1

Boundary-Aware Multi-Domain SubspaceDeformation

Yin Yang, Weiwei Xu, Xiaohu Guo, Kun Zhou, Baining Guo

Abstract—In this paper, we propose a novel framework for multi-domain subspace deformation using node-wise corotationalelasticity. With the proper construction of subspaces based on the knowledge of the boundary deformation, we can use theLagrange multiplier technique to impose coupling constraints at the boundary without over-constraining. In our deformationalgorithm, the number of constraint equations to couple two neighboring domains is not related to the number of the nodes onthe boundary but is the same as the number of the selected boundary deformation modes. The crack artifact is not present inour simulation result and the domain decomposition with loops can be easily handled. Experimental results show that the singlecore implementation of our algorithm can achieve real-time performance in simulating deformable objects with around quartermillion tetrahedral elements.

Index Terms—model reduction, domain decomposition, FEM, deformable model

F

1 INTRODUCTION

MOdel reduction is an important technique foraccelerating the physics-based simulation of

deformable objects. The basic idea is to project thehigh-dimensional equation of motion to a carefullychosen low-dimensional subspace to construct a re-duced model. Traditional global subspace methods,however, cannot handle the object’s local deformationbehaviors well unless a large number of basis vectorsare used, which in turn would cancel out the bene-fit of acceleration. Multi-domain subspace techniquesprovide a good solution to this problem by parti-tioning the deformable object into multiple domainsand constructing reduced models for each domainindependently. The advantages are two-fold. First,local deformation behaviors can be well captured witha modest number of basis vectors for each domain.Second, local simulation of each domain can moreflexibly handle deformable objects of complex, seman-tic geometries. Users can easily specify the numberof degrees of freedom (DOFs) and the types of basesfor each domain to accommodate hybrid simulationresults.

The key challenge in applying multi-domain sub-space techniques to deformable object simulation is

• Yin Yang and Xiaohu Guo are with the Department ofComputer Science, The University of Texas at Dallas, TX. E-mail:{yinyang|xguo}@utdallas.edu

• Weiwei Xu is with Hangzhou Normal University, Hangzhou, China.E-mail: weiwei.xu.g@gmail.com

• Kun Zhou is with State Key Lab of CAD and CG, ZhejiangUniversity, Hangzhou, China. E-mail:kunzhou@acm.org

• Baining Guo is with Microsoft Research Aisa, Beijing, China. E-mail:bainguo@microsoft.com

the seamless coupling of the neighboring domain-s at their boundary interface. Recently, two cou-pling methods have been developed for multi-domainsubspace deformations using the reduced nonlinearSt.Venant-Kirchhoff (StVK) deformable model. Kimand James [1] proposed to couple the domains usingpenalty (spring) forces for character skinning. Withan increasing number of basis vectors in the reducedmodel, the cracks become invisible visually. However,this method requires pre-determined motions of localframes for each domain, and the length of the timestep is usually small due to the possible large penal-ty forces at the boundary interface. Another multi-domain subspace deformation method in [2] relies onshape matching and mass lumping at the boundaryinterfaces to handle the coupling issue. Nevertheless,cracks might still be visible if the deformation goeslarge and this problem can be remedied by geometricblending operations at the post-simulation stage. Thismethod works well for the domain decompositionwith tree-like hierarchies and small domain interfaces.However, seamless coupling of multi-domain sub-space deformations with arbitrary domain decompo-sition remains a technical challenge.

The goal of this paper is to develop a seamless cou-pling method for multi-domain subspace deformationin the framework of node-wise corotational elasticity.To this end, we propose a boundary-aware modeconstruction method that characterizes the deforma-tion subspace of each domain through its boundarydeformations. Instead of posing coupling constraintson boundary node pairs, which can easily lead toover-constraints for large-scale meshes, our algorithmformulates these constraints with rigid and soft bound-ary modes, which form a compact representation ofboundary deformations. In this way, we can apply theLagrange multiplier technique to solve the boundary

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coupling constraints without over-constraining. Theboundary modes are computed by solving the staticequilibrium equations as in component mode synthesis(CMS) [3]. The large deformation of each domain issimulated using the modal warping technique [4].

With our algorithm, cracks are avoided in the sim-ulation result and the domain decomposition withloops can be naturally handled as well. We show thecapability of our multi-domain subspace deformationalgorithm by simulating a variety of large-scale de-formable objects. User manipulation, such as directlyconstraining the node position or the rotation of thedomain boundary, is also supported in our algorithmto ease the animation production.

The rest of the paper is organized as follows.Sec. 2 reviews the related work. Sec. 3 describes howto construct deformation modes from the boundarydeformation. The deformation algorithm and directmanipulation methods are described in Secs. 4 and5. Experimental results and limitations are discussedin Sec. 6. Finally, we conclude in Sec. 7.

2 RELATED WORK

Physics-based simulation of deformable objects hasbeen an active research topic in computer graphicssince 1980s. A comprehensive survey can be foundin [5], [6]. Terzopoulos et al. [7], [8] proposed afundamental framework to simulate 3D deformationsbased on the theory of elasticity. The ordinary differ-ential equations (ODEs) describing the dynamics ofdeformable models can be numerically solved withthe finite element method (FEM) and generate realisticdeformations. High computational cost is a drawbackof FEM especially for the finite element meshes oflarge size. In order to make deformable models morepractical for interactive applications, numerous workshave been proposed and have greatly advanced re-lated areas. Multi-resolution [9] or adaptive simula-tion [10], [11] uses hierarchical or adaptive bases ofthe deformation to accelerate the computation. Thesetypes of techniques use the high-level bases to repre-sent general deformations and the low-level or refinedbases for more detailed deformations when necessary.Similarly, embedded mesh, mesh coarsening or con-trolling lattice [12], [13], [14], [15], [16] handle thedeformation with auxiliary coarsened grids.

Corotational elasticity and its variations are widelyused in computer graphics for fast simulation of largescale deformations. It was first introduced by Mulleret al. [17], [18] via stiffness warping and widely adoptedin various applications [19], [20], [21], [22]. It has beenextended to thin shell [23] and meshless [24] sim-ulation. Warp-canceling corotation has recently beenproposed to improve the approximation accuracy ofstiffness warping to element-wise corotational elas-ticity [25]. Although the stiffness matrix can be keptconstant in node-wise corotational methods as in [4],

[17], it is still hard to directly apply it to the real-timesimulation of a large scale mesh as in our case, sincethe pre-factorization of the large scale stiffness matrixmight take hours and sometimes not plausible on adesktop PC due to the memory limitations.

