontinuous ob-jectives. The common ground for homogenization and shapeannealing is truss design for maximal efficiency, as shown ina comparison by Reddy and Cagan (1993). Shape annealinghas been extended in the present work to the generation of de-signs that not only take into account structural efficiency butalso explore the trade-off among the additional design goalsof utility, economy, and elegance.

    3. METHOD

    Shape annealing, a simulated annealing optimization on ashape grammar representation, was introduced as a designtechnique for the generation of optimally directed designs ofshape (Cagan & Mitchell, 1993). Optimally directed designis an approach to design optimization that directs the designgeneration toward the numeric range of a global optimum. Theshape annealing method was applied to the structural topol-ogy layout problem and was shown to be capable of gener-ating optimally directed structural topologies that avoidgeometric obstacles and satisfy buckling and stress criteria

    (Reddy & Cagan, 1995). This approach was extended to in-clude a simple model of economy in the optimization througha search for optimal groupings of members by cross-sectionalarea (Shea et al., 1996). Given a problem specification, the op-timal number of groups and the cross-sectional area of eachgroup were determined during the design process. By limit-ing the number of distinct member sizes through groups, atrade-off was created between achieving the minimum weightof the structure and having fewer distinct member cross-sectional areas with a corresponding economy of scale.

    3.1. Algorithm: Shape annealing

    The shape annealing method, as applied to structural de-sign, builds structures by using a shape grammar (Stiny,1980), optimizes the structures with the stochastic optimi-zation method of simulated annealing (Kirkpatrick et al.,1983; Swartz & Sechen, 1990; Ochotta, 1994), and ana-lyzes the structures by using the finite element method.2 Thealgorithm used follows the diagram shown in Figure 1. First,an initial structure is generated from a minimal connectionof truss members between the applied loads and supportpoints of the problem specification. If economy is includedas a design goal, the members in a design are grouped ac-cordingly. Second, the structure is loaded and analyzed withthe finite element method. The cost function is then evalu-ated by using the optimization model that is based on the

    2 Interfaces for MSC/NASTRAN™ and FElt (Gobat & Atkinson, 1994)have been built. FElt was used as the FEA tool for the examples presentedin this article.

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  • 382 K. Shea and J. Cagan

    design goals of the problem, the structural behavior of thedesign, and the geometric constraints. The initial design isautomatically accepted. Third, a rule from the shape gram-mar is applied to the structure to create a new design that isthen analyzed, and its cost is calculated. The costs of thenew design and the previous design are used in the simu-lated annealing algorithm to determine whether to acceptthe new design or revert to the previous design. A betterdesign is always accepted, but a worse design may be ac-cepted based on a probability function (Metropolis et al.,1953). A rule from the shape grammar is applied again tothe structure to create a new design, and the process con-tinues iteratively until the annealing schedule terminates orthe design has converged, or frozen. At the end of the designprocess, some members in the design may have the mini-mum allowable cross-sectional area and can be removed at thedesigner's discretion as long as the structure remains stable.However, a designer may choose to leave the members in thedesign because they add negligible weight but may provideadditional visual benefits. Details of the shape annealingmethod implementation can be found in Shea et al. (1996).

    The cost function used in the optimization process is aweighted sum of the computational models of the designgoals (i.e., efficiency, economy, elegance, and utility) andthe constraint violations (i.e., stress, Euler buckling, and over-lap of geometric obstacles). The objective weights are set apriori by the designer and represent the relative trade-off ofeach objective to the weight of the structure, which has aweighting factor of one. Extending the shape annealingmethod to support multiple independent loading conditionsresults in the following cost function:

    cost = 2) (objective_weight,• objective,

    ^ constraint_weight,;• I ^ constraint_violation,;=iL \«r=i

    ion,

  • Innovative dome design shape annealing 383

    4. DOME DESIGN

    Geodesic domes, originally invented by R. Buckminster Fuller(1954), have been used for a wide range of purposes includ-ing temporary exposition structures, scientific test centers, andhousing (Baldwin, 1996). Domes incorporate many interest-ing design goals because they are considered the strongest,lightest, and most efficient building system (Prennis, 1973).As domical structures, geodesic domes enclose the maxi-mum space with minimum surface area in which the layoutof the structure is influenced by both the desired span andheight as well as the method of layout. Economic design goalsinclude minimizing the number of different strut lengths andthe number of distinct cross-sectional areas as well as mini-mizing the surface area to present the least possible surfaceto the weather and thus minimizes heat loss (Baldwin, 1996).The objective function formulated for dome design mini-mizes both the weight of the structure and the surface area andmaximizes the enclosed volume. In addition, economy isadded to the design optimization model through dynamicgrouping of members by cross-sectional area (Shea et al.,1996) or length. Area groups are a group of members with thesame cross-sectional area, and length groups are a group ofmembers with lengths that fall within a small range based ona specified tolerance. The number of groups is then used tocalculate a group penalty that is added to the objective func-tion. This group penalty is formulated to increase in magni-tude throughout the annealing schedule and to have a stronginfluence with an increase in the number of groups.

    A computational model of elegance can be based on for-mal elegance, which is elegance derived from formal quali-ties such as symmetries, proportion, rhythm, uniformity, andvariety that are based purely on geometric properties. El-egance derived from material or associative aesthetic values(Mitchell, 1990) is not considered in the design generation be-cause this type of elegance is founded in designer prefer-ences and interpretation. The assessment of this type ofelegance is left to the designer when viewing the structural lan-guage of solutions. In previous work, the form of a structurewas based on functional considerations alone through mod-els for structural efficiency and economy except when a geo-metric constraint on symmetry was imposed. In this work, anaesthetic model of visual uniformity is incorporated into theoptimization model to reflect the geodesic dome design goalof a uniform breakdown of a sphere. Thus, an aesthetic valueis calculated for a design from the standard deviation of thelength of all members in a design. A design with a more uni-form breakdown and thus a lower standard deviation of lengthis considered to be of greater aesthetic value than a design witha more random breakdown. The addition of an aesthetic de-sign goal motivates the design of aesthetically pleasing struc-tures that are not based solely on behavioral properties andgeometric constraints.

    The objective function consists of a summation of the de-sign goals in which each design goal has a corresponding

    weighting factor that is set a priori by the designer to rep-resent the relative trade-off with structural weight, whichhas a weighting factor of one. The objective function usingall design goals for dome design is

    objective = mass + I weight, •1

    enclosed volume

    + (weight2-surface area)

    + (group_weight-group_penalty),

    where the group penalty is defined as

    group_penalty = «,

  • K. Shea and J. Cagan

    Fig. 2. Two of Buckminster Fuller's geodesic dome patents: Building Con-struction, 1954 (a) and Hex-Pent, 1970 (b).

    different from standard layouts because, although they arebased on geodesic breakdowns, the designs themselves arenot required to adhere to geodesic patterns. However, de-signs generated from the grammar can approach geodesicpatterns.

    Subdividing the members of a dome serves to achievebehavioral, economic, and utilitarian design goals. For ex-ample, dividing a member in half decreases the length ofeach individual member, which in turn decreases the re-quired cross-sectional area of compression members. How-ever, the total length of the two new members is greater than

    2v triacon breakdown

    Class I

    2v alternate breakdownClass II

    Fig. 3. Classes of geodesic dome breakdowns.

    the single old member because the bisection point is pro-jected onto the specified ellipsoid, thus forming two legs ofa triangle above the old member. Therefore, to reduce theweight of the structure and increase the efficiency, subdivid-ing creates a trade-off between decreasing cross-sectionalarea and increasing member length. Adding the utilitariangoals of maximizing enclosure space and minimizing sur-face area create additional trade-offs in subdividing mem-bers. A higher frequency of subdivisions allows the structureto approximate a sphere more closely, which will lead to amaximum enclosure space for minimum surface area.

