Multiscale Heterogeneous Modeling with Surfacelets

Transcription

1 759 Multsale Heterogeneous Modelng wth Surfaelets Yan Wang 1 and Davd W. Rosen 2 1 Georga Insttute of Tehnology, yan.wang@me.gateh.edu 2 Georga Insttute of Tehnology, davd.rosen@me.gateh.edu ABSTRACT Computatonal desgn of strutures and materals requres multsale models to represent geometry and physal propertes. In ths paper, we propose a multsale heterogeneous model, dual-rep, to represent geometry and property dstrbuton mpltly. A new bass funton, alled surfaelet, s proposed to apture boundary nformaton more effently than tradtonal wavelets. Zoom operatons are defned to support multsale desgn. A method of materals model reonstruton from mages s also developed. Keywords: omputatonal materals desgn, wavelet. DOI: /adaps INTRODUCTION In the most reent deade, omputer-aded materals desgn has beome a rtal omponent of produt desgn, sne produt nnovaton heavly depends on the avalablty of new materals. Future omputer-aded desgn (CAD) systems should support multsale geometry and materals modelng. There are hallenges to represent strutures wth omplex geometry and topology n materals desgn. For nstane, super-porous strutures exst n natural and man-made materals. The topology s muh more omplex than the regularly used engneerng shapes. If we zoom n to an even smaller sale, materals are no longer ontnuous and the geometry beomes the materal struture. The heterogenety of materal propertes needs to be modeled expltly f we want to desgn materals wth very fne-graned detals. Exstng CAD geometr modelng based on explt and parametr representatons s not effent n spefyng the omplex topology. In multsale CAD systems, rangng from nano (10-9 ~10-8 m), meso (~10-7 m), mro (10-6 ~10-4 m), to marosales (>10-3 m), the spefatons of materal ompostons and dstrbutons are as mportant as geometr shapes. At the tradtonal marosale n engneerng desgn, geometry and materals an be separately spefed. However, at the mro- and meso-sales, materal ompostons beome mportant n funtonal realzaton, suh as n ompostes and funtonally graded materals. At the nano-sale, geometry and materal property beome one-to-one orrespondent to eah other, suh as n rystals and metal-organ frameworks. Therefore, we need a onse and prese form to apture materal ompostons along wth geometry so that engneers an desgn materals n an nteratve envronment.

2 760 Even though mult-resoluton geometry representatons (e.g., subdvson and wavelets) have been developed, the geometr nformaton s only at the same length sale; from ths nformaton, representatons at dfferent levels of detal (LOD) an be generated. The major dfferene between multsale and mult-resoluton modelng s that n a mult-resoluton envronment, the omplete model nformaton s avalable at the hghest resoluton, whle n multsale modelng, the hghest resoluton need not be generated, exept when and where t may be needed. For nstane, a multsale model does not requre the entre Boeng 787 arplane model to be avalable at the atom sale f one omponent s requred at ths sale. Instead, n a multsale envronment, t s neessary to render the model at a spefed sale when and where a spefed regon of a model an be rendered at a desred sale for a spef purpose, suh as vsualzaton, fnte-element analyss, or manufaturng proess plannng [1]. In ths paper, we propose a new multsale geometr and materals modelng method that uses an mplt representaton to effently aptures nternal and boundary nformaton. Ths new approah an effetvely represent multsale omplex strutures as well as materal ompostons and dstrbutons. It serves as the foundaton for modelng struture-property relatonshps for materals desgn. We all our modelng approah dual representaton or dual-rep, sne t represents both nternal and boundary dstrbutons of materals n a unfed mplt form. Ths new model s proposed to support the multsale zoom proess. Fg. 1 llustrates how suh a zoom proess s mportant n representng multsale materals suh as bone strutures. The external boundares an be represented as ontnuous 2D red urves n Fg. 1-(a). As we zoom n, the zero-wdth boundary beomes a doman wth nternal materal dstrbutons, as n Fg. 1-(b). In other words, explt boundares wth zero wdth n the prevous sale now beome ontnuous dstrbutons of materals wth fne-graned features nsde the boundary strutures themselves. On the other hand, when zoomed n, t beomes more onvenent to model some nternal regons where relatvely hgh gradents of dstrbutons exst (.e., wth large ontrast between blak and whte olors n the mages) by boundary representaton, as n Fg. 1-(). Wth further magnfaton of ontnuous regons to the atom level, the polyrystallne struture emerges, where gran boundares and defets an be modeled effently wth boundary representatons agan, as n Fg. 1-(d). Therefore, boundary and nternal are omplementary strutures. The purpose of the dual-rep model s to support suh role exhanges of boundary and nternal strutures at dfferent length sales. The enablng ore part of the dual-rep model s a new bass funton alled surfaelet. (b) Boundary beomes ontnuous doman (a) 2D bone struture wth boundary (n red) and nternal materal dstrbutons () Boundary appears n ontnuous doman (d) Polyrystallne materals wth gran boundares Fg. 1: An llustraton of dual-rep model for multsale geometry and materals modelng wth boundary and nternal strutures exhanges at dfferene sales.

