METAMORPHOSIS OF PERIODIC SURFACE MODELS

Transcription

1 Proceedings of ASE 2009 Internationa Design Engineering echnica Conferences & Coputers and Inforation in Engineering Conference IDEC/CIE 2009 August 0 Septeber 2, 2009, San Diego, Caifornia, USA DEC2009/DAC-870 EAORPHOSIS OF PERIODIC SURFACE ODES Cheng Qi and Yan Wang Departent of Industria Engineering & anageent Systes University of Centra Forida Orando, F 286 ABSRAC A phase transition is a geoetric and topoogica transforation process of aterias fro one phase to another, each of which has a unique and hoogeneous physica property. Providing an initia guess of transition path for further physica siuation studies is highy desirabe in aterias design. In this paper, we present a etaorphosis schee for periodic surface (PS) odes by interpoation in the PS paraeter space. he proposed approach creates utipe potentia transition paths for further seection based on three soothness criteria. he goa is to search for a sooth transforation in phase transition anaysis.. INRODUCION Coputer-aided nano-design (CAND) is an extension of coputer based engineering design traditionay at buk scaes to nano scaes. he genera target of odeing and siuation in nanoateria design is to search stabe and reaizabe structures and conforations with the inia tota syste energy. Geoetry optiization is the centra thee in ost of the nanoscae siuations. For the widey used oca search agoriths, siuation resuts are sensitivey dependent on the initia conforation. odeing ethods, which aow for the efficient construction of initia geoetries that are reasonaby cose to goba optia soutions, are iportant to iprove both convergence rate and accuracy of prediction. hus, enabing efficient structura description and editing is one of the key research issues in CAND. In the previous research [, 2], an ipicit surface odeing approach known as periodic surface (PS) ode is proposed. Periodic surfaces are either oci or foci. oci surfaces are fictiona continuous surfaces that pass through discrete partices in D space, whereas foci surfaces can be ooked as isosurfaces of potentia or density in which discrete partices are encosed. he PS ode aows for paraetric construction fro atoic scae to eso scae. Reconstruction of oci surfaces fro crystas [], surface degree operations to support fine-grained odeing [4, 5], and feature-based approach for crysta construction [6, 7] were aso studied. In this paper, we propose a surface orphing or etaorphosis approach for PS odes. his geoetry transforation is very usefu to siuate and visuaize phase transition processes in studying functiona aterias. A phase transition is a geoetric and topoogica transforation process of aterias fro one phase to another, each of which has a unique and hoogeneous physica property [8, 9]. ransforation of PS odes can hep to visuaize structure changes and provide initia estiations of transition paths. For exape, Figure (a) shows a BaSi 2 structure in its cubic phase and the corresponding foci surface ode, which encoses Ba atos. It has properties of seiconductor. Figure (b) shows a ayered phase of BaSi 2, which has properties of eta. he interest of phase transition anaysis is to search the goba optia transition path between the two phases with the inia potentia energy change. Structure transforation based on geoetric anaysis such as the one in Figure 2 can provide an initia guess of transition path for further physica siuation studies. In this paper, the type of phase transition which we are interested in is the oveent of any atos that resuts in a continuous change between two different crysta structures. he exapes of such type of phase transition can be diffusioness transforations and artensitic transforation. Any other types of phase transition such as a eting transforation or a freezing transforation are not in the scope of this paper. We reasonaby assue that the transition between two phases is processed in a region where a the physica conditions in that region are unifory distributed. Hence, we ignore the infuence of physica treatents such as teperatures and pressures, and ony focus on the etaorphosis of geoetry itsef. A surface orphing approach for PS odes is proposed in this paper. hree soothness criteria are proposed to quantify structure changes for pathway seection. he ain contribution of this paper is the unique surface orphing ethod of our PS ode based on interpoation in the PS paraeter space. Our ethod can provide utipe potentia transition paths for choices with different criteria. Copyright 2009 by ASE