    A shape grammar for dome design based on geodesic pat-terns for the standard breakdowns discussed is presented inFigure 4. This grammar was developed for an open domewithout a covering skin, which implies that the only allow-able shapes are triangles to ensure structural stability. Thelines in the shape grammar represent truss members, andthus only truss topologies can be generated using this gram-mar. The rules of the grammar described below are dividedinto shape and topology modification rules. The shape mod-ification rules take a free point and move it some distanceor change the diameter of a member cross section based onthe labels + or —, which indicate the direction the changeis made. These labels are set based on the behavior of eachmember: If a member is below both the stress and bucklinglimits, the label is set to — to decrease the cross-sectionalarea; otherwise, the label is set to +. Shape modificationrules and the shape annealing algorithm perform shape andsizing optimization of a structure with a fixed topology. Thetopology modification rules take a shape within the designand transforms it into a different configuration of shapesbased on geodesic patterns. A free line, denoted by the la-bel / , is an exterior line in the design. Note that the topol-ogy rules are created in pairs so that any modification canbe reversed, except for rule 5, which is its own reverse. Theapplication of a topology reversal rule is monotonic, and intheory any previous design state can be reached through thesequential application of reverse rules.4 The grammar rulesare fully parametric and use the following geometric con-straints based on considerations of good structural design:

    4 Spatial emergence does not exist in the representation presented, sothat any rule is always monotonic.

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  • Innovative dome design shape annealing 385

    (x+dx, y+dy, z")

    where: (x + dx)2+(y+ dx)z

    , 2height \span V

    z = — — J ( ^ ) 2 - ( x + dx)2-(v-dy)2span V 2

    rulel

    rule 2

    Shape Modification Rules

    f f trule 3

    f r frule 4

    ruleS

    rule 6 rule 7

    ° free point• fixed pointf free line

    Topology Modification Rules

    Fig. 4. Dome grammar.

    1. no two members may intersect without a joint,

    2. members cannot overlap, and

    3. the minimum angle between members is 10°.

    The shape grammar shown generates a two-dimensionaltruss layout that is projected onto a curved plane defined byan ellipsoid of a given height and span. The base of the domeis circular, defined by a uniform span in both the x and ydirections. These parameters specified by the designer actas functional constraints on the shape grammar because they

    define the desired enclosure space. An ellipsoid rather thana sphere is used because much vertical enclosure space of-ten is wasted with spherical domes; nevertheless, an ellip-soid can represent a sphere if so desired. An additionalparametric constraint is placed on the shape modificationrule to keep all points of the design within the circular baseof the defined ellipsoid. Figure 5 shows a top-down view oftwo dome layouts generated by hand from the grammar; theview illustrates that the grammar is capable of generatingstandard geodesic forms through the sequential application

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  • 386 K. Shea and J. Cagan

    Generation of 2v Alternate Breakdown

    Fig. 5. Generation of standard forms.

    of shape grammar rules. For the triacon pattern layout, thespecified rule is applied under reflected quarter symmetry;for the alternate pattern layout, the specified rule is appliedunder rotational quarter symmetry.

    5. RESULTS

    The dome grammar presented in Section 4 will now be usedto generate solutions for two dome design problems. The firstproblem compares the shape annealing method with tradi-tional shape optimization, and the second problem exploresinnovative dome design at both a spatial and functional levelbased on geodesic patterns. The shape annealing method is astochastic method that does not guarantee global optimalitybut rather uses design exploration to produce a variety of gooddesigns of essentially equal quality, that is, objective func-tion value. Spatial dome layout is investigated through an op-timally directed search by using the geometric design goalsof geodesic structures, that is, to approximate closely the de-fined ellipsoid by maximizing enclosure space and minimiz-ing surface area. This problem then will be extended to thefunctional layout of domes and will explore the effect ofchanging the semantics included in the objective function, ordesign goals, on the spatial and functional styles and qualityof solutions generated. For all designs, generated points in theplanar layout are projected onto the surface defined by an el-lipsoid to generate three-dimensional structures. For all de-signs presented, a top-down view and a three-dimensionalview are shown in which the width of the lines represents thediameter of the members uniformly scaled across all figures.