3 761 In the remander of the paper, we gave a bref bakground revew of relevant work n Seton 2. The proposed surfaelet funtons are ntrodued n Seton 3. The dual-rep multsale heterogeneous model based on surfaelets s desrbed n Seton 4. Fnally, a model reonstruton approah s desrbed n Seton 5. 2 BACKGROUND In ths seton, we gve a bref overvew of relevant work to our proposed multsale materals modelng approah, nludng heterogeneous modelng n Seton 2.1 and mult-resoluton geometry representatons n Seton Heterogeneous Modelng Heterogeneous modelng usually denotes the modelng of geometry and materal omposton. Some researhers may nlude non-manfold topology under the term, whh we allow, but do not emphasze. Two general approahes to heterogeneous modelng have been proposed [2]: dsretzed and non-dsretzed approahes. In the frst ategory, materals and geometry are modeled separately, suh as mesh-based and voxel-based methods, where geometry s approxmated by volume meshes or voxels [3], and materal dstrbutons are determned by topology optmzaton or numeral nterpolaton from ontrol features [4]. Other researhers appled voxel-based representatons that utlzed spatal oupany enumeraton of part geometry. Agan, materal omposton nformaton was appled to ether ndvdual voxels or nterpolated over sets of voxels usng a part s boundng surfae [5]. General ellular deompostons have also reeved onsderable attenton [6]. A general ellular deomposton-based approah ntegrated physal property dstrbutons nto the geometry+materal model [7] that others have nvestgated. Some researhers have generalzed the ellular modelng approah to nlude manufaturng proess-related Loal Composton Control (LCC) elements [8]. In non-dsretzed approahes, some researhers have separated the representaton of materal ompostons and propertes from the underlyng part geometry [9]. Others have utlzed mplt modelng approahes, whh has advantages n that a ommon mathematal model s used for both geometry and materal omposton [10]. Shapro and oworkers have appled the theory of R- funtons to show how materal omposton [11] an be performed usng mplt modelng approahes. The advantage of ther approah s the unfyng nature of mplt modelng to model geometry, materal omposton, and dstrbutons of any physally meanngful quantty throughout a part. A dfferent group proposed a method based on hypertextures [12] that provdes more ntutve user ontrols, aordng to the developers. Smlar to the mplt modelng approahes, materal ompostons were spefed on part surfaes and smlar types of dstane measures were used to ompute ompostons nternal to parts. The lmtaton of all work desrbed here s that the models are not multsale; at best some an be alled mult-resoluton, spefally the methods n the frst general approah based on geometr deompostons. Even then, the resolutons are lmted to the overall part geometry and the partular deomposton nto ells, voxels, or mesh elements. In ontrast, our proposed approah s both multsale and mult-resoluton, where part representatons an be generated at the desred sze sale and resoluton. 2.2 Mult-resoluton Geometry Representatons There have been varous efforts on mult-resoluton geometry representaton. The bas deas are based on funtonal approxmatons n the Hlbert spae. Choosng approprate bass funtons that have mult-resoluton propertes s the ore researh ssue. The most studed ones are B-splnes and wavelets B-Splne B-splne s wdely used as the bass funton n geometr modelng and smulaton. Through knot nserton and degree elevaton [13], B-splne urves and surfaes an be refned wth more ontrol ponts ntrodued. Reversely, ontrol ponts an be redued n approxmatons.