2 (a) BaSi 2 structures in cubic phase (b) BaSi 2 structures in ayer phase Figure. BaSi 2 and corresponding foci surfaces Figure 2. Foci surface transition fro BaSi 2 cubic phase to ayer phase In the reainder of the paper, Section 2 gives a brief overview of reated work in surface orphing with ipicit and vouetric representations. Section reviews the basis of the periodic surface ode and its atrix for. Section 4 describes two surface orphing schees, which are direct inear interpoation and interpoation in the PS paraeters space. Section 5 proposes three soothness criteria for choosing surface orphing paths. 2. D EAORPHOSIS OF IPICI SURFACES Research effects in etaorphosis initiay focused on two diensiona iages. For iage orphing, there have been extensive investigations [0,, 2]. A direct extension fro 2D etaorphis to D was proposed by itta []. he D objects were represented by utipe 2D iages and 2D orphing techniques were used to orph between their 2D representations. he D interediate objects were thus reconstructed fro the resuting 2D iages. In D etaorphosis, azarus and Verroust [4] suarized and categorized a the agoriths in transforation between two shapes into two ajor approaches, voue based approaches and boundary based approaches. he voue based approaches focus on interpoations between two shapes in voxe representation. Pasko and Savchenko [5] defined a etaorphosis between two genera ipicit surfaces by direct inear interpoation of the corresponding vouetric vaues. he ethod was entirey autoatic but acked the contro over the transforation. Wyvi [ 6 ] presented a skeeton based approach to aow users to seect pairs of corresponding skeetons. he transforation coprised severa interpoations in associated potentia fieds or soft objects of paired skeetons. Kau and Rossignac [ 7 ] deveoped an interpoation agorith based on inkowski sus of two sets of vouetric data. he ethod was further extended for transforation between a set of convex poyhedra using Bézier interpoation and inkowski sus [8]. Gain and Akkouche [9] proposed an agorith for soft objects buit fro skeetons of convex shapes, which was a ixed approach between skeeton and inkowski sus. he paired skeetons were interpoated with inkowski sus, and soft objects were bended by interpoations. hen the fina interpoated shapes were the inkowski sus between the interpoated soft objects and skeetons. Barbier et a. [20] extended Gain s work by reoving the iitation to convex poygona eeents of arbitrary diension. he new approach presented a vast variety of shapes incuding curves, surfaces, and voues such as boxed or cone-spheres that ay be used as skeeta eeents. Hughes [2] proposed an approach with interpoation of two shapes in the Fourier doain. he approach interpoates the ow frequencies of the initia and end shapes whie the high frequencies of the end shape are increentay added in. he advantage of this approach is to avoid the shape distortion caused by direct interpoation of high frequency coponents. A siiar approach was proposed by He et a. [22], but it was based on D waveets. he initia vouetric data was decoposed with utipe eves of resoution, and the interediate surfaces were reconstructed after interpoation of ow frequencies. he spatia inforation within each frequency band of waveets enabed a sooth transition. o enabe oca contros, erios et a. [2] proposed a featurebased voue etaorphosis which aowed user to specify features in one shape corresponding to the other. hese paired features can be transfored fro one to the other during the orphing process. Cohen-Or and evin [ 24 ] proposed a orphing agorith based on three-diensiona distance fied. he initia and end shapes were first defored by soe pointto-point warping functions which were designed to enforce topoogica correspondence and geoetrica properties. he technique interpoates the distance vaues of each voxe and reconstructs the