Another series of contributions are based on modalanalysis (MA), which is a well-developed techniquewidely used in engineering areas. The MA utilizes theeigen decomposition to project the full deformationspace to the vibrations of different frequencies [26],[27], [28], [29], [30]. The eigenvectors associated withlow vibration energies are discarded as they are be-lieved to have less contribution to the final defor-mation. In order to handle rotational deformations,Choi and Ko [4] proposed a technique called modalwarping. The curl of the linear displacement field isused to estimate nodal rotation and warp the dis-tortion induced by using linear modal bases. Alter-natively, nonlinear deformation can also be capturedwith modal derivatives [31] which extends the lineardeformation subspace to the parabolic subspace. On-the-fly subspace construction [32] provides anotherdirection to accelerate the simulation. The subspacebases are the recently-simulated displacements andvary during the simulation with the extra cost of ex-amining residual error periodically. Geometry-basedshape matching [33], [34] provides an alternative forfast computation of soft 3D volumes. Unfortunately,shape matching is not able to incorporate the materialproperties intuitively and increasing the number ofDOFs does not necessarily lead to a more accuratesimulation.

Local subspace methods such as [1], [2], [35] can becategorized as domain decomposition methods (DDMs).The input mesh is decomposed into mutually disjointdomains. The model reduction is then applied to eachdomain. Because the domains are always connectedand interacted with the neighbors, the domain’s defor-mation is not unconstrained like a single deformableobject. The key technical challenge of local subspacemethods is to impose domain coupling with lowcosts when there exist a large number of boundaryDOFs. Such inter-domains constraints should be con-sidered during the selection of deformation modesin order to construct a reasonable local subspace ateach domain. Huang et al. [35] use node-pair positionconstraints for domain coupling and a pre-computedforce-displacement matrix to accelerate matrix-vectormultiplication. Barbic and Zhao [2] adopt a passiveinter-domain deformation mechanism with rigid in-terface fitting when the parent domain deforms. Thisframework works well for domain decompositionwith tree-like hierarchies, and the pre-computationtime can be significantly reduced when a large num-ber of domains are of the same geometry. However,due to the rigid interface assumption, it is not suitablefor domains connected through large soft interfaces.Kim and James [1] use spring forces to avoid the

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stiffening induced by the subspace coupling. Mean-while, additional damping forces are also provided tosuppress boundary jigging. This coupling strategy iseffective and simple to implement. It targets characterskinning where the motion of the local frame ofeach domain must be pre-determined. These DDMtechniques, unfortunately do not fully incorporate theboundary condition during the construction of thelocal subspace. As a result, the displacements at theduplicated domain interfaces may experience slightincompatibility and crack or interpenetration couldappear. However this problem can be easily fixed bya simple blending treatment at post-simulation stageas in [2].

Our algorithm complements existing methods. It isinspired by CMS [3] to construct the modes at thestatic equilibrium while we perform model reductiongeometrically at the boundary DOFs to avoid theover-sized boundary problem. The domain couplingconstraint can be directly imposed on the reducedcoordinates without the over-constraining problem.Boundary modes constitute the static deformation of thedomains. The extra internal vibrational deformationis also captured with interior modes that do not par-ticipate in the domain coupling. Domains are alwaysexactly coupled and the post-simulation processing isavoided.

3 BOUNDARY-AWARE MODE CONSTRUC-TION

The input finite element mesh (tetrahedral mesh inthis paper) is called a host mesh and is to be di-vided into multiple sub-meshes or domains. We en-force the face-connectivity between a pair of neigh-boring domains such that they must share at leastone triangle face if considered connected1. The modesare simply the pre-computed domain displacementswhich serve as local subspace bases. Our boundary-aware mode construction characterizes the deforma-tion subspace of each domain through its bound-ary deformation. Unlike the classic CMS method [3]which provides complete boundary freedom by as-signing each boundary DOF an individual mode,we reduce the boundary freedom with the use ofgeometrically-constructed bases. After that, the corre-sponding boundary modes of the domain are comput-ed through solving the static equilibrium with linearelasticity. Besides boundary modes, internal modesare also incorporated for enriched local deformation.

1. The face-connectivity avoids the singularity of the sub-stiffnesscorresponding to internal DOFs, during the mode computation.Because the interface’s displacement is always constrained duringthe computation of domain modes, as long as interface is able todetermine the rigid body motion of the domain, the internal submatrix is always non-singular.

3.1 DOF ClassificationA domain with n nodes has total of 3n DOFsas each node has independent freedoms along x,y and z . The DOFs that are shared with neigh-boring domains are called boundary DOFs denot-ed by set B. All the other DOFs are called in-ternal DOFs (even they may be located at thesurface of the mesh) and are denoted by set I.

Fig. 1. A three-domain bar.

A DOF is either in Bor in I and cannot bein B and I at the sametime. If a domain has kneighboring domains, Bis further grouped intok subsets, B1, B2,...Bk.Each subset holds DOFsthat are shared with thesame neighbor and iscalled an interface. Fig. 1shows an illustrative 2Dexample of a bar model.The middle domain hasthe boundary with twointerfaces. Let Φ denote the modes of a single domain.According to the DOF types, it can be decomposedinto two parts Φ = [Φ>I |Φ>B ]>. As illustrated in Tab.1,in the following sections where the detailed modecomputation is explained, we use subscript to denotethe DOF type and superscript for mode type.

.

Φ Mode matrixΦR Rigid (R) boundary mode matrix (Sec. 3.2)ΦS Soft (S) boundary mode matrix (Sec. 3.2)ΦN Normal (N) mode matrix (Sec. 3.3)ΦI Inertia (I) mode matrix (Sec. 3.3)ΦI Mode submatrix corresponding to internal (I) DOFsΦB Mode submatrix corresponding to boundary (B) DOFsΦBk

Mode submatrix corresponding to DOFs on interface Bk

TABLE 1Matrix notation used in the mode computation.

3.2 Boundary ModesThe boundary modes are the domain’s displacementat static equilibrium status when it is imposed toexternal boundary displacements (e.g., displacementat B). Such equilibrium can be expressed in the formof Ku = f , where the domain’s stiffness matrix isdenoted by K. u and f represent the displacementand forces of the domain. The unknown respondingdeformation at I is computed by solving the follow-ing equilibrium:

KII KIB1... KIBk

KB1I KB1B1 ... KB1Bk

...KBkI KBkB1 ... KBkBk

[

ΦIΦB

]=

[0FB

].