4 762 A more general mult-resoluton representaton s subdvson urves and surfaes [14, 15], whh are based on the B-splne refnablty. That s, the B-splne bass funton an be wrtten as a lnear ombnaton of the translated and dlated opes of tself. Therefore, by reursvely refnng ontrol ponts based on some topologal and geometr rules, the ontrol polygon or mesh onverges to the lmt B-splne urve or surfae. Reently, more subdvson shemes have been developed for surfaes, suh as Loop [16], Butterfly [17], 3 [18], 2 [19], et., wth arefully hosen subdvson rules Wavelets In the doman of 2D shape representatons, wavelets are among the most popular mult-resoluton representatons. Smlar to Fourer analyss, wavelet analyss s to represent and approxmate sgnals (or funtons n general) wth orthogonal or non-orthogonal bass funtons. Both methods an be used to unover frequeny omponents of sgnals. Both an be used to represent mult-resoluton subspaes and fast algorthms are avalable for both. However, nstead of snusodal funtons n Fourer analyss, the funtonal spae for wavelet analyss s deomposed based on a salng funton t and a wavelet funton t wth one dmensonal varable t for mult-resoluton analyss. Wavelets are self-smlar and an be saled up and down. More spefally, the wavelet funton (0.1) t a,b a 1/2 a 1 t b s saled by a salng (dlaton) fator a and translated by a translaton fator b. Although ertan forms (e.g. Haar, Daubehes, Morlet, et.) have been used extensvely, t s atually general and an be ustomzed for spef problems. Smlar operatons an be appled to salng funtons. The most mportant feature of wavelets s that they are loalzed n both real (tme) and reproal (frequeny) spaes due to the property of regularty and vanshng moments. Wth these propertes, wavelets together wth salng funtons an be used to approxmate sgnals and funtons n a ompat form (ompared to the tradtonal Fourer analyss), wth loal features represented more effently. Wavelets provde a general approah for mult-resoluton funtonal analyss. In geometr modelng doman, the wavelet transforms were used to desrbe planar urves wth multple resolutons [20]. Combnng subdvson and lftng, a so-alled normal mult-resoluton approxmaton was developed to nterpolate urves and surfaes [21]. Some new wavelets have also been onstruted to represent geometres. For nstane, Wang et al. [22, 23] developed a translatonnvarant B-splne wavelet funton for urve representaton, whh an be omputed reursvely wth dfferental operatons. The above dfferent mult-resoluton modelng approahes apture geometr nformaton n dfferent levels of detal. However, they do not meet the requrement of multsale desgn senaros, where geometr nformaton as well as materal propertes should be aptured n a way so that users an retreve ether low-granularty global strutures or hgh-granularty loal nformaton wthout renderng the global pture wth the greatest detals. Our proposed multsale representaton of geometry and materals s based on a hybrd approah of mplt and boundary representatons. The general dea s to represent materals dstrbuton n a doman and ts boundary wth a ombnaton of two types of bass funtons. Whle regular wavelets an be used n nternal dstrbutons, a new bass funton s ntrodued to model materals dstrbutons n boundares, whh s alled surfaelet. It s a generalzaton of rdgelet and urvelet, as ntrodued n the next seton Wedgelet, Curvelets, and Surflet Wavelets perform well for objets wth pont sngulartes n dmenson 1. However they are not effetve n dealng wth edge dsontnutes n dmenson 2. Several so-alled dretonal wavelet approahes have been proposed to solve ths ssue, nludng wedgelet [24] and urvelet [25], as well as ther lose relatves suh as rdgelet [26], ontourlet [27], beamlet [28], and platelet [29]. The wedgelet approah parttons 2D spae nto squares as buldng bloks whh have lne segments. 2D mages then an be approxmated by a olleton of spefally hosen wedgelets. The urvelet funton s an extenson of the standard wavelet funton, whh ombnes the onepts of neural networks, Radon transform, and wavelets. It was developed to ompress mages ontanng

5 763 ontnuous lne or urve segments, where the standard wavelets are not effent. The dea of the urvelet was orgnally started as rdgelet, where an angular element s ntrodued n the wavelet funton as r a,b, a,b x os y sn a 1/2 a 1 x os y sn b (0.2) so that the parameters nlude salng, translaton, and orentaton. Later, urvelet was onstruted n the Fourer spae n order to have better ontrols of resoluton n the frequeny doman. Wth the mult-resoluton deployment of rdgelets, a parabol salng onstrant of wdth length 2 an be ntrodued so that urvelets are no longer lnes wth nfnte lengths. If wavelets an be thought of as fat ponts wth ertan wdths of loal support, urvelets are fat needles where orentaton nformaton an be aptured. In 3D analyss, Yng et al. [30] extended 2D urvelet transform to 3D wth smlar frequeny spae tlngs. Smlarly, Lu and Do [31] extended ontourlets to three dmensons n a dsrete spae. Chandrasekaran et al. [32] extended wedgelets to hgh-dmensonal spae and approxmate funtons wth polynomal buldng bloks, alled surflets, nstead of lnear buldng bloks n wedgelets. In ths paper, we propose a new modelng approah to apture multsale materal dstrbutons wth a new surfaelet bass funton. The proposed surfaelet an be vewed as a 3D generalzaton of the above dretonal wavelets. Also dfferent from the purpose of the above efforts, whh s to develop effent approahes to represent and ompress edge and surfae sngulartes n 2D or 3D mages n the format of pxels or voxels, we are nterested n modelng both ontnuous dstrbutons and surfae sngulartes effently n a unfed mplt form. In ontrast to the above dretonal wavelet bases whh are onstruted n the Fourer doman, we would lke to have dret ontrols n Euldean spae. In order to support dual-rep that allows for multsale materals representaton, a mxture of wavelet and surfaelet bass funtons s hosen. Wavelet bass funtons are used for ontnuous dstrbutons, whereas surfaelet bass funtons are used for surfae sngulartes or boundares. Ths approah enables us to apture both nternal and boundary domans n a unfed form so that seamless zoom operatons between sales an be performed. Our new bass funton of surfaelets s a generalzaton of rdgelets. In the tradtonal rdgelets, lne or plane-type boundares an be represented effently. However, ts support s not loalzed along the rdge dreton, where x os y sn s kept onstant. Therefore, parttonng of the Euldean spae nto smaller domans s requred n the ases that boundares are devated far from lnear sngulartes. As a result, artfally dsontnuous boundares may be ntrodued. Our new surfaelet funtons generalze the shape sngulartes to hgher orders suh as quadrat surfaes. Ths allows for more flexbltes of boundary representaton. The surfaelet s desrbed n Seton 3. 3 SURFACELET We need to represent both nternal and boundary materal dstrbutons n multsale and multresoluton modelng. The exhange between boundares and nternal strutures suh as the ones n Fg. 1 annot be aptured effently n the forms of regular wavelets or splnes. Therefore, a dfferent approah s requred. Boundares an be vewed as surfae sngulartes that are dsontnuous n one dreton whle ontnuous n other two n 3D spae. They an be represented loally as planes, ylnders, ellpsods, et. Therefore, we propose surfaelet bass funtons as buldng bloks of boundares n multsale modelng. We defne a general surfaelet bass funton as r a,b,p a 1/2 a 1 b,p r where r x,y,z (0.3) s the loaton n the doman n the Euldean spae, : s a wavelet funton, a s a non-negatve salng fator, : b,p 3 s a surfae funton so that x,y,z b,p 0 mpltly defnes a surfae, wth the translaton fator b and the shape parameter