interediate surfaces out of the interediate distance fied. urk and O Brien [25] deveoped a etaorphosis schee to transfor between two shapes by creating a variationa ipicit function in a higher diension after artificiay introducing one extra diension. hus, the parae sices based on the extra diension represented the transforation sequence. urk and O Brien [26] further proposed another etaorphosis based on ipicit surfaces creations. In this ethod, a set of constraints which were fro scattered data of a surface created an ipicit function. he transforation between two shapes was defined by the ipicit surfaces buit fro the ixed constraints of the two shapes. Various interpoation approaches have been deveoped. Fausett et a. [27] deonstrated a technique to transfor between severa D shapes of different topoogy by bi-inear interpoation in a higher diensiona space. Fang et a. [28] presented a continuous fied based orphing agorith. In this agorith, a copex surface was construed by severa poyhedra skeetons. Basic continuous fied for each skeeton was created using variationa interpoation. he interediate surfaces between two shapes were approxiated by the iso- 2 Copyright 2009 by ASE

3 surface of the goba fied which is fused by the bended basic fieds of the two shapes. reece et a. [29] deveoped an agorith to iprove the transforation between two shapes by ensuring that no part of each surface reains disconnected during the orph. he orph was guided by correspondence of sphere representations of the two shapes, which was apped fro their distance fied voue representations. Optiizationbased approaches were aso taken. Cong et a. [0] deveoped an approach for shape etaorphis under the constraints of initia and end surface functions. With the p-apacian equation, a series of reguarized ters based on the gradient of the ipicit function was generaized. he approach soves the tie dependant ipicit function which iniizes the supreu of the gradient during the orph. Bao et a. [] presented a orphing process between two hoeoorphic point-set surfaces by optiizing an energy function. Aong boundary based approaches, Sun et a. [2] proposed to interpoate poyhedra odes using intrinsic shape paraeters, such as dihedra anges and edge engths. azarus and Verroust [] deveoped a etaorphosis for cyinder-ike objects by constructing poyhedra eshes using paraeterization for two shapes. he paraeters incude D axes for discretization, vertices, edges and faces. he shape transforation was defined by the interpoation of the paraeterization. Chen and Parent [4] introduced a user interactive agorith to extract paraeters for two D objects represented by panar contours. Weighted averaging of these paraeters defined the shape transforation. Kanai et a. [5] presented an agorith for D geoetric etaorphosis between two objects based on haronic ap. 2D ebeddings were created by haronic ap for the two D shapes with adjacent reations preserved. Inbetween shapes were created by erging of the two 2D ebeddings. Different fro the above, the etaorphosis approach deveoped in this paper is based on the interpoation in the PS paraeter space. his provides certain eves of contro for our PS odes. In addition, two soothness criteria and one heuristic orphing ethod are proposed for pathway seection.. PERIODIC SURFACE A periodic surface is generay defined as = = ( ) ψ() r = μ cos2 πκ ( p r ) = 0 (.) where κ is the scae paraeter, p [,,, ] = a b c θ is a basis vector, which represents a basis pane in the -space E, r = [ x, yzw,, ] is the ocation vector with hoogeneous coordinates, and μ is the periodic oent. We usuay assue w = if not expicity specified. It eans the diensions x, y and z are in the sae scae. he degree of ψ () r in Eq.(.) is defined as the nuber of unique periodic basis vectors in set { p }, deg ( ψ ( r) ): = { p }. he scae of ψ () r is defined as the nuber of unique scae paraeters in set { } sca ψ ( r ) : = κ. We usuay assue the scae paraeters are natura nubers ( κ N ). Each basis vector can be regarded as a set of parae 2D subspaces in E, which pays an iportant roe in interactive anipuation of PS odes. he PS ode described in Eq.