(1)

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The domain’s stiffness matrix is grouped and orderedcorresponding to the classification of DOFs (e.g.,boundary or internal DOFs) and it is constant underlinear elasticity. fB is the external force applied at theboundary to drive the boundary displacement. If adomain has more than one interface (e.g., k > 1), themodes are computed independently for each interfacesuch that the imposed boundary displacement hasnonzero values only at one interface each time andall the other interfaces are fixed. Consequently, let ΦBi

be the nonzero interface displacement for Bi, and ΦBhas the block-diagonal-like form as:

ΦB =

ΦB1

ΦB2

. . .ΦBk

.Rigid boundary modes The rigid boundary modes,denoted by ΦR, represent the domain’s deformationwhen its interfaces only have rigid body motion.Correspondingly, ΦR

Bihas six columns, where the

first three columns in ΦRBi

represent three linearlyindependent translational displacement fields overBi. A natural choice is to use unit displacementsalong each axis as x = [1, 0, 0]>, y = [0, 1, 0]> andz = [0, 0, 1]> for each interface node. The other threecolumns are the rotational modes constructed by as-signing each interface node the displacements alongthe tangent directions of the rotations around threelinearly independent rotation axes (i.e. x, y and z).We set the centroid of the interface ci as the rotationpivot. Such tangential rotation modes are only ableto represent infinitesimal rotations for linear elasticity.We later show in Sec. 5 that they can also representlarge rotations with the help of modal warping. Foran interface node p whose rest position is p, thecorresponding 3 × 6 block in ΦR

Bihas the structure

like:

ΦRp,p∈Bi

= [x|y|z|(p−ci)×x|(p−ci)×y|(p−ci)×z]. (2)

By expanding the first line of Eq. 1 (which correspond-s to the internal DOFs of the domain) and substitutingΦBi with ΦR

Bi, ΦRI can be computed through:

ΦRI = −K−1II [KIB1

ΦRB1|KIB2

ΦRB2| . . . |KIBk

ΦRBk

]. (3)

Stacking ΦRI and ΦR

B yields the rigid boundarymodes:

ΦR =

[ΦRI

ΦRB

]. (4)

The number of rigid boundary modes is 6k which on-ly depends on the number of neighbor domains. Therigid boundary modes represent the general deforma-tion of the domain and they are always chosen as thedomain’s subspace bases in our implementation.Soft boundary modes Soft boundary modes com-plement the rigid boundary modes by incorporating

the deformations that are induced by non-rigid in-terface displacements. Assigning each boundary DOFan individual boundary mode will lead to a verybig system if there are many boundary nodes. Thiscompromises the original purpose of using modelreduction. Instead, the soft boundary modes are de-signed to only capture the most notable interfacegeometry, which is similar to regular modal analysisof deformable model. For each interface, we computeits manifold harmonic bases [36] (denoted by H), bysolving the generalized eigen problem of the Laplacianmatrix of the interface: −QH = ΛBH, where Q,Bare square symmetric matrices with size nBi , thenumber of nodes on interface Bi. Elements in Q,Bare computed with:{

Qa,b =cot(βa,b)+cot(β

′a,b)

2Qa,a =

∑bQa,b

and

{Ba,b =

|t|+|t′|12

Ba,a =∑

t∈St(a) |t|6

,

where t, t′ are two triangles that share the edge (a, b)with area |t| and |t′|. βa,b and β′a,b denote the twoangles opposite to the edge (a, b) in t and t′. St(a)stands for the set of triangles incident to a.

Each Harmonics basis is a vector of size nBi. It is

spanned to to represent the interface displacements inx, y and z directions respectively as:

ΦSBi

= H⊗ I, (5)

where ⊗ denotes the Kronecker product and I ∈ R3×3

is the identity matrix. The complete set of harmonicbases spans the full space of the interface displace-ment and the harmonic bases corresponding to thelarge eigenvalues can be discarded to reduce thenumber of interface freedoms since they representthe detailed geometry features of the interface. Suchproperty of harmonics is also utilized for mesh defor-mation with model reduction [37], [38].

Redundant bases could be induced if an inter-face displacement represented with harmonic basesis close to rigid. Therefore, for a given harmonic-spanned interface displacement φSBi

∈ R3nBi×1, we

check the residual error of projecting φSBionto the

existing rigid interface subspace spanned by ΦRBi

:

e =‖ [ΦR

Bi]([ΦR

Bi]>[ΦR

Bi])−1[ΦR

Bi]>φSBi

− φSBi‖

‖ φSBi‖

. (6)

If the error e is small enough (e.g., < 1% in our im-plementation), it indicates that a certain combinationof the existing rigid interface displacements is able torepresent φSBi

well, and φSBiis opt out. It is noteworthy

that the first harmonic basis has zero eigenvalue. Thecorresponding three bases computed by Eq. 5 areidentical to the three translational modes. As a result,they are always excluded.

The unknown internal displacement ΦSI with re-

spect to the interface harmonics can be computedsimilar to Eq. 3:

ΦSI = −K−1II [KIB1

ΦSB1|KIB2

ΦSB2| . . . |KIBk

ΦSBk

]. (7)

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By stacking boundary and internal parts computedin Eq. 5 and Eq. 7, the soft boundary modes areassembled as:

ΦS =

[ΦSI

ΦSB

]. (8)

3.3 Internal Vibrational ModesBoundary modes are suitable for capturing staticdeformation driven by the un-accelerated boundarydisplacement. Enriched deformation at internal DOFsis necessary when the inertia force associated withboundary displacement is considered. Following anidea similar to [29], we include additional internalvibrational modes in the framework for detailed localdeformation at internal DOFs (I). Internal vibrationalmodes always have zero values at the boundary D-OFs. Therefore, they are not responsible for domaincoupling.Normal modes A natural choice for computing thevibrational mode is via solving the generalized eigenproblem of internal DOFs (so the modes look likefixed at the boundary):{

(KII −ΛMII)ΦNI = 0

ΦNB = 0

, (9)

