Real-time hybrid solid simulation: spectral unification of deformable and rigid materials

COMPUTER ANIMATION AND VIRTUAL WORLDSComp. Anim. Virtual Worlds 2010; 21: 151–159Published online 29 May 2010 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/cav.373...........................................................................................Real-time hybrid solid simulation:spectral unification of deformable andrigid materials

By Yin Yang, Guodong Rong, Luis Torres and Xiaohu Guo∗..........................................................................

A novel framework is proposed in this paper to simulate hybrid solids with deformable andrigid materials in real-time. Both types of materials are uniformly integrated into onespectral simulator. Based on the modal warping technique, we employ a new constraintstrategy which eliminates the accumulation of approximation errors at the boundaryinterfaces, thus naturally gluing different materials. We also utilize the GPU to acceleratethe run-time computation when updating the geometry of the hybrid solid—the mostexpensive step in this framework. This work provides a general-purpose solution ofsimulating hybrid objects in real-time, even for large-scale models. Copyright © 2010 JohnWiley & Sons, Ltd.

KEY WORDS: GPU; hybrid simulation; modal analysis; physically-based animation

Introduction

In physically-based animation, objects are expected to ef-ficiently exhibit realistic motions. Different types of ob-jects typically require particularly designed simulatingstrategies. The diversity of these techniques which nicelyresolve each individual physical phenomenon raises newchallenges if we want to have a mixed scenario that ei-ther various simulators are to be incorporated1–3 or if theobject under simulation itself is a compound of differentphysical features.4–7 The challenge originates from theintrinsic discrepancies among various physical solverswith specialized techniques. The difficulties go furtherfor the hybrid body simulation because of the ambiguityof physical properties at the connecting regions of differ-ent materials which calls for a sophisticated constraintstrategy.

Inspired by such requirements and challenges we pro-pose a new spectral framework to simulate hybrid solidsconsisting of both deformable and rigid materials. In thisframework, heterogenous materials correspond to subdynamic systems that evolve in a fully coupled mannerin the spectral domain. The spectral reduction of degreesof freedom (DoF), as the biggest benefit of spectral simula-

*Correspondence to: X. Guo, Assistant Professor, Departmentof Computer Science The University of Texas at Dallas. E-mail:[email protected]

tion technique (also well known as modal analysis) resultsin a significant speed-up over its spatial counterpart.

In addition, the uniform simulation strategy at all thesubsystems allows us to construct a unified spectral sim-ulator for the whole hybrid solid. We assemble the globalspectral displacement and rotation matrices, and usethese matrices to convert spectral displacement to spatialdisplacement. Because every three rows in these globalmatrices correspond to a node on the hybrid solid, theseglobal matrices based operations can be naturally paral-lelized with graphics processing units (GPU). Our experi-ments (Table 1) show that the utilization of the GPU con-tributes about an extra threefold improvement in termsof time performance. Meanwhile, the symmetric and con-sistent formulation smooths the whole theory derivationand effectively facilitates the programming implementa-tion.

We extend the modal warping method8 to make thesimulation robust to rotation. This technique providesan elegant approach to approximate nodal rotation ac-companied with deformation using linear-strain-tensor-based computation. This rotation is also used in our forti-fied constraint formulation ensuring correct interactivityamong subsystems. System degeneration induced by thereinforced boundary condition is alleviated by limitingthe number of boundary constraints without introduc-ing much boundary inconsistency (Figure 3). As a result,real-time simulation of hybrid solids with deformable

............................................................................................Copyright © 2010 John Wiley & Sons, Ltd.

Y. YANG ET AL............................................................................................

Offline computation (seconds) Online computation (FPS)

Model #Tetra. #Modes Build Solve Initialize CPU GPUM, K, C eigenproblem linear system simulation simulation

Bar1 323 100 0.008 1.547 0.006 269.1 845.2Bar2 5372 200 0.419 4.617 0.026 31.9 89.2Gargoyle 10 000 50 1.547 5.101 0.022 20.3 64.9Armadillo 13 852 100 2.931 6.823 0.035 14.3 43.6Dragon 32 959 50 12.209 8.781 0.05 10.3 35.6

Table 1. Time performance.

Figure 1. The hybrid Armadillo model with two rigid bonesinside the right leg.

and rigid materials is made possible by utilizing only asmall number of modal bases. Figure 1 shows the resultof bending one leg of the Armadillo model with 13 852tetrahedra and 100 spectral bases, where the hybrid leghas two rigid “bone” regions inside.