(.) can aso be represented by a PS paraeter atrix as shown in Figure. he PS paraeter atrix contains four sub-atrices ( κ, μ, p and 0). Each eeent μ in μ aong with the -th eeent κ in κ and the -th row p in p defines a cosine function μ cos( 2 πκ ( p r )). Periodic surfaces are thus odeed by the su of these cosine functions. Switching the first couns or the ast rows of the PS paraeter atrix does not change the periodic surface. here are totay!! possibe cobinations of couns and rows in the sub-atrix μ, and!! different PS paraeter atrices represent the sae periodic surface. his property is iportant when we appy interpoation in the PS paraeter space, as discussed ater in Section 4.2. κ κ κ 0 μ μ μ p κ = rows μ p μ μ μ p μ2 μ μ2 p2 μ μ μ p κ, ( ) { } couns Figure. PS paraeter atrix PS ode is epoyed for surface creation because periodic structures are ubiquitous in natura aterias. Crysta structures are one of the good exapes that certain cobinations of atos are appeared periodicay in the D space. Due to its periodic property, a cosed-for equation is abe to represent a periodic surface. Hence, it is an efficient too for periodic structure odeing. 4. ORPHING SCHEES In this section, two orphing schees are discussed. hey are direct inear interpoation of vouetric data and interpoation in the PS paraeter space. In the first ethod, vouetric data is generated by interpoating between those of the initia and the end surfaces. he interediate surfaces are the isosurfaces based on the interpoated vouetric data. In the second ethod, inear interpoation is appied between the paraeters of the initia and the end surfaces. Hence, the interediate surfaces are created by cosed-for PS odes. Copyright 2009 by ASE

4 4. Vouetric Interpoation inear interpoation is widey used in geoetric orphing process. It can be appied in the orphing of periodic surfaces. For the initia surface ψ () = 0 r and the end surface ψ () r = 0 2, vouetric interpoation between these two surfaces can be defined as ( λψ ) ( r) + λψ 2( r ) = 0, where λ [0,]. As an exape, Figure 4 iustrates the effect of the interpoation between the P surface and I-WP surface. he PS paraeter atrices for P surface and I-WP surfaces are shown as foows P surface paraeter atrix: I-WP surface paraeter atrix: λ = 0.05 λ = 0.0 λ = 0.5 λ = 0.20 λ = 0.25 λ = 0.0 λ = 0.5 λ = 0.40 λ = 0.45 λ = 0.50 λ = 0.55 λ = 0.60 λ = 0.65 λ = 0.70 λ = 0.75 λ = 0.80 λ = 0.85 λ = 0.90 λ = 0.95 λ =.00 Figure 4. inear interpoation between P surface and I-WP surface he ethod of inear interpoation of vouetric data in the three diensiona space is straight forward. However, ony one transition path is possibe by appying this ethod if the initia and the end surfaces are known. here is no contro over the transforation process because the transition path can not be ocay odified. Creating correspondences between certain oca regions is not possibe. In addition, the surface transforation ay not be continuous. For instance, in the exape of Figure 4, the topoogy changes draaticay fro λ = 0.5 to λ = he six hoes ceary as features vanish siutaneousy in the interediate surface. Siiary, the features of hoes reappear in the interediate surface when λ = he figure aso shows that the transition path is hardy unifor because the topoogy changes ore in the first haf of the path than in the second haf. o investigate ore possibe transition paths, the ethod of interpoation in the paraeter space of PS odes is proposed in Section Interpoation in the paraeter space of PS odes An aternative orphing schee is to interpoate in the PS paraeter space between the initia and the end surfaces. ore specificay, for two surfaces = = 2 2 ( ) ψ () r = μ' cos2 πκ'( p' r ) = 0 (4.) 2 = = ( ) ψ () r = μ'' cos2 πκ''( p'' r ) = 0 (4.2) the interpoated surface is = = ( ) ψ() r = μ cos2 πκ ( p r ) = 0, where μ = ( λ) μ ' + λμ '' κ = ( λ) κ' + λκ'' p = ( λ) p' + λp'' (4.) λ [0,] = ax(, 2) = ax(, 2) κ ' 0 In the PS paraeter atrix for, et A = ' ' for ψ () r μ p κ '' 0 and B = '' '' for ψ 2 μ p () r. he interpoation is aso defined as ( λ) A+ λb, where λ [0,]. In other words, the interpoation is a process to ineary transfor fro one atrix to another, as iustrated in Figure 5. Here, the PS paraeter atrices of the initia and the end surfaces are to be extended to the sae size before interpoation if they are in different sizes. Figure 6 iustrates the atrix extension. In Figure 6(a), if < 2 and < 2, the couns for the scae paraeters κ 's can be extended fro to 2, and the rows for the basis vectors can be 4 Copyright 2009 by ASE