Eq. 9 is actually performing the linear mode analysis(LMA) over I. It can be understood as choosing thedeformation modes that increase the system’s energyleast [39] with the additional imposed constraints thatthe boundary DOFs are fixed.Inertia modes Using normal modes is a good choicewhen there is no clue about what internal deforma-tion is going to happen. However, in multi-domaindeformation, the internal deformation occurs mostlikely as the consequence of the accelerated bound-ary movement. To better account for such boundary-triggered internal deformation, we pre-compute thecorresponding internal deformation based on theknowledge of the boundary displacements (whichare defined with boundary modes) and the resultingdeformation modes are called inertia modes. Theyare called so because inertia modes are computedby solving the equilibrium with inertia forces thatcorrespond to the boundary acceleration:[

KII KIBKBI KBB

] [ΦII(1)0

]= MΦR +

[0

FIB

], (10)

where, ΦR represents the acceleration along the di-rections of the rigid boundary modes. Soft boundarymodes can also be included in Eq. 10. However, fromour experience, the inertia forces associated with softboundary modes are often much smaller than theforces associated with rigid boundary modes. So theyare discarded for model reduction. The first order ofinertia modes is computed as:

ΦI(1) =

[ΦII(1)0

]=

[K−1IIMIIΦ

RI

0

]. (11)

Similarly, if the acceleration associated with displace-ment ΦI

(1) is not neglected, another set of deformationcan be computed with the same form of equilibriumas in Eq. 10. Successive blocks of higher order inertiamodes are built with following recurrence relation-ship:

ΦI(k+1) =

[ΦII(k+1)

0

]=

[K−1IIMIIΦ

II(k)

0

]. (12)

When the size of inertia subspace is close to full spacesize, the inertia modes could bring redundancy to thesubspace bases. However, high order inertia deforma-tion has a much less contribution to the final internaldeformation. Accordingly, the high order inertia modeare discarded. For the sake of stability, one may applya mass-based Gram-Schmidt orthonormalization overthe inertia modes after each block iteration. Never-theless, in our experiment, the system is stable evenwithout mass-orthonormalization.

Another advantage of inertia mode is its faster pre-computation: the inverse of KII is shared in thecomputation of boundary modes (e.g., Eq. 3 and 7).In fact, inertia mode ΦI

I spans a block Krylov subspace:

ΦII = [AΦR

I |A2ΦRI |A3ΦR

I | . . .], (13)

where A = K−1IIMII . Krylov subspace is a well-known numerical method to compute the generalizedeigen problem as defined in Eq. 9. We start the poweriteration from the rigid boundary modes. From thispoint of view, inertia modes can be understood as atuned version of normal modes as the inertia forcesare pre-known and boundary-driven. In practice, wechoose to use inertia modes instead of normal modesbecause they can be computed more efficiently thannormal modes.

Fig. 2. The assumption of rigid interface [2] inducesartifact when the interface is wide and the neighbordomains are soft. Using our soft modes yields smoothand natural deformation. Same number of modes (30per domain) are used in the comparison while 5 softboundary modes are used to capture the boundarydeformation. The external force is shown as an arrow.

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3.4 Discussion

Rigid interface vs. soft interface When the interfaceis small, a rigid body motion can well approximateits displacement. In this case, it is reasonable to makethe assumption of the interface rigidity as in [2]. Suchapproximation can be achieved in our framework byonly adopting rigid boundary modes and internalvibrational modes. However, in some situations wheredomains share broad and flexible interfaces, suchassumption of interface rigidity could induce visualartifacts because pure rigid interface is not sufficientfor propagating enough deformation across domains.As shown in Fig. 2, the rigid interface fitting usedin [2] leads to the discontinuous shape while softboundary modes generates much smoother deforma-tion across the host mesh and the sharp edges ofthe cube remain continuous. It may be possible touse higher order blending to smooth the deformationbut it requires extra post-simulation computation likemoving least square (MLS) embedding [40].

Fig. 3. Subspaces spanned by different types ofmodes constitute the layered deformation. Final de-formation of the domain can be understood as thesuperposition of deformations from each subspace.Complete set of modes span the full space.

Layered subspaces In our boundary-aware modeconstruction framework, the local domain’s deforma-tion organized in layers. For each domain, we canconsider its final deformation as the superposition ofthree subspace deformations as shown in Fig. 3. First,we slowly move the interface without changing itsshape. Necessary domain deformation is generated inorder to keep the domain coupled with its neighbors.This portion of deformation is represented with rigidboundary modes. After that, some non-rigid interfacedisplacements are further generated to better accom-modate the deformed boundary with soft boundarymodes. Finally, internal vibrational modes captureadditional local internal deformation. Complete setof boundary modes and internal vibrational modesconstitute the full space of the domain’s deformation.

Low-rank2 boundary modes capture dominate inter-face displacements and low-rank internal vibrationalmodes describe majority internal vibrations with fixedboundary condition. From this figure, we can see thatour boundary-aware mode construction strategy care-fully performs model reduction on boundary modesand internal vibrational modes separately. By doingso, neighbor domains always have consistent lowdimensional interface displacement which is purelydecided by the geometry feature of the interface (viacomputing its harmonic bases and rigid motion). Wewill see in Sec. 4.2 that such mechanism facilitates thesubspace domain coupling so that the interface con-straint can be directly enforced over the generalizedreduced coordinate.Our method vs. free vibrational LMA LMA providesthe most natural deformation bases of the uncon-strained vibration space in the case that the domain’sdeformation is completely unknown [39]. It is done bysolving a generalized eigen problem (KΦ = ΛMΦ)defined over the entire domain [26]. Many modelreduction techniques construct subspaces based onthis method [4], [31], [41]. However, simply choos-ing the low frequency LMA modes may not be thebest solution for the multi-domain deformable object,because domains are always coupled and not sim-ple unconstrained free deformable bodies. Boundarymodes are specially designed for the subspace domaincoupling and in general, they are not always thedomain’s vibrations of lowest frequency. In Fig. 4, weplot the frequency distribution of three types of modesmentioned above over the spectrum of the domain’sfree vibration. The y axis in the plots is the projectionof the modes onto the LMA bases of different frequen-cy. We can see in the figure that the rigid boundarymodes have more low frequency components. How-ever, some higher frequency components still exist.On the other hand, soft boundary modes have a widerdistributed spectrum which means that they havemore high frequency vibrational components. Simi-larly, internal vibrational modes have fixed boundary.As a result, they also have wide distribution over thefree vibrational spectrum.