    5.1. Comparison with structural optimization

    The first problem is based on the shape optimization of aspace truss presented by Pederson (1973). Figure 6 showsthe fixed layout for Pederson's shape optimization problemand the corresponding initial layout for shape annealing. Theobjective of the design is to maximize efficiency of the struc-ture subject to four independent loading conditions appliedin the z direction:

    1. - 3 X 105 Nat joint 1,2. -3 .9 X 105 N divided equally among all free joints,

    3. -1 .5 X 105 N at joint 1 and - 1 X 105 N at joints 4and 5, and

    -1 .5 X 105 N at joint 1 and - . 7 X 105 N at joints2-4.

    4.

    Parametric constraints on the problem are as follows: Joint1 is restricted to move only in z, which changes the overallheight of the dome and thus the ellipsoid on which the pla-nar layout is projected. Joints 2-5 are restricted to movealong the orthogonal axis on which they lie in the initiallayout. In addition, the design is required to maintain 1/8symmetry. The material properties are modulus of elasticity(£, 2.1 X 10" N/m2), specific weight (v, 77,008.5 N/m3),allowable tensile stress (cr', 1.3 X 108 N/m2), allowablecompressive stress {ac, -1.04 X 108 N/m2), proportionalstress limit (

  • Innovative dome design shape annealing

    y

    387

    30 m

    30 m

    Shape Optimization Fixed Layout Initial Shape Annealing Layout

    Fig. 6. Structural optimization comparison.

    CompressiveTensile Elastic Compressive Plastic

    Allowable cr-aForce

    where

    a = cross-sectional area,

    pL +

  • 388

    Fig. 7. Symmetric dome designs: (a) weight = 66,757 N, height =7.62 m; (b) weight = 67,679 N, height = 7.45 m; (c) weight = 68,503 N,height = 6.33 m.

    K. Shea and J. Cagan

    5.2. Innovative geodesic dome design

    This section explores the application of geodesic patternswith shape annealing for both spatial and functional domedesign.

    5.2.1. Spatial dome design

    The spatial breakdown of an ellipsoid for maximum en-closure space and minimum surface area will now be inves-tigated. The purpose of this investigation is 1) to verify thatthe grammar, through random, iterative application of rules,is capable of generating geodesiclike spatial designs and 2)to determine the achievable spatial limits of enclosure spaceand surface area with a maximum of 50 members in the de-sign. The problem dimensions are shown in Figure 9, wherepoint 1 at the center of the structure is fixed at a height of9.25 m, thus defining an ellipsoid with a span, or circularbase, of 30 m diameter and a height of 9.25 m. The designshown in Figure 10a has an enclosure space of 3555 m3 anda surface area of 10.07 m2. The design shown in Figure 10bhas an enclosure space of 3528 m3 and a surface area of9.77 m2. Geodesic patterns can be seen in these designs ashighlighted in Figure 10a, thus verifying the ability of thedome grammar to generate geodesic-like dome designs un-der random application of rules.

    5.2.2. Spatial and functional dome design

    We will now apply the dome grammar to the design ofdomes for maximum efficiency, economy, utility, and el-egance. As discussed in Section 4, the purpose of a domestructure is to provide a maximum amount of enclosed spacefor an exposition structure or housing, for example, and pro-vide low covering and energy costs by minimizing the sur-face area. We will now investigate how adding design goalsto the optimization objective function influences the qual-ity and appropriateness of the generated designs. The ma-

    30 m

    30 m

    Fig. 8. Asymmetric dome designs: (a) weight = 65,129 N, height5.43 m; (b) weight = 80,334 N, height = 5.31 m.

    Initial Shape Annealing Layout

    Fig. 9. Initial shape.