5 extended fro to 2 for surface ψ () r by setting μ ' = 0, κ' = κ'' and p' = p '' for [ +, 2] or [ +, 2]. In Figure 6(b), if < 2 and > 2, the couns for scae paraeters of surface ψ () r can be extended fro to 2 by setting μ ' = 0 and κ' = κ'' for [ +, 2]. At the sae tie, the rows for basis vectors of surface ψ () r can be extended fro 2 2 to by setting μ '' = 0 and p'' = p ' for [ 2 +, ]. κ' κ' κ' 2 0 μ' μ' μ' p ' μ' μ' μ' p ' μ' 2 μ' μ' 2 p ' 2 μ' μ' μ' ' p Figure 5. atrix transforation (a) Figure 6. atrix extension 0 κ'' κ'' κ'' 2 0 μ'' μ'' μ'' p '' μ'' μ'' μ'' p '' μ'' 2 μ'' μ'' 2 p '' 2 μ'' μ'' μ'' '' p As entioned in Section, switching rows or couns respectivey within the ast rows or the first couns of the PS paraeter atrix does not change the periodic surface. However, it akes the interpoated periodic surfaces different when the ethod of interpoation in the PS paraeter space is appied, as seen in Eq.(4.). hus, for two surfaces ψ () r in Eq.(4.) and ψ () 2 r in Eq.(4.2), there are possiby ax (, 2)!ax (, 2)! 2! 2! different transition paths between the initia and end surfaces defined in Eq.(4.). he objective is to seect the soothest transition path aong a these potentia candidates. hus, quantitative criteria for soothness are needed. In the next section, two soothness criteria are proposed SOOHNESS CRIERIA In this section, we propose two soothness criteria, inia space fied change and inia surface energy change, which wi be described in Sec. 5. and Sec. 5.2, respectivey. he first criterion is based on the potentia energy point of view, which assues the tota change of space energy wi be (b) 0 inia during the orphing process. On the other hand, the second criterion is based on the surface energy point of view. It assues the optia transition paths wi ake the tota change of surface energy inia. In addition, a heuristic ethod of surface orphing is proposed in Sec inia Space Fied Change In this criterion, it is assued that the optia transition path is the one with the inia change of space fied accuuativey during the orphing process. Since the vouetric vaues associated with PS odes represent the potentia energy in the three-diensiona space, the criterion of inia space fied change is to easure the tota change of potentia energy caused by the orphing process. Hence the transition path with the inia tota vouetric vaue change is considered to be the soothest one. ore accuratey, in the doain D = [ 05. x 05., 05. y 05., 05. z 05.,], the space fied change (SFC) between two periodic surfaces ψ () a r and ψ b () r is defined as D ( ψ ( ) ψ ( )) 2 SFC b r a r dr. Suppose ψ 0 () r is the initia surface, ψ n () r is the end surface and ψ k () r s (k =, 2,, n ) are the n interediate surfaces, the tota SFC (SFC) is cacuated by n ( ψ ( ) ( )) 2 t r ψt r dr (5.) SFC = t = D hus, the objective of this criterion is to iniize the SFC. In order to find the transition path with the inia SFC, we need to search and evauate a possibe candidates. he foowing exape shows the orphing process fro the P surface to I-WP surface using this criterion. abe shows the corresponding PS paraeter atrices of the P surface and I-WP surface after the atrix extension. In this exape, = 2 =, = and 2 = 9, as shown in abe. he degree of the P surface is extended to the sae as that of the I-WP surface by setting P' = P '' and μ ' 0 =, for = 4,5,...,9. he tota nuber of the transition paths is ( ) ( ) ax,!ax,9! = 504. For each of the n interediate! 9! fraes (n=20) in the transition path, we cacuate the su square of vouetric vaue changes. A resoution of in the discretized space is used. he SFC is found according to Eq.(5.) using step size he optia transition path is the one with the inia SFC. he best pair of PS paraeter atrices based on this criterion is isted in abe. Figure shows the optia transforation sequence. 5 Copyright 2009 by ASE