Related Work

In deformable model, the linear strain tensor is com-monly used for the sake of computational efficiency.However, it cannot properly handle rotational deforma-tions. Some researches9,10 treated the deformations as lin-ear elasticity and rigid motion separately. However, theglobally extracted rotation is still not precise enough forlarge rotations. The domain decomposition method (DDM)11

was proposed as a solution to this problem where therotation is computed at each subdomain. Alternatively,Muller et al.12 invited stiffness warping where the stiffnessis warped according to an estimated nodal rotation.

Modal analysis, on the other hand, gives a new per-spective to deformation. Pentland and Williams13 firstproposed a basic solution using modal analysis to simu-

late a deformable body. It was pointed out that the modesassociated with higher resonance have less effect on theshape of the object. This property is utilized in almostall the research related to modal analysis because by dis-carding the high frequency components, the original sys-tem is reduced to a new subspace of much smaller size.Choi and Ko8 transplanted the warping idea into spectralspace. In their work, rotation is calculated as the curl ofthe displacement field expressed using spectral displace-ment. Hence, similar to stiffness warping, linear-strain-tensor-based computation is able to approximate rota-tional deformation as well in spectral space. This tech-nique is further extended to meshless simulation14 andthin shell simulation.15 Besides algorithms, the rapid de-velopment of graphics hardware also provides new re-search space on this topic.16,17 With more powerful pro-gramable graphics hardware, the time performance canbe improved even further.

Mixed simulation of multiple physical simulatorsis more involved, as the interaction and coordinationamong the different simulators generate more complex-ity. We roughly classify this type of research into threecategories. References [2–4] are several top paradigmsof the first category where multiple physically heteroge-nous scenarios are simulated simultaneously. Couplingof these objects as well as the simulators behind themis the main objective of this type of research. As a re-sult, collision or contact detection is a must because thesimulators are not truly coupled until the objects are in-teracting with each other. Another category of researchfeatured in References [5,18,19], on the other hand, fo-cuses on only one object that more or less demonstratesmixed physical characteristics. Certain hierarchical aux-iliary structures in addition to the original mesh such ascoarse meshes or grids are commonly employed to facil-itate simulation. Although computation based on lowerresolution meshes is much faster, this advantage is some-

............................................................................................Copyright © 2010 John Wiley & Sons, Ltd. 152 Comp. Anim. Virtual Worlds 2010; 21: 151–159

DOI: 10.1002/cav

REAL-TIME HYBRID SOLID SIMULATION...........................................................................................

how overshadowed by the additional expense of con-verting, embedding or mapping between multiple reso-lutions. The last category6,7,10 puts considerable effortson physically solving the internal connections amongdifferent objects or components. Unfortunately, the ro-tation at the interface between heterogeneous materialsis not clearly addressed: Galoppo et al.10 used the uniformrigid body rotation during simulation. This approach, toour understanding, could lead to some unnatural resultswhen there is large rotational deformation and in Ref-erences [6,7], certain constraint based formulation wereintroduced without explicitly comments on rotation han-dling while we explicitly handle such rotation based onfinite strain theory. In the following sections, the detailedbuild-up of our framework is described.

Spectral Simulation ofDeformable and Rigid

Regions

We use the subscript d to denote the values for deformableregions. The Euler–Lagrange equation of a deformablebody in R3 discretized using the finite element method(FEM) is:

Mdud + Cdud + Kdud = fd (1)

where ud is a time-dependent vector representing thespatial displacement of the object; Kd and Md are thestiffness matrix and mass matrix, respectively; fd is the ex-ternal force; Cd is the Rayleigh damping matrix as com-monly adopted, which is the linear combination of Kd

and Md , i.e., Cd = ξMd + ζKd , where ξ and ζ are twoweighting constants.