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  • Innovative dome design shape annealing 389

    Table 1. Dome design cases

    Fig. 10. Spatial layout of domes.

    terial properties used for these designs are modulus of elas-ticity (E, 2.1 X 10"N/m2), specific weight (v, 7850 N/m3),allowable tensile stress (a1, 1.3 X 108 N/m2), and allow-able compressive stress (crc, -1.04 X 108 N/m2). The buck-ling limit is calculated by using the Euler buckling formula,Pcr = wEa/I. The initial layout for the problem is shown inFigure 9, where the two simultaneous loads applied are self-weight of structural members and -300,000 N applied atthe center, joint 1, in the z direction. A sequence of solu-tions will be shown in which each case expands the objec-tive function to incorporate additional design goals, asdescribed in Table 1. The designs are shown in Figures 11-18, with the corresponding design metrics listed in Table 2.Costs are listed for each design shown to compare multipledesigns from the same case. Convergence statistics for 12designs generated for each case, with standard deviationsshown in parentheses, are shown in Table 3. Because sim-ulated annealing is a stochastic method that can get trappedin local optima, it is standard practice to generate three de-signs and select the best design. Using this heuristic, a sec-ond set of statistics, shown in Table 4, is calculated fromfour sets of three designs using the same 12 designs gener-ated for the statistics in Table 3. These designs were gener-ated in approximately 80 min, each using a DEC alpha.

    6. DISCUSSION

    Shape annealing has been shown to be capable of generat-ing traditional solutions to structural design problems pro-vided the design generation is properly constrained. Whenthese constraints are removed, shape annealing generatesfunctional yet spatially innovative solutions for the samedesign problem. Comparing the solutions in Figure 7a andFigure 7b shows that the topologies of the two designs are

    Case

    1

    2

    3

    4

    5

    6

    7

    8

    9

    Design Goals

    Utility = maximum enclosure spaceEconomy = minimum surface areaEfficiency = minimum weightUtility = maximum enclosure spaceEconomy = minimum surface areaEfficiency = minimum weight

    Efficiency = minimum weightUtility = maximum enclosure spaceEfficiency = minimum weightUtility = maximum enclosure spaceEconomy = minimum surface areaEfficiency = minimum weightUtility = maximum enclosure spaceEconomy = minimum surface area

    minimum member area groupsEfficiency = minimum weightUtility = maximum enclosure spaceEconomy = minimum surface area

    minimum member length groupsEfficiency = minimum weightUtility = maximum enclosure spaceEconomy = minimum surface areaElegance = visual uniformityEfficiency = minimum weightUtility = maximum enclosure spaceEconomy = minimum surface area

    minimum member area groupsElegance = visual uniformity

    Loading

    None

    Self-weight

    Self-weight and centerapplied load

    Self-weight and centerapplied load

    Self-weight and centerapplied load

    Self-weight and centerapplied load

    Self-weight and centerapplied load

    Self-weight and centerapplied load

    Self-weight and centerapplied load

    only slightly different. Previous work on the generation oftransmission towers (Shea et al., 1996) that compared shapeannealing of planar trusses with shape optimization solu-tions also resulted in a small number of distinct topologies.The solution shown in Figure 7c has a different topologybut results in an increase in weight, leaving the designer todecide which design is preferable. Although constrainingthe problem may lead to finding only a few distinct solu-tions, these solutions are still quite different from the stan-dard geodesic forms. Constraining the problem generatessolutions that a designer may expect, but the strength of thismethod is design exploration that allows the method to gen-erate a variety of distinct novel solutions. When the sym-metry constraints are removed to allow for asymmetricdesigns, functionally feasible solutions with drastically dif-ferent topologies are generated.

    When the layout problem is not well constrained, the de-sign space increases, providing more possibilities of designsto explore. Semantics can be added to the generation processthrough both the optimization model and the grammar itselfto increase the level of problem knowledge based on designgoals that serve to focus the search for appropriate designs.Expanding the objective function to reflect the design prob-lem more accurately, reinforces good design decisions. For

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  • 390 K. Shea and J. Cagan

    Fig. 11. Dome design for self-weight. Fig. 12. Objective = weight.