6 abe. he PS paraeter atrices of P and I-WP surface for inia space fied change criterion P surface I-WP surface λ = 0.05 λ = 0.0 λ = 0.5 λ = 0.20 λ = 0.25 λ = 0.0 λ = 0.5 λ = 0.40 λ = 0.45 λ = 0.50 H ( ψ ) ψ ψ 0 K = 4 ψ (5.2) where ψ is gradient and H ( ψ ) is Hessian atrix. It is assued that the optia transition path wi ake the accuuated change of surface energy inia during the orphing. We ap a periodic surface fro its vouetric space into its surface energy space using Eq.(5.2) in the doain D = [ 05. x 05., 05. y 05., 05. z 05.,]. he surface energy change (SEC) between two periodic surfaces ψ () a r and ψ b () r is defined as ( ( ) ( )) 2 SEC Kb r Ka r dr. Suppose that ψ 0 () r is the D initia periodic surface, ψ n () r is the end periodic surface and ψ () r s (k =, 2,, n ) are the n interediate periodic k surfaces. he tota SEC (SEC) is cacuated by the Eq. (5.). hus, the objective of this criterion is to iniize the SEC. In order to find the one with the inia SEC, we need to search and evauate a the possibe transition paths. n ( K ( ) ( )) 2 t r Kt r dr (5.) SEC = t = D Figure 8 shows the optia orphing sequence fro P surface to I-WP surface. In this exape, a resoution of in the discretized space is used. Step size is chosen as 0.20 so that the SEC is based on the accuuated surface energy change of five surfaces or five steps. abe 2 ists the best pair of PS paraeter atrices based on this criterion. λ = 0.55 λ = 0.60 λ = 0.65 λ = 0.70 λ = 0.75 λ = 0.05 λ = 0.0 λ = 0.5 λ = 0.20 λ = 0.25 λ = 0.80 λ = 0.85 λ = 0.90 λ = 0.95 λ =.00 Figure 7. he optia transition path based on inia SFC criterion 5.2 inia Surface Energy Change Curvature in geoetry is generay known as the aount by which a geoetric object deviates fro being fat. In differentia geoetry, surface energy can be defined as E = ( k ) 2 + k2 da 4 where k and k 2 are the two principe curvatures. In this criterion, we consider the Gaussian curvature of a periodic surface as an indicator of surface energy. Godan [6] deveoped the curvature foruas for ipicit surfaces. herefore, for a periodic surface in Eq. (.), the Gaussian curvature can be cacuated by λ = 0.0 λ = 0.5 λ = 0.40 λ = 0.45 λ = 0.50 λ = 0.55 λ = 0.60 λ = 0.65 λ = 0.70 λ = 0.75 λ = 0.80 λ = 0.85 λ = 0.90 λ = 0.95 λ =.00 Figure 8. he optia transition path based on inia SEC criterion 6 Copyright 2009 by ASE

7 abe 2 he PS paraeter atrices of P and I-WP surface for inia surface energy change criterion P surface I-WP surface Heuristic orphing Criteria he cacuation of either inia tota change of space fied or inia tota change of surface energy is a tie consuing process, because the vouetric data or Gaussian curvature of a interpoated interediate surfaces needs to be cacuated in a vouetric doain for a possibe transition paths. In this section, we propose a heuristic orphing criterion in the case of = 2 = and κ' = κ'' = in Eq. (4.) and Eq. (4.2) for an easy-to-copute soution. In fact, the conditions of = and κ = in Eq. (.) can aways be et if we utipy a the scae paraeters κ s with their corresponding p s to create new basis vectors. he purpose of this criterion is to reduce the tie copexity to deterine an acceptabe sooth transition path by avoiding evauations of a the interediate surfaces. here are three assuptions behind the heuristic orphing criterion. First, since periodic oents μ 's are the agnitude of cosine functions, they are considered to be scae reated paraeters. Second, since basis vectors P 's define the frequency and phase of cosine functions, they are considered to be shape reated paraeters. Finay, it is assued that the orphing process is sooth if the changes of basis vectors are as itte as possibe. For two surfaces ψ () r and ψ () 2 r in Eqs. (4.) and (4.2) with κ ' 0 = 2 = and κ' = κ'' =, et A = ' ' for ψ () r μ p κ '' 0 and B = '' '' for ψ 2 () r. Both A and B are ( + ) 5 μ p atrices, where = ax(, 2 ). We generaize the heuristic orphing criterion as foows.! Step. For a the possibe transition paths, find! 2 the one with inia p' p ''. = Step 2. If ore than one transition paths are found in step, further choose the one with inia inia 2 p ' if > 2. = p '' if < 2 or = Step. If ore than one transition paths are found in step2, arbitrariy choose any one of the. For the exape to find a transition path fro the P surface to I-WP surface, we appy this heuristic