6 764 m vetor p determnng the loaton and shape of surfae sngularty respetvely. Partularly, a 3D rdgelet that represents plane sngulartes s defned as r a,b,, a 1/2 a 1 os os x os sn y sn z b where 0,2 and / 2, / 2 are angular parameters orrespondng to rotatons around z- (0.4) and y-axes n the Euldean spae. Here the shape parameter vetor s p,. Smlarly, a surfaelet that represents ylndral sngulartes an be defned as a,b,,,r1,r 2 r a 1/2 a 1 r 1 os os x os sn y sn z b 2 r 2 sn x os y 2 (0.5) A surfaelet that represents ellpsodal sngulartes s defned as a,b,,,r1 r,r 2,r 3 a 1/2 a r os os x os sn y sn z b r 2 sn x os y 2 r 3 sn os x sn sn y os z 2 (0.6) The sosurfaes of these three surfaelets for plane, ylndral, and ellpsodal sngulartes are shown n Fg. 2, where the Mexan hat bass t 2 1/2 3 1 t 2 2 e t2 2 /2 (0.7) s appled and 0.8. (a) 3D rdgelet ( a 0.2, b 0.1, 11 /6, (b) Cylndral surfaelet ( a 0.2, b 0.1, 11 /6, /4, r1 1, r2 4 ) Fg. 2: Isosurfaes of surfaelets. () Ellpsodal surfaelet ( a 0.2, b 0.1, 11 /6, / 4, r, r2 4, r3 9 ) 1 1 The parameters of surfaelets an be geometrally nterpreted as follows. For 3D rdgelets as n Fg. 3-(a), any pont on a plane os os x os sn y sn z t has the same evaluaton of the wavelet funton a 1 t b. Therefore, the sosurfaes of Eqn.(0.4) are planes. Smlarly, for ylndral surfaelets n Fg. 3-(b), all ponts on a ylnder r os os x os sn y sn z b 1 2 r 2 sn x os y 2 2 r 0 have the same evaluaton of wavelet funtons n Eqn.(0.5). For ellpsodal surfaes n Fg. 3-(), all ponts on the ellpsod r os os x os sn y sn z b 1 2 r 2 sn x os y 2 r 3 sn os x sn sn y os z 2 r 2 have the same evaluaton n Eqn.(0.6). 0

7 765 z z z y n n n r r r y y b b b x x (a) 3D rdgelet (b) Cylndral surfaelet () Ellpsodal surfaelet Fg. 3: Geometr nterpretaton of surfaelets. x Let ẑ denote the Fourer transform of are hosen suh that the Parseval relatonshps and the admssblty ondton z t and z denote the omplex onjugate of z. If 1 t t dt 2 d t s (0.8) 2 1 d 1 (0.9) are satsfed, based on the gener surfaelet bass funton n Eqn.(0.3), an L 2 funton reprodued by where f r p 0 f r an be a,b,p a,b,p r dadp db (0.10) a,b,p f r a,b,p r dr (0.11) 3 For the 3D rdgelet n Eqn.(0.4), the surfaelet oeffents a,b,, s an be effently alulated n the Fourer doman, sne 1 a, b,, 3 a, b,, r f r dr a, b,, f d (0.12) 2 as the result of Eqn.(0.8). We further onvert the Cartesan oordnate to the spheral oordnate. 1 Let u os os, os sn, sn, os os, os sn, sn u, and a u r b t. Then b at a a b e d a t e d at a e t e dt 1/2 1 u r 1/2 at b 1/2 a, b,, u r r 1/2 b a e a Therefore, for some spef values of and, we have 1 1/2 b a ae f a, b,, d (0.13) 2 The 3D rdgelet oeffent a, b,, thus an be obtaned by the one dmensonal nverse Fourer transform of Fourer transform of 1/2, a a f,, along the radal lne, f r. 1/2 a, where f s the 3D a may be regarded as a wndow n the frequeny doman. Therefore, the omputaton of surfaelet oeffents an be done effently by: (1) 3D fast fourer