    Table 2. Design metrics for Figures 10-18

    Case Figure Objective to Minimize" CostWeight

    (N)Enclosed Volume

    (m1)Surface Area

    (m2) No. Groups

    11

    2

    2

    3

    34

    4

    55

    6

    6

    7

    7

    8

    8

    9

    10a10b

    lla

    lib

    12a

    12b

    13a

    13b14a

    14b

    15a

    15b

    16a

    16b

    17a

    17b

    18a

    18b

    (ii',/volume) + (uvsurface area)

    (iv,/volume) + (vv2- surface area)

    weight + (w,/volume) + (uvsurface area)

    weight + (>V|/volume) + (w2-surface area)

    weight

    weight

    weight + (w, /volume)

    weight + (H-,/volume)

    weight + (iv,/volume) + (w2-surface area)

    weight + (w,/volume) + (HV surface area)

    weight + (iv,/volume) + (w2-surface area)

    + group_penalty(area)

    weight + (iv,/volume) + (w2-surface area)

    + group_penalty(area)

    weight + (vv,/volume) + (vv2-surface area)

    + group_penalty(length)

    weight + (\v,/volume) + (>v2-surface area)

    + group_penalty( length)

    weight + (iv,/volume) + (uvsurface area)

    + o-(length)

    weight + (iv,/volume) + (vv2'surface area)

    + o-(length)

    weight + (>V|/volume) + (iv2-surface area)+ group_penalty(area) + cr(length)

    weight + (vv, /volume) + (vv2-surface area)+ group_penalty(area) + a( length)

    3741

    3789

    4311

    4626

    6584

    7477

    10,591

    10,597

    10.877

    11,305

    18,371

    N/AN/A203

    4056584

    7477

    6791

    6889

    6449

    6538

    9343

    33,687

    32,875

    13,940

    17,984

    43,637

    29,018

    30,261

    42,489

    21,211

    11,203

    10,756

    14,726

    43,781 16,889

    3658

    3555

    3101

    2980

    1933

    1792

    2641

    2699

    2860

    2566

    2751

    2337

    2317

    2368

    3098

    2799

    2277

    2475

    10.079.77

    8.82

    8.666.85

    5.78

    8.34

    8.569.06

    8.698.94

    7.33

    7.89

    8.06

    8.53

    7.70

    6.46

    7.48

    N/AN/AN/AN/AN/AN/AN/AN/AN/AN/A4

    7

    6

    8

    (7 =

    (J —

    8cr =

    7a =

    1.36

    1.50

    1.96

    2.48

    m

    m

    m

    m

    V , = weight,, iv2 = weight2.

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  • Innovative dome design shape annealing 391

    Fig. 13. Objective = weight + (w, /enclosure space). Fig. 14. Objective = weight + (iv,/enclosure space) + (uy surface area).

    example, when comparing the results in Table 4 for cases 3and 4, adding the goal of maximizing volume not only in-creases the enclosure space but also decreases the weight, im-plying that for this case these design goals cooperate. Designgoals also can compete to create a trade-off, as illustrated whencomparing cases 4 and 5. With the addition of minimizing sur-face area to the objective function, the weight increases whilethe enclosure space decreases, but the design goal of mini-mizing surface area was improved to only a small degree. Thisresult may indicate to the designer that an explicit minimi-zation of surface area may not be necessary if the previous so-

    lutions were found to be satisfactory. It should be noted thatsmall declinations of the objective values also can be attrib-uted to the increase in the difficulty of the optimization prob-lem from the addition of design goals. Adding economy to theobjective function through member grouping in cases 6 and7 creates further competition that has adverse effects on weightand enclosure space and improves the surface area. The de-sign goal of elegance based on an aesthetic value determinedfrom spatial uniformity creates designs that are visually morefamiliar and intuitive to the designer and have the additionalbenefit of increasing the enclosure space at the expense of

    Table 3. Design statistics for 12 designs

    Case

    12

    34

    56

    7

    8

    9

    Objective lo Minimize

    (M'I/volume) + (iiy surface area)weight + (u>,/volume) + (ny surface area)weightweight + (iv,/volume)weight + (it',/volume) + (iiysurface area)weight + (M>,/volume) + (iiysurface area)