criterion. In step, 48 transition paths aong the tota of 504 possibe transition paths are found. In step 2, 5 transition paths are further seected as candidates. Finay in step, we arbitrariy choose one transition path, as shown in abe. Figure 9 shows the heuristic transition path with the step size of abe he PS paraeter atrices of P and I-WP surface for heuristic orphing criterion P surface I-WP surface λ = 0.05 λ = 0.0 λ = 0.5 λ = 0.20 λ = 0.25 λ = 0.0 λ = 0.5 λ = 0.40 λ = 0.45 λ = 0.50 λ = 0.55 λ = 0.60 λ = 0.65 λ = 0.70 λ = 0.75 λ = 0.80 λ = 0.85 λ = 0.90 λ = 0.95 λ =.00 Figure 9. Heuristic orphing process fro P surface to I-WP surface 7 Copyright 2009 by ASE

8 5.4 Resut discussions We presented three quantitative soothness criterions in this section and exapes are given. Here, we ony focus on etaorphosis of geoetric aspect of a shape. Any physica conditions such as teperature which are possibe to cause deviations between the actua transition path and the cacuated optia path are ignored in this stage. Since the type of phase transition that we are interested in is the oveent of any atos which resuts in changes between two crysta structures, it is reasonabe to assue that a physica conditions ike teperatures and pressures are unifor throughout the regions under consideration. In the inia space fied change criteria, the best transition path is the one with the east change of fied in the D space. Observing the shape transforation in Figure 7, we find the isosurfaces with isovaue zero in the transition path are reasonabe to expect the inia space fied change subjectivey. he inia surface energy change criteria is to find the transition path which causes the east change of surface Gaussian curvature, an indicator of the degree of bend of a shape. According to the resut shown in Figure 8, the shape transforation appears to be ess change in surface bend than those in Figure 7 and Figure 9. he heuristic orphing criteria does not have physica interpretation, but it is faster in cacuation than the other two because it is no need to cacuate vouetric data for each possibe transition path. he resut shown in Figure 9 shows the shape transforation is reasonaby sooth. One of the advantages of these soothness criteria is to provide a quantitative way so that the best transition path can be deterined objectivey. In addition, the ethod returns different outputs for different criteria. A proper seected criterion is needed ony for seecting a transition path rather than another phase transition schee. However, the ajor concern of the inia space fied change and surface energy change criteria is cacuation efficiency because a possibe aternative transitions ust be evauated before the optia path can be concuded. A heuristic criterion is proposed for quick and acceptabe soution, which ay not be the optiu. 6. SUARY AND FUURE WORK he artice presents a etaorphosis ethod between two PS odes, using interpoation in the PS ode paraeters space. he paraeter atrix representation of periodic surfaces is used so that the shape transforation is generated by atrix transforations. he etaorphosis of periodic surfaces creates utipe transition paths between two shapes. For further seections, different soothness criteria are used to contro the orphing process. wo soothness criteria, the inia space fied change and the inia space energy change, are proposed in this paper. o reduce cacuation tie, a heuristic criterion is aso presented. he in-between shapes generated by the proposed ethod and soothness criteria are natura and satisfactory. One of the future extensions is to set up corresponding contro points of the two periodic surfaces over the transforation. It ay yied ore sooth, hence ore natura, transition path throughout the etaorphosis. Another potentia extension is to add a fixed nuber of partices as a constraint during the orphing process. he consideration of the constraint wi ake it ore reaistic to ateria design. ACKNOWEDGEEN his work is supported in part by the NSF grant CI REFERENCES [] Wang, Y. (2006) Geoetric odeing of nano structures with periodic surfaces. ecture Notes in Coputer Science, Vo.4077, pp.4-56 [2] Wang, Y. 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