The boundary modes (both rigid and soft) are com-puted for each interface independently. That is, whenimposing a certain displacement to an interface, wekeep other domain interfaces fixed. This strategy hastwo advantages: 1) it guarantees that the subsets ofboundary modes corresponding to different interfacesare linearly independent to each other and; 2) the dis-placement of an interface of the domain is determinedonly by the modes associated with the interface. Theinternal vibrational modes have vanishing values at

2. We say low-rank which typically refers to the modes that areselected with high priority, such as rigid boundary modes and softboundary modes associated with low frequency harmonics. Forinternal vibrational modes, it means low-order inertia modes ornormal modes with small eigenvalues.

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Fig. 4. The shapes of each type of warped modes associated with right interface of the domain. The left interfaceis assigned with zero displacements.

the interfaces and only contribute to the deformationat I. Thus they are always linearly independent to theboundary modes and do not participate in the domaincoupling.

4 DEFORMATION ALGORITHM

In this section, we briefly describe the reduced Euler-Lagrange formulation for multi-domain deformablebodies and the subspace domain coupling. Then weintroduce how to generalize the reduced node-wisecorotational elasticity [4] for multi-domain deforma-tion.

4.1 Equation of MotionFor a single domain i, the Euler-Lagrange equation onthe reduced coordinate [q]i can be written as:

[Mq]i[q]i + [Cq]

i[q]i + [Kq]i[q]i = [fq]

i, (14)

where [Mq]i, [Cq]

i and [Kq]i are the reduced mass,

damping and stiffness matrices. They are constantunder linear elasticity. For the case of the commonly-used Rayleigh damping, [Cq]

i is a linear combinationof [Mq]

i and [Kq]i. [fq]i is the reduced external force.

Domain displacements [u]i and reduced coordinate[q]i are related by the equation: [u]i = [Φ]i[q]i, where[Φ]i contains the selected modes of the domain. It isnoteworthy that since we are not using the regulareigenvectors as local subspace bases, the reducedequations are not decoupled and need to be solvedusing direct linear solver.

For a dynamic system with multiple domains, theglobal reduced mass, damping and stiffness matri-ces are block-diagonal: Mq = diag([Mq]

i), Cq =diag([Cq]

i) and Kq = diag([Kq]i). Similarly, the global

mode matrix Φ also has block-diagonal-like structure

Fig. 5. The structures of global mode matrix and itsnonzero diagonal submatrix at an individual domain.Shadowed blocks are nonzero submatrices.

as shown in Fig. 5. The global reduced displacemen-t/velocity/acceleration q/q/q is the column vectorstacking [q]i/[q]i/[q]i at all domains. We do not applymass-orthogonalization to the modes as done in [31]because the boundary modes have clearly-specifiedgeometric properties which can be further utilizedfor interactive manipulation. We do not experienceany stability issues in our experiment (with time step1/30 sec and implicit Newmark intergration).

4.2 Boundary Coupling

The locking issue and over-constraining problem arewell resolved in our framework. First, neighboringsubspaces are always compatible because the modesare computed based on the pre-defined boundary dis-placements, and the possible interface displacementsat each subspace are the same. Second, the interface

8

displacement is no longer a high dimensional variableas it is expressed with a set of reduced geometricbases. Without loss of generality, assuming there aretwo domains α and β sharing an interface, a validdomain coupling requires keeping the duplicated in-terface DOFs at both domains overlaping during thesimulation. Let [Φ]α and [Φ]β denote the modes su-perset at the domains. Then the interface constraintcan be expressed as:

[ΦBi ]α[q]α − [ΦBi ]

β [q]β = 0, (15)

where [q]α and [q]β are the reduced modal displace-ment in domains. In general, the local interface indexis different from a domain to another. To simplify thenotation, we just use subscript Bi

for both domainswhich can be considered as a global index of theinterfaces on the host mesh.

Eq. 15 indicates that an interface displacement mustbe able to be represented with the reduced coordi-nates at both domains. In other words, the interfacedisplacement has to be within the intersection ofthe subspaces spanned by [ΦBi

]α and [ΦBi]β . If the

intersection is only a small portion of the originalsubspaces, the regions nearby the boundary may ap-pear to be “locking” as some deformations are filteredby the neighboring subspace. Time-varying subspaceconstruction [32] may alleviate such an effect but doesnot guarantee eliminating it. In order to have a naturaldeformation across the interface, the subspaces at thedomains should be compatible at the interface DOFs.That is, any subspace interface displacements of α canalso be represented within the subspace at β. Anotherproblem of using Eq. 15 lies in the fact that the numberof boundary constraints depends on the number of thedomain’s boundary DOFs. For a host mesh of largescale, a high dimensional boundary constraint couldeasily turn the system into an over-constrained one.

In our framework, each interface is clearly asso-ciated with a subset of boundary modes and thedisplacement of the interface is only determined bythe corresponding reduced coordinates. Suppose do-main d0 neighbors to k domains (d1, d2, ... dk, k ≥ 1)at interfaces B1,B2, ...Bk, the boundary constraint forinter-domain coupling can be directly enforced at thereduced domain coordinates that correspond to theinterfaces for each pair of neighbor domains:

[qB1 ]

d0 − [qB1 ]d1 = 0

[qB2 ]d0 − [qB2 ]

d2 = 0...

[qBk]d0 − [qBk

]dk = 0

, (16)

where the notation like [qB1]d0 stands for the subset

of the reduced displacement at domain d0 that corre-sponds to interface B1.

4.3 Large Scale DeformationThe linear elasticity based model reduction describedabove is not able to simulate large deformations.This is because 1) the Cauchy’s strain tensor used isa linear strain tensor which generates inappropriatestrain under rotations; and 2) the linear combinationof the modes is not able to represent the intermediaterotational displacement. As a result, the rotationaldeformation must be specially handled. We adopt anode-wise corotational formulation with model reduc-tion as in [4].

The curl of the linear deformation field is used toapproximate the local rotation at each node. For finiteelements with a linear shape function, it can be pre-computed with the subspace modes. At each timestep, we assemble a block-diagonal warping matrix R.Each 3× 3 diagonal block of R represents the currentnodal warping. We refer the reader to the literature [4]for a detailed derivation of R. Because of the domaindecomposition, interface nodes are duplicated at theneighbor domains. This means that the number ofrows of the global mode matrix Φ is larger than thenumber of DOFs on the host mesh. Accordingly, weassemble an auxiliary elementary matrix E, such thatthe rows of Φ corresponding to the duplicated bound-ary nodes are picked only once. This operation impliesthat the computation of the curl at the interface nodetakes all the neighbor domains into account, whichensures the smoothness of the warped deformationu:

u = REΦq. (17)

E is fixed when the domain decomposition is doneand the product of EΦ can be pre-computed. Theupdate of nodal displacements is “local” as modesfrom other domains do not contribute to the finaldisplacement of the node3. Accordingly, in real im-plementation, only local matrix-vector products arenecessary to compute the displacement of the nodes.