8 766 f r to obtan f ; (2) radal samplng of f transform (FFT) of along the radal lne by varyng wth a fxed angular ombnaton of, to form a new Radon doman of,, ; (3) applyng the wndow funton 1/2 a a to eah 1D Radon radal doman,, fnally applyng the 1D nverse FFT to eah one of the fltered 1D Radon domans. for a partular angle; and (4) For ylndral surfaelets n Eqn.(0.5) and ellpsodal surfaelets n Eqn.(0.6), the oeffents are not as smple as n Eqn.(0.13). But we stll an ompute based on Eqn.(0.12) n the frequeny doman, where s vewed as a gener wndow funton. a, b,, 4 DUAL-REP MULTISCALE HETEROGENEOUS MODEL Based on the proposed surfaelet bass funtons, we model omplex and multsale geometres n an mplt form for mult-materals representaton and analyss. The materal dstrbutons or onentratons are frst spefed expltly at a fnte number of postons. Then the dstrbutons wthn the doman an be nterpolated based on seleted bass funtons. We are nterested n modelng both nternal and boundary dstrbutons n a multsale and multresoluton representaton. As mentoned n Seton 2.2.3, wavelets are effent n representng onedmensonal pont sngulartes but not two- or three-dmensonal ones. In ontrast, surfaelets are effent n two- or three-dmensonal sngulartes but not one-dmensonal ones. Therefore, a ombnaton of wavelet and surfaelet bass funtons s hosen. Wavelet bass funtons are used for ontnuous nternal dstrbutons, whle surfaelet bass funtons are used for boundary dstrbutons. The dual-rep model s llustrated shematally n Fg. 4. A grd s ntrodued and the orrespondng nodal values are fxed n the modelng spae. Nodes n the grd serve as ponts at whh geometry and materal ompostons are sampled. They serve as the enters of referenes for wavelet and surfaelet funtons, eah of whh has a sope of loal support. Gven some sampled values of materal propertes f at the nodes r n a grd spae, where 1,, n, we an approxmate materal propertes wthn as n f r f aj,b k,p l a,b,p r r (0.14) 1 a b p j k l where a j, b k, pl are the respetve dsretzed parameter values of a, b, p n Eqn.(0.3). When the sampled nodes r s (denoted by n Fg. 4) are wthn the nternal regon, we hoose the bases n Eqn.(0.14) as regular wavelet tensor produts as a,b,p r r a,b x x a,b y y a,b z z wthout dretonal nformaton. But when the sampled nodes (denoted by ) are lose to boundares, we hoose a, b, p r r n Eqn.(0.14) to be the surfaelet funtons. In ths dual-rep model, wavelets have retangular supports n apturng dstrbutons. In ontrast, surfaelets have the support of long and ellpsodal shapes. Wth the ombnaton of wavelets and surfaelets, we an approxmate property dstrbutons effently for both nternal and boundary regons. In an nteratve desgn envronment, f desgners spefy the materal propertes f at the nodes r s (9 nodes n the example of Fg. 4), and the shapes of the bass funtons assoated these nodes a, b, p r r, the property dstrbuton f wthn the doman then an be determned.

9 767 δω Ω 4.1 An Example of Dual-Rep Model Fg. 4: Wavelet and surfaelet support llustrated n a 2D spae. Fg. 5 shows a smple example of dual-rep models n 3D spae wth a ombnaton of wavelet and surfaelet bass funtons. The olored sosurfaes ndate the dstrbuton of two materals, rangng from red to green. There are 27 grd ponts n the doman. Two of the grd ponts ( 1,27 ) are assoated wth ellpsodal surfaelets (labeled by ), whereas the others are wth regular wavelets (labeled by ). Both funtons are based on the Mexan hat bass n Eqn.(0.7) where 0.5, and the dstrbuton funton s 26 f x,y, z f a,ba 3/2 a 1 x x a 1 y y a 1 z z 2 2 y y 1 2 4x x f 1 a,ba 1/2 a z z 1 (0.15) 27 f 27 a,b a 1/2 a x x 27 2 y y z z 27 2 where b 0, (1) 27 0, (1) /2, (27) 0, a,b 3 2 /a ( 1,,27 ). The materal property values at the nodes are lsted n Tab. 1 wth rows and olumns orrespondng to the postons n Fg. 5. Fg. 5 shows the sosurfaes of the dstrbutons wth dfferent sale fators ( a 8, 4,2,1, 0.5, 0.25 ). It an be seen that the dstrbutons are relatvely smooth for large sale fators, sne there are sgnfant overlaps between support regons of bass funtons. As the sale fator gradually redues, the grd ponts beome more solated slands. As a result, the sale fator provdes dret ontrols n multsale and mult-resoluton modelng. The zoom operatons based on the sale fator are ntrodued n Seton Zoom-In Operatons wth Multple Sales A multsale model should support zoom-n operatons suh that geometr, materal, and property models at dfferent sze sales an be seamlessly ntegrated. When the desgner zooms to a loser vew of a regon usng the zoom-n operator, the number of grd ponts, and therefore the grd resoluton, nreases loally. In ontrast to mult-resoluton representatons where the same model s used at dfferent levels of resoluton, a multsale representaton uses dfferent models. The zoom-n proess requres that a separate model s ntrodued to approxmate loal propertes at a smaller sale. There are two steps n buldng new models at a fner sale. One s hoosng new grd ponts and values, the other s hoosng new bass funtons. In hoosng new grd ponts and values, we an use