    + group_penalty(area)

    weight + (u'|/volume) + (iiysurface area)+ group_penalty(length)

    weight + (»»,/volume) + (iiy surface area)+ cr(length)

    weight + (ii>,/volume) + ( u y surface area)+ group_penalty(area) + o-(length)

    Weight(N)b

    N/A1407(1301)11659(3425)8983 (2046)9722 (2784)

    15,826(3269)

    22,157(4471)

    12,137(1381)

    16,901 (2833)

    Enclosed Volume

    (mY

    3408(149)2733(219)1631 (337)2588(127)2486(145)2312(247)

    2286(153)

    2844(211)

    2455 (224)

    Surface Area(m2)h

    9.42 (0.36)8.02 (0.5)5.91 (1.06)8.34 (0.45)7.98 (0.6)7.37 (0.86)

    7.68 (0.52)

    8.26 (0.53)

    7.48 (0.80)

    No. Groups'1

    N/AN/AN/AN/AN/A10(4)

    13(7)

    o- = 2.01 m (0.48 m)

    8(1)a = 2.82 m (0.63 m)

    V , = weight,, M>2 = weightybStandard deviations are shown in parentheses.

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  • 392 K. Shea and J. Cagan

    Fig. 15. Objective = weight + (iv,/enclosure space) + (>v2-surfacearea) + group_penalty(area): designs with four (a) and seven (b) distinctcross-sectional areas.

    Fig. 16. Objective = weight + (w,/enclosure space) + (vv2-surfacearea) + group_penalty(length): designs with six (a) and eight (b) lengthgroups.

    increasing the weight. Including all design goals, efficiency,economy, utility, and elegance, in case 9 results in an in-crease in weight when compared with case 7, which did notinclude elegance but an increase in enclosure space and a de-crease in the surface area were gained.

    When comparing the designs in Figure 14 with the de-signs in Figure 17 and the designs in Figure 15 with thedesigns in Figure 18, most engineering designers would tendto prefer the more uniform designs based on their sense ofthe relation between elegance and functional efficiency. The

    interpretation of designs is based on the level of knowledgeand the intuition of the designer as well as their aestheticvalues. A practical designer may look at the regular designsand automatically be able to interpret how the structureworks, whereas the functionality of the irregular patternsmay be more difficult to comprehend. An advantage ofcomputer-generated designs is that functionality is based onanalysis rather than on intuition. For a visual designer, whilethe visual uniformity of designs may appeal to their senseof aesthetics, the irregular patterns may be seen as more

    Table 4. Design statistics for best one of three designs (four sets of three designs)

    Case

    1 (spatial layout)2 (self-weight only)3456

    7

    8

    9

    Objective to Minimize8

    (iv,/volume) 4- (w2-surface area)weight + (w,/volume) + (vv2- surface area)weightweight + (w,/volume)weight + (w,/volume) + (vv2-surface area)weight + (w,/volume) + (w2- surface area)

    + group_penalty(area)weight + (\V|/volume) + (>i>2-surface area)

    + group_penalty(length)weight + (w,/volume) + (w2-surface area)

    + cr(length)weight + (>v,/volume) + (vv2-surface area)

    + group_penalty(area) + o-(length)

    Weight(N)b

    N/A592 (353)8019(1538)6863 (367)8536(1558)14,786(4120)

    18,540(2348)

    11,110(293)

    15,542(2416)

    Enclosed Volume(m-Y

    3545(91)2912(157)1983(223)2713(55)2559 (205)2313(340)

    2358(119)

    2965 (128)

    2547 (266)

    Surface Area(m2)b

    9.71 (0.28)8.42 (0.4)6.71 (0.81)8.60 (0.29)8.29 (0.67)7.28(1.41)

    8.13(0.22)

    8.46 (0.57)

    7.85 (0.63)

    No. Groups'1

    N/AN/AN/AN/AN/A8(3)

    7(1)

    a = 1.77 m (0.41 m)

    7(1)o- = 2.78 m (0.23 m)

    V , = weight,, w2 = weight2.bStandard deviations are shown in parentheses.