5 DIRECT MANIPULATION

Manipulation of deformable objects is important foruser interactivity and animation production. Our sys-tem enables the user to manipulate the deformableobject through applying constraints to the nodes orthe interfaces.

If a node p is constrained to a specified position.The corresponding position constraint equation is:RpΦpq = c, where Rp and Φp represent the warpingmatrix and three-row mode matrix at the constrainednode. c is the desired node position. Rp is a time

3. Each three-row block in the global mode matrix (Φ) thatcorresponds to a node on the FE mesh is sparse (as shown in Fig. 5).If it is an internal node, the block has nonzeros at the columnscorresponding to the domain that owns this node. If it is a boundarynode, the block has nonzeros at the columns corresponding to itsneighbor domains. In either case, the update only needs the reducedcoordinates of related domains instead of the entire global q.

9

Fig. 6. Enforcing anchor nodes on the left end of thebar model using the Lagrange multiplier method: wecan clearly see the locking region without using PCmodes. Both bars are simulated with 100 modes intotal.

varying matrix and we use the warping matrix in theprevious time step to approximate the current warp-ing matrix. Therefore, a linear constraint equation canbe used:

Φpq− R−1p c = 0. (18)

For subspace dynamics, the system could becomeover-constrained if multiple nodes are constrained bythe user for interactive manipulation as each con-strained node consumes 3 DOFs from the system.Therefore, a mechanism is needed to prevent thesystem from being over-constrained while keeping thesystem size small. In addition, subspace displacementmay not be able to represent all the user-specifiedpositions of the nodes, which could also lead to thelocking problem.

In order to solve this issue, we design a new typeof mode called the position constraint (PC) mode. Thedesired PC mode should be 1) linearly independent tothe existing modes included at the domain and 2) ableto represent any displacements for the constrainednodes. The latter requirement ensures the compensat-ing PC modes alone are able to fulfill the constraintsso that other existing modes do not need to sacrificetheir own freedom. We denote the constrained DOFswith set C. Any DOFs in I, if chosen in C are removedfrom I. Imposing a unit displacement to each DOF inC while keeping the remaining DOFs in C as well asthe ones in B fixed yields: KII KIC KIB

KCI KCC KCBKBI KBC KBB

ΦPCI

IPCC0

=

0FPCCFPCB

, (19)

and PC modes ΦPC can be computed through:

ΦPC =

ΦPCI

IPCC0

=

−K−1IIKICIPCC0

, (20)

where IPCC is the identity matrix standing for the unitdisplacement added at C. The PC modes maintainthe size of the subspace by introducing compensatingmodes to the system to preclude the system’s DOFsfrom being “drained out” by the user constraints. It

effectively resolves the locking problem induced bythe position-constrained nodes (Fig. 6).

Our framework also supports direct manipulationof the interface by specifying its orientation. Such kindof manipulation enables the user to adjust the relativeorientation among domains with deformation. Let Rbe the user specified rotation for a boundary Bi andits axis-angle representation is (a, θ). p denotes therelative position of an interface node p with respectto the rotation pivot (e.g., the interface centroid). Thecorresponding reduced coordinate qrotBi

∈ R3×1 onthe three tangential rotational modes is qrotBi

= θaand the unwarped displacement is u = Φrot

p qrotBi=

θ[a]×p, where [a]× is the skew-symmetric matrix of a.With Rodrigues’s formulation, the desired displacement(Rp− p) and the warped displacement (Rpu) can bewritten as:

Rp− p = ([a]2× + sin(θ)[a]× − cos(θ)[a]2×)p, (21)

and

Rpu = (I + [a]2× + [a]×1−cos(θ)

θ − [a]2×sin(θ)θ )u

= ([a]2× − cos(θ)[a]2× − sin(θ)[a]3×)p ,(22)

respectively. Note that [a]× = −[a]3× holds for skew-symmetric matrix [a]×. Therefore, the left hand sidesof Eq. 21 and Eq. 22 are essentially equivalent to eachother such as:

Rp− p = RpΦrotp qrotBi

. (23)

This means that, for the rotation R with axis-anglerepresentation as (a, θ), as long as qrotBi

= θa, we canalways have the desired rotational displacement withwarping. Hence, the corresponding interface orienta-tion constraint simply becomes:

qrotBi= θa. (24)

All the constraints including boundary constraint(Eq. 16), position constraint (Eq. 18) and interfaceconstraint (Eq. 24) are imposed through Lagrangemultiplier method. The constraint equations form aconstraint matrix, which is a constant if the con-strained DOFs are not changed during the simula-tion. Using Newmark integration, the reduced Euler-Largange equations of all the domains can also beconverted to a linear system which is solved at eachtime step based on the imposed constraint [42].

6 EXPERIMENT RESULTS

Accuracy analysis We first show a comparative ex-periment of simulating a bar model which is droppingunder gravity (−9.8 m/s2) with one end fixed. Asshown in Fig. 7(a), from right to left the simulatorsused are nonlinear full space (single domain StVK,the ground truth), nonlinear single domain subspaceintegration with modal derivative [31], deformation

10

(a)

(b)

Fig. 7. Comparative simulation among various simula-tors.

substructuring [2] (nonlinear multi-domain simula-tor), modal warping [4] (single domain, linear simu-lator with deformation warping) and our frameworkwith and without soft boundary modes. The model isevenly divided into three domains for multi-domainsimulators. All the simulators use the same materialparameters (e.g., Young’s Modulus and Poisson’s Ra-tio). In order to ensure the accuracy of the nonlinearsimulator, we did not limit the maximum iterationnumber at each time step, and the threshold is set as1e−6. The mass and stiffness damping coefficients areset as 0.5 and 0.08 respectively. The time step intervalis 0.01 sec and the average implicit Newmark timeintegration [43] is used for all simulators. The averagevertical displacements of vertices at the free end of themodel are recorded and shown in Fig. 7(b). All thesubspace simulators have the same number of bases(55 bases). The relative L2 error for each simulatorw.r.t. ground truth is 7.36% for modal derivative,13.21% for modal warping, 13.04% for substructuring(the displacements from duplicated boundary DOFsare averaged), and 10.35% and 8.45% for our method(without and with soft boundary modes). Inducinginterface flexibility to the system with soft boundarymodes increases the accuracy of the simulation.Coupling comparison With corotational displace-ment correction, we can produce a similar deforma-tion to other state-of-art nonlinear methods. One ad-vantage of our method is that the multiplier-enforcedcoupling always guarantees a seamless domain con-nection while the boundary-driven subspace avoid-