10 768 dfferent subdvson rules. For nstane, f the Catmull-Clark sheme s appled, the property value of the nserted node wll be the weghted average of neghborng nodes, and the weghts are based on topologal relatonshps wth ts neghbors. Based on users spefatons, nterpolatng subdvson shemes an be hosen for hard onstrant on grd pont values, whereas approxmatng shemes an be appled for soft onstrant (a) a 8 (b) a 4 () a 2 (d) a 1 (e) a 0.5 (f) a 0.25 Fg. 5: Isosurfaes of materal dstrbutons n Eqn.(0.15) wth dfferent sales. f19 0.2, f20 0.2, f f22 0.1, f23 0.4, f f25 0.3, f26 0.5, f f10 0.8, f11 0.2, f f13 0.5, f14 0.2, f f16 0.2, f17 0.1, f f1 1.0, f2 0.5, f3 0.1 f4 0.6, f5 0.4, f6 0.2 f7 0.3, f8 0.2, f9 0.4 Tab. 1: Materal values spefed at the grd ponts n Fg. 5. In hoosng new bass funtons for a dfferent model, we need to fnd those bases that are sutable for both the newly nserted and orgnal nodes for the new model at a more detaled level, dependng on the nature of subdomans. Mappng proedures are needed to generate new models. For mappngs from ontnuous doman to ontnuous doman, hoosng new bass funtons s straghtforward. The same wavelets wth dfferent sales an be used. Smlarly, for boundary to boundary mappng, the same surfaelets are appled. The omplexty les n the ases of ontnuous to boundary and boundary to ontnuous mappngs. Contnuous to boundary mappngs our when nternal boundares/nterfaes emerge, suh as n the transton from Fg. 1-(a) to Fg. 1-(). Conversely, boundary to ontnuous mappngs our n transtons suh as the one from Fg. 1-(a) to Fg. 1-(b), when nternal boundares dsappear or are smoothed out wth a fner LOD.

11 769 For the ase of soft onstrants, eah step of the zoom-n proess has wavelet or surfaelet nterpolaton nvolved. When a subdoman s hosen, new grd ponts n the subdoman need to be spefed. For eah new grd pont, the property value s kept the same as n the prevous sale. More formally, suppose that at the j -th sale the property dstrbuton wthn ts doman j s f j n j j r f,j a,b,p,j j aj,b k,p r r l 1 a b p j k l We would lke to fnd a new model f j 1 n j 1 j 1 r f,j 1 a,b,p,j 1 j 1 aj,b k,p r r l 1 a b p j k l wthn a subdoman suh that a dstane funton j 1 j j 1 j f f s r r dr j 1 2 r j s mnmzed, gven that the property values at the new grd ponts j r 1 s are j 1 j j 1 r 1,, 1 r j 1 f f nj The neessary ondton of optmalty s s 2 f 1 r f r f j 1 a,b,p,j 1 j 1 r r dr 0 for all a,b,p, That s n j 1,j 1 aj,b k,p l j 1 r r j 1 ' j 1 ',j 1 f ' a ',b ',p',j 1 a,b,p dr,j 1 a ',b ',p ' j '1 a ' b ' p' 1 f j j 1 j 1 r r (0.16) for all a,b,p, Ths lnear system an be solved effently f the number of grd ponts and the levels of dsrete a, b, p values are reasonable and the support of wavelet bass funtons s loalzed. g r,j 1 j 1 a,b,p r r dr (0.17) For the ase of hard onstrants, the values of the smaller sale model at some spefed postons r ( g 1, 2, ) have to exatly math those of larger sale model. The dstane funton beomes g h f j 1 r f j 2 r dr j 1 g f j 1 r g f j r g (0.18) g where s are the Lagrange multplers that onvert the onstraned optmzaton problem to a nononstraned one. The neessary ondton of optmalty then s f j 1 r f j r f j 1 a,b,p,j 1 j 1 r r dr j 1,j 1 f g rg j 1 a,b,p r j 1 0 for all a,b,p, g f j 1 r g f j r g for all g or