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  • Innovative dome design shape annealing 393

    (a)

    Fig. 17. Objective = weight + (vv,/enclosure space) + (w2-surfacearea) + trlcng,h.

    unique with striking visual impact. The variance of de-signer intuition and preference motivates the need for se-mantics to be incorporated directly into the grammar to allowfor further parametric control of the rules. In the designspresented, semantics have been used to ensure the genera-tion of valid structural topologies and as global design goalsto coerce the generation of desired designs. Semantics also

    Fig. 18. Objective = weight + (it^/enclosure space) + (vv2-surfacearea) + group_penalty(area) + o-|cnglh: designs with seven (a) and eight(b) distinct cross-sectional areas.

    can be used to control local aesthetic and functional prop-erties of subdesigns. These local semantics can be formu-lated from the preferences and aesthetic values of thedesigner to explore the range of possible solutions withinthe designer's style.

    Returning to the application of standard geodesic pat-terns with shape annealing, it can be seen that for spatiallayout, case 1, layout under self-weight, case 2, functionallayout for visual uniformity, cases 8 and 9, and geodesicpatterns were approached. For functional design not includ-ing an aesthetic measure, although the random nature of thedesign process was not expected to generate these break-downs exactly, local patterns of the modeled geodesic break-downs can be seen. Triacon patterns occurred morefrequently than with alternate patterns. Local control of aes-thetics as discussed previously also could allow for designsthat stylistically fall between the traditional geodesic-likeforms and the extreme asymmetric forms.

    7. CONCLUSIONS

    Shape annealing can generate a range of designs with dif-ferent styles that solve a given structural problem whengranted the latitude for design exploration. A shape gram-mar has been developed based on standard rules for geode-sic dome design and has been applied to the design of bothtraditional and novel domes, thus illustrating the capabilityof shape annealing to generate three-dimensional dome de-signs. It was shown that adding functional semantics to theobjective function provides search knowledge in the gener-ation of appropriate solutions. Two areas of future work are1) the extension of the method to fully independent three-dimensional layout and 2) an investigation into local con-trol of aesthetic properties through semantics that will allowfor more appropriate and elegant asymmetric solutions.

    ACKNOWLEDGMENTS

    The authors would like to thank the Engineering Design ResearchCenter, an NSF research center at Carnegie Mellon University, andthe National Science Foundation under grants DDM-9300196 andEID-9256665 for providing funding for this work. We also thankBill Mitchell and George Stiny for their discussions of this work.

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    Kristina Shea is a design engineer at the ComputationalDesign Lab of the Department of Mechanical Engineeringat Carnegie Mellon University. She received her B.Sc. in1993, her M.Sc. in 1995, and received her Ph.D. in 1997,all in Mechanical Engineering at Carnegie Mellon. Her prin-cipal research interest is the creation of computational en-gineering design methods and tools. Special interests includedesign theory and methodology, grammatical design, de-sign optimization and generation, and artificial intelligencein design.

    Jonathan Cagan is an Associate Professor of MechanicalEngineering at Carnegie Mellon University. His research,teaching, and consulting are in the areas of design theory,methodology, automation, and practice. He received his B.Sc.in 1983 and M.S. in 1985 from the University of Rochesterand his Ph.D. in 1990 from the University of California atBerkeley, all in Mechanical Engineering. Dr. Cagan is therecipient of the National Science Foundation's NYI Award(1992) and the Society of Automotive Engineer's Ralph R.Teetor Award for Education (1996). He is a member of thePhi Beta Kappa, Tau Beta Pi, and Sigma Xi National HonorSocieties and of the ASME, AAAI, SAE, and AAEE Pro-fessional Societies. He is a registered Professional Engi-neer and has worked in industry as an Applied ResearchEngineer.

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