s the locking and over-constraint problem. Fig. 8illustrates the coupling effect using our algorithmand the penalty force based method [1]. The elbowjoint of the arm is a one DOF joint and rotatingwith angular velocity 0.2 rad/sec. Both quasi-staticnonlinear modes and modal derivatives are used fornonlinear subspace bases. The cracks at the elbowvanish when increasing the number of modes to 40.The simulation time step has to be limited to smallvalues (e.g., 0.01 sec) because of the usage of highlystiff springs. On the other hand, our method excelswith the larger time step (0.3 sec) and smaller size ofsubspace without any cracks or stability issue. p

Another comparison is with deformation substruc-turing using rigid interface fitting [2] (Fig. 9). In thiscomparison, the tyrannosaurus model is decomposedinto 17 domains and the rank for each domain is 30(modal derivatives for substructuring and rigid/softboundary modes for our method). We notice thatnear-rigid fitting used in substructuring [2] couldalso induce cracks at the domain interfaces as thedeformation and interface region goes large imme-diately after the simulation. Post-simulation process-ing with interface blending is able to relieve thisartifact [2]. With soft boundary modes, our methodavoids this problem and does not need any extra post-simulation amendments. More importantly, simpleblending could still introduce artifacts of discontinuityas shown in Fig. 2 which requires more advancedgeometrical blending at post-simulation stage in eachtime step.

Our method is essentially using linear elasticity andthe system matrix is constant during the simulation.The runtime performance is faster than nonlinearsimulators: in the first comparison shown in Fig. 8,the overall frame per second (FPS) is 172 using ourmethod and 89 using implicit springs [1]; in thesecond comparison as in Fig. 9, our overall FPS is34 compared to 18 using substructuring [2]. In theimplementation of [2], we perform three Newton-Raphson iterations at each time step. If we reduce thisnumber to one, the FPS for the tyrannosaurus modelincreases to 28 with higher residual error.Simulation with manipulation Large deformationsare well handled in our framework. Fig. 11 shows asunflower model with 47, 164 elements and 13 do-mains. We apply external forces at the top of theflower. Large deformations are generated at the stalkand the leaves have relatively smaller local vibrationaldeformations. Unlike deformation substructuring [2],our framework does not rely on hierarchical domaindecomposition. Deformable objects with looped do-main connection can be simulated with our methodnaturally. Fig. 10 shows the result of simulating asailboat model with seven loops and 253,998 elements.We apply scripted forces at the masts and the bodyof the boat. Inertia modes of order two are assignedto the canvases to have enriched local deformation

11

Fig. 8. Coupling comparison between our method and nonlinear multi-domain with penalty method [1].

Fig. 9. Coupling comparison between our method and deformation substructuring using rigid interface fitting [2].

effects with the applied wind field. Domain loop canalso be handled with spring coupling [1]. However,for looped multi-domain deformation using substruc-turing [2], using implicit penalty force between do-mains could break the sequential simulation order ofdomains and bring more complexity to the system.

Supporting manipulations is straightforward withconstraints in our framework. In Fig. 12, the fish mod-el consists of 111, 196 elements and ten domains. It ismanipulated by the user in real-time. One constrainednode is used to control the body motion of the fish,while the orientations of the interfaces connecting therear fin and fish head are also manipulated. In ourimplementation, a damped constraint equation [44] isused in order to avoid sharp impulse-like constraintforces and increase the stability of the system.

Performance Detailed time performances can befound in Tab. 2. Our system is implemented with aWindows 7 PC equipped with a 2.26GHz Intel XeonCPU (only a single core is used for simulation) and12GB RAM. Because the domains are coupled withhard constraints, we must solve the system entirely(similar to coupling with implicit penalty forces as

Fig. 11. Simulation of a sunflower model with largedeformation.

in [1]) instead of sequentially solving each domainindividually [2]. However, linear elasticity with a con-stant system matrix needs only to be pre-conditionedonce before simulation, and solving the system is notthe bottleneck of the framework. For example in oursailboat model, solving the pre-factorized linear sys-tem takes less than 1ms which is only 5% computationtime of one time step. The global mode matrix Φ has ablock-diagonal structure and the run-time nodal dis-placement updates (e.g., using Eq. 17 to compute thenodal displacement on the mesh) are actually “local”as it only depends on the number of modes in the do-

12

Fig. 10. Simulation of a sailboat model with 253,998 elements, 24 domains and seven loops (highlighted withred circles) in real-time (21 FPS). The total number of modes used is 830. The domains corresponding to themasts and the body of the boat have stiffer material while the canvases have softer material. We applied scriptedforces at the masts and the body of the boat and the wind force field at the canvases (shown as arrows in thefigure).

Model statistics Domains Precomputation Runtime PerformanceModel

# elements # vertices # r # domains boundary inertia conditioning system disp. FPS

Sunflower 47, 164 12, 659 260 13 3.46 s 0.3 s 0.03 s 0.06 ms 8 ms 67

Fish 111, 196 27, 075 340 10 11.76 s 6.3 s 0.03 s 0.08 ms 18 ms 41

Tyrannosaurus 121, 976 32, 009 510 17 15.3 s 9.4 s 0.14 s 0.14 ms 25 ms 34

Sailboat 253, 998 48, 998 830 24 43.5 s 6.2 s 0.17 s 0.6 ms 30 ms 21

TABLE 2Time performance. # r : total number of modes; boundary : pre-computation time for boundary modes (rigid andsoft ones); inertia: pre-computation time for inertia modes; conditioning: time for preconditioning linear system;system: time for solving reduced linear system at each time step; disp: time for computing displacement from

reduced coordinates; FPS: overall frame per second.

Fig. 12. Interactive manipulation on the fish model.

main rather than the global subspace size. Hence ourframework is on average d times faster than modalwarping [4] where d is the number of domains. Interms of pre-computation, our method is significantlyfaster than global subspace method which generallyrequires solving a very large eigen problem. It couldtake up to several hours and consume considerableamount of memory, while local subspace bases pre-

computation is orders of magnitude faster and can befinished within seconds. In the table, we record thepre-computation time for “fresh” computation of eachtype of mode including calculating the inverse theinternal stiffness matrix (KII), which is the most time-consuming part in the mode computation. In fact, thisonly needs to be done once for each domain (if inertiamodes are used). As a result we can pre-factorize KIIbefore mode computation and the pre-computationfor domain modes can be further shortened.