12 770 s n j 1 j 1 f ',j 1 ' j 1 a ',b ',p' r r ',j 1 j 1 a,b,p r r dr,j 1 j a ',b ',p ' f j 1,j 1 rg a,b,p r j 1 '1 a ' b ' p ' 1 g g f j,j 1 r j 1 a,b,p r r dr for all a,b,p, j 1 (0.19) n j 1 j 1 ',j 1 f a j 1 rg ' ',b ',p ' r,j 1 ' a ',b ',p ' f j r g for all g '1 a ' b ' p ' Fg. 6 s a smple one-dmensonal example to llustrate the zoom-n operaton. The orgnal model wth a 0.25, b 0.0, 1.0, 0.5, 0.1, 0.5, 1.0 f x f a,b a, b 5 a 1/2 a 1 x x b, 1.0, 0.5, 0.0, 0.5,1.0 x and the orrespondng f, and s the Mexan hat wavelet n Eqn.(0.7) wth Fg. 6-(a) shows the orgnal model, where blue rles denote the grd ponts x s and values f s. Applyng the soft onstraned zoom-n operaton to a subdoman 0.5, 0.0 and solvng Eqn.(0.17), we obtan a smaller sale model as shown n Fg.6-(b), where the blue rles ndate the new grd ponts n the smaller sale. Note that the atual model values at the grd ponts do not neessarly math the spefed ones n the soft onstraned ases. If hard onstrants are appled on two boundary ponts of the subdoman as ndated by the red rles n the orgnal model as n Fg. 6-(), we obtan the new small sale model of Fg. 6-(d) by solvng Eqn.(0.19). Fg. 6-(f) s a dfferent small sale model when hard onstrants are mposed on all fve new grd ponts as shown n the orgnal model n Fg. 6-(e). In all three small sale models, the same grd ponts are hosen. The respetve oeffents of the resultant small sale models are lsted n Tab. 2. Fg. 7 s a three-dmensonal example of zoom-n operaton. The orgnal model s shown n Fg. 7- (a). When 1/64 of the doman s seleted to represent detals, a smaller sale model as shown n Fg. 7- (b) s bult based on Eqn.(0.19) wth 27 onstranng grd ponts. 5 MODEL RECONSTRUCTION Besdes dret spefaton of dstrbutons n multsale materals modelng, t s also desrable that models an be reonstruted from exstng materals n the sense of reverse engneerng. Dual-Rep models an be reonstruted from mages of nano- or mro-strutures. We an use nano- and mrosale mages suh as those from atom fore mrosopy and eletron mrosopy to haraterze a materal s struture. The boundares of grans or phases an be deteted from the mages. In addton, the shadng and ts gradents n the mages provde the nformaton of materal dstrbutons. Gven that albraton of mages s done n the pre-proessng step, a mappng from shadng n mages to propertes an be aheved. In Seton 3, we have shown that reonstruton of surfaelet models based on the Fourer transform s possble. In ths seton, we take a dfferent surfaelet transform approah for model reonstruton. A slghtly dfferent vew of the surfaelet transform s that the surfaelet oeffents an be obtaned by Radon alke transforms. As llustrated n Fg. 8-(a), the Radon transform n a doman Ω s the ntegral along the plane (represented as the dash lne n 2D), whh s perpendular to a lne wth an angle. The plane and the lne nterset at a pont whh has the radal dstane of from the orgn.

13 771 (a) orgnal model () orgnal model (e) orgnal model (b) small sale model (d) small sale model (f) small sale model (soft onstraned) (hard onstraned at 2 (hard onstraned at 5 Fg. 6: A one-dmensonal example ponts) to llustrate the zoom-n operatons. ponts) Orgnal Model Coeffents of Sale 1: 1, , , , ,1 0.5 Unonstraned Sale 2 Coeffents 1, , , , , Zoom-In Model Constraned grd ponts x 0.5 x Sale 2 Coeffents 1, , , , , Constraned grd ponts x 0.5, x x 0.25, x 0.125, x Sale 2 Coeffents 1, , , , , Tab. 2: The example of zoom-n operaton n Fg. 6. (a) orgnal model (b) zoom-n smaller sale model Fg. 7: A three-dmensonal zoom-n operaton example.

14 772 Varyng the radus results n a vetor of ntegral values I, wth some spef angles and. That s, I, f x,y,z x os os y os sn z sn dx dydz (0.20) where s the Dra delta funton. The rdgelet transform then s the 1D wavelet transform of the surfae ntegral values I, on the plane as n, a,b (0.21) I a,b,,, In general, our gener surfaelet transform s the 1D wavelet transform of the surfae ntegrals. For the ylndral surfaelet n Eqn.(0.5), the surfae ntegral s 2 I,,r0,r 1,r 2 f x,y,z r os os x os sn y sn z 1 r 2 sn x os y 2 2 r 0 For the ellpsodal surfaelet n Eqn.(0.6), the ntegral s r 1 os os x os sn y sn z I,,r0,r 1,r 2,r 3 f x,y,z r 2 sn x os y 2 r 3 sn os x sn sn y os z y y 2 2 r 0 2 dx dydz dx dydz (0.22) (0.23) x x Ω Ω (a) rdgelet transform (b) ylndral and ellpsodal surfaelets Fg. 8: Geometr nterpretatons of parameters n surfaelet transforms. The materals model an be reonstruted from mages based on the surfaelet transform defned by Eqns.(0.20)-(0.23). The proess of omputng surfaelet oeffents from 3D mages s llustrated n Fg. 9. From a 3D materals dstrbuton, the surfae ntegrals on surfaelets are alulated and arranged n a 3D matrx wth,, and as ndes. Then 1D wavelet transforms along the axs dreton are performed for all s and s. The results are surfaelet oeffents for a partular angle.

15 773 y x surfae ntegrals 1D wavelet transforms Ω Fg. 9: The general proess of surfaelet transforms. From sannng eletron mrosope (SEM) mages of SO2-MnO2 based erams [33] shown n Fg. 10-(a), the materal model an be reonstruted based on the surfaelet transform. Fg. 10-(b) shows the examples of surfae ntegrals based on the 3D rdgelet n Eqn.(0.4), where dfferent olors denote the magntudes. The domnant gran boundary orentatons an be easly dentfed. Fg.10-() shows the surfaelet oeffents after the wavelet transform. Here, the Haar wavelet s used. The low-pass oeffents are stored n the upper half of the 3D matrx, whereas hgh-pass oeffents are n the lower half. Some fne detals and loal features are revealed n the hgh-pass oeffents. y z x μ β surfae ntegral wth rdgelet α 1D wavelet transform wth Haar λ β α (a) SEM mages of SO2-MnO2 erams [31] Domnant gran boundary orentatons (b) wedgelet surfae ntegral More detaled nformaton () surfaelet oeffents Fg. 10: An example of materals model reonstruton based on 3D rdgelet. As a seond example to llustrate, a model of polyarylontrle-based nanofbers [34] an be reonstruted based on the ylndral surfaelet. The SEM mages are shown n Fg. 11-(a). The orgnal 3D mages n the Euldean spae thus are transformed to the reproal spae. The examples of surfae ntegrals are shown n Fg. 11-(b). The surfaelet oeffents are n Fg. 11-(), as n Eqn.(0.5). Here, the parameters of the ylndral surfaelet are r 1 r 2 1 and radus r The enters of rular nanofbers an be easly loated from the surfae ntegrals and oeffents. Note that not

16 774 only the orgnal materal model wth the largest number of resoluton models are also readly avalable, where the number of those terms wth small oeffents. ( ) a, b, p s n Eqn.(0.14) an be used, lower ( ) a, b, p s s redued by uttng off 6 SUMMARY In ths paper, we ntrodue a new multsale heterogeneous representaton, dual-rep, to support omputatonal geometry and materals desgn. The dual-rep model represents both nternal and boundary dstrbutons wth a unfed mplt form. Surfaelets as new bass funtons are proposed to apture boundary nformaton, whh are more effent than the tradtonal wavelets, whereas wavelets are used to represent nternal dstrbutons. Wth the unfed bass funtons, zoom-n operatons are developed to enable nteratve multsale spefatons. Surfaelet transform allows surfaelet oeffents to be dretly omputed from mages n model reonstruton. y z x μ β surfae ntegral wth ylndral surfaelet α 1D wavelet transform wth Haar λ β α (a) SEM mages of polyarylontrlebased nanofbers [32] Centers of rular nanofbers (b) ylndral surfae ntegrals () surfaelet oeffents Fg. 11: An example of materals model reonstruton based on ylndral surfaelets. REFERENCES [1] Rosen, D. W.: Thoughts on herarhal modelng methods for omplex strutures, Computer- Aded Desgn & Applatons, 6(3), 2009, [2] Kou, X. Y.; Tan, S. T.: Heterogeneous objet modelng: A revew, Computer-Aded Desgn, 39(4), 2007, [3] Bhashyam, S.; Shn, K. H.; Dutta, D.: An ntegrated CAD system for desgn of heterogeneous objets, Rapd Prototypng Journal, 6(2), 2000, [4] Koeng, O.; Fadel, G.: Applaton of genet algorthms n the desgn of mult-materal strutures manufatured n rapd prototypng, Pro. Sold Freeform Fabraton Symposum, pp , Austn, TX, August 9-11, [5] Wu, Z. W.; Soon, S. H.; Feng, L.: NURBS-based volume modelng, Internatonal Workshop on Volume Graphs, , 1999.

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