7 CONCLUSION

We have developed a seamless coupling method formulti-domain subspace deformation in the frameworkof node-wise corotational elasticity. With boundary-aware mode construction method, the boundary cou-pling constraints can be directly imposed on thereduced coordinates, efficiently avoiding the over-constraining problem. Our algorithm can achieve real-time simulation performance for large-scale meshesand can support direct manipulation of deformableobjects.

13

Limitation When there are many boundary interfacesin the domain decomposition, the number of bound-ary deformation modes can be large. This leads tolinear growth in the required number of multipliersand accordingly the dimension of the system matrix.It can be handled by separating the variables in thelinear constraint equations into independent and de-pendent variables, and constructing the system matrixonly using the independent variables to avoid explicituse of Lagrange multipliers. Node-wise corotationalformulation produces ghost forces if the domains areunconstrained when local displacement at each nodeis warped. Fortunately, such effect is not seen in ourexperiment because constraints always exist in theframework (e.g., interface constraints or position con-straints). The hard constraints suppress such artifactand the results are physically plausible.

In subspace deformation techniques, the time forsystem resolving is small. We found that the displace-ment warping step, i.e. the estimation of rotation ateach node, costs more than 50% of the time in onesimulation step. We plan to reduce the time throughadaptive rotation estimation. It can be done by onlyestimating the rotations of sparsely sampled nodesand checking whether the rotations can be directlyused at the remaining nodes.

Unconstrained deformable object is only partiallysupported with this framework. Because of the linearelasticity used, the underlying rigid body motionsof the domains are only the first order approxima-tion of the real rigid body dynamics. Fortunately,modal warping is able to correct the distorted rotation.Another method is following the similar strategy asin [31], [45] to explicitly couple the rigid body motionand the deformation of the domains.Future work An interesting research direction isto investigate a way to apply the boundary-drivenmode construction method to the coupling of multi-domain subspace deformations using nonlinear StVKdeformable model in the spirit of modal derivativeframework. It is possible to construct a first or higherorder approximation of the change of boundary de-formation modes using Eq. 1.

We also plan to investigate a parallel multi-domainsubspace deformation algorithms. This calls for cou-pling constraints to be handled in each domain sepa-rately. Another interesting research direction is to ap-ply the multi-domain subspace deformation techniqueto other application areas, such as fabrication-awaredesign, to provide fast simulation results when theuser edits the geometry model.

ACKNOWLEDGMENTThe author would like to thank anonymous reviewersfor their constructive comments and suggestions. YinYang and Xiaohu Guo are partially supported by theNSF of USA under Grants Nos. CNS-1012975 and IIS-1149737. Weiwei Xu is partially supported by NSFC

(No. 61272392). Kun Zhou is supported by NSFC (No.61272305).

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Yin Yang is currently a Ph.D student andresearch assistant with Department of Com-puter Science, The University of Texas atDallas. His research focuses on Physics-based animation/simulation and related ap-plications, scientific visualization and medicalimaging. For more information, please visithttp://www.utdallas.edu/˜yinyang.

Dr. Weiwei Xu is a professor at HangzhouNormal University in the department of com-puter science. Before that, Dr. Weiwei X-u was a researcher in Internet GraphicsGroup, Microsoft Research Asia from Oc-t. 2005 to Jun 2012. Dr. Weiwei Xu hasbeen a post-doc researcher at Ritsmeikanuniversity in Japan around one year from2004 to 2005. He received Ph.D. Degree inComputer Graphics from Zhejiang University,Hangzhou, and B.S. Degree and Master De-

gree in Computer Science from Hohai University in 1996 and 1999respectively. He has published more than 20 papers on internationalconference and journals, include 9 papers on ACM Transactionon Graphics. His main research interests include digital geometryprocessing and computer animation techniques. He is now focusingon how to enhance the geometry design algorithm though theintegration of physical properties.

Xiaohu Guo received the PhD degree incomputer science from the State Universi-ty of New York at Stony Brook in 2006.He is an associate professor of computerscience at the University of Texas at Dal-las. His research interests include computergraphics, animation and visualization, withan emphasis on geometric, and physics-based modeling. His current researches atUT-Dallas include: spectral geometric anal-ysis, deformable models, centroidal Voronoi

tessellation, GPU algorithms, 3D and 4D medical image analysis,etc. He received the prestigious National Science Foundation CA-REER Award in 2012. He is a member of the IEEE. For moreinformation, please visit http://www.utdallas.edu/˜xguo.

Kun Zhou is currently a Cheung Kong Distin-guished Professor in the Computer ScienceDepartment of Zhejiang University, and amember of the State Key Lab of CAD&CG.Before joining Zhejiang University, he wasa Leader Researcher of the graphics groupat Microsoft Research Asia. He received hisBS degree and PhD degree in computerscience from Zhejiang University in 1997 and2002, respectively. His research interests in-clude shape modelling/editing, texture map-

ping/synthesis, real-time rendering and GPU parallel computing.

Dr. Baining Guo is a Deputy Managing Di-rector at Microsoft Research Asia, where healso leads the graphics lab. Prior to joiningMicrosoft Research in 1999, he was a se-nior staff researcher with the MicrocomputerResearch Labs of Intel Corporation in SantaClara, California. Dr. Guo graduated fromBeijing University with B.S. in mathematics.He went to Cornell University for his graduatestudy from 1986 to 1991 and obtained M.S.in Computer Science and Ph.D. in Applied

Mathematics. Dr. Guo is an IEEE fellow and an ACM fellow.Dr. Guo’s research interests include computer graphics, visu-

alization, and natural user interface. He is particularly interestedin studying light transmission and reflection in complex, texturedmaterials, with applications to texture and reflectance modeling. Healso worked on real-time rendering and geometry modeling. Dr. Guowas on the editorial boards of IEEE Transactions on Visualizationand Computer Graphics (2006–2010) and Computer and Graphics(2007 – 2011). He is currently an associate editor of IEEE Com-puter Graphics and Applications. He has served on the programcommittees of numerous conferences in graphics and visualization,including ACM Siggraph and IEEE Visualization. Dr. Guo holds over40 US patents.