The generalized eigenproblem defined as:

Kd�d = Md�d�d (2)

is solved, where �d is called the modal displacement matrixwhose columns are the eigenvectors; �d is a diagonalmatrix and the diagonal elements are the correspondingeigenvalues. With the computed eigenvectors, the spatialdisplacement ud can be expressed as:

ud = �dqd (3)

where qd is called the spectral displacement. All the spatialunknowns are to be re-expressed in the spectral version

of Equation (1):

Md qd + Cd qd + Kdqd = fd (4)

where fd = ��d fd is the spectral force. A convenient side-

effect is that Md = I (I is the identity matrix), Kd = �d

and Cd = (ξI + ζ�d) are all diagonalized.The rotation is handled using the modal warping8

method. With this technique, the rotation at each nodein the deformable region is extracted and tracked by a3×1 rotation vector w, whose direction and magnitudedenote the rotation axis and angle, respectively. The ro-tation vector can be computed as the curl of the displace-ment field. Then we can build a modal rotation matrix �

as a counterpart of � such that:

wd = 12

(∇×)H�dqd = �dqd (5)

Here H is the shape function of the tetrahedra mesh, wd

is a 3 × n vector concatenating the rotation at every node.When advancing simulation, we embed a local coordinateframe at each node. Now the motion within one time stepof the deformable region can be re-expressed as:

Md qd + Cd qd + Kdqd = ��d (R�

d fd) (6)

where Rd is the rotation matrix computed from wd .q, q, and q correspond to the local quantities as men-tioned. The final spatial displacement used for updatingthe geometry of the deformable region can be computedthrough:

ud = Rd�dqd (7)

where Rd is the accumulated rotation at each node fromthe beginning of simulation to the current time step. Thedetailed derivation can be found in Reference [8].

Clearly, fewer modes used bring more deformationdetails loss. However, in our experiment, it is typicallyhard to tell by sight such difference and some previousresearches8,13 also justify the high visual plausibility ofusing spectral simulation.

Because both M and K are positive semi-definite ma-trices, all the eigenvalues from Equation (2) are greaterthan or equal to zero. Interestingly, there are always sixeigenvalues equal to zero. This implies that these sixspectral modes represent a dynamics with a zero spectralstiffness matrix which, naturally, results in a zero-strainover the body. Such dynamics is precisely the rigid bodymotion and the number of modes equals to the number


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Y. YANG ET AL............................................................................................

of DoF needed in the spatial case, i.e., three for the posi-tion and three for rotation. Based on the above analysis,the formulation for simulating the rigid region can bedirectly incorporated as:

Mrqr + Crqr + Krqr = ��r (R�

r fr) (8)

Similarly, we use the subscript r to represent the valuesof rigid regions. The number of spectral bases employedis the only difference between deformable and rigid re-gions.

Without loss of generality, we suppose the hybrid ma-terial consists of one deformable region and one rigidregion. We can then unify all subsystems into one as:

{Md qd + Cd qd + Kdqd = ��

d (R�d fd)

Mrqr + Crqr + Krqr = ��r (R�

r fr)(9)

where fd and fr include not only the external forces ap-plied to each subsystem but also the internal forces be-tween the subsystems.

Boundary Handling

The explicit computation of these internal forces is nottrivial. However, these forces can be naturally expressedin the form of constraints which make the duplicatedboundary nodes overlap at the interface: ub

d = ubr , where

ubd and ub

r stand for the nodal displacements from the de-formable region and the rigid region, respectively. Onemust note that regions of the same type sharing one ormore triangle faces are counted as one. Thus the conceptof boundary is only the interface between a pair of de-formable and rigid regions. Using Equation (7) we have:

EbdRd�dqd = Eb

rRr�rqr (10)

where Ebd and Eb

r are the mapping matrices. The displace-ment at the boundary nodes are picked out by pre-multiplying Eb to the global displacement u such that:ub = Ebu. The mapping matrices can be easily built oncethe partition of the deformable region and the rigid re-gion is finished.

Equation (10), however, does not yield expected re-sults. Our experiments show that the strain accumu-lates at the boundary and suddenly releases periodically,which causes oscillation at the boundary and the systemis highly unstable under rotation. The reason behind thisproblem is the accumulated rotation term, R in Equa-tion 10. According to finite strain theory, the deformationcan be considered as a combination of a linear elastic de-formation and an infinitesimal rotation. Though the firstterm is computed accurately with the linear tensor, therotation term is approximated using modal warping. Theerrors generated will accumulate and make the systemunstable. Accordingly, we propose a new constraint for-mulation following the same idea of finite strain theorythat decomposes the boundary constraint into two typesof subconstraints of linear elastic and rotation, respec-tively. In the other words, the change of both linear de-formation and rotation within one time step are forcedto be equal from two neighboring regions. The modaldisplacement matrix and modal rotation matrix facili-tate the extraction of the linear deformation and rotationsymmetrically:

{Eb

d�dqd = Ebr�rqr

Ebd�dqd = Eb

r�rqr

(11)

If the warping technique is to combine the linear rotationand deformation together, this constraint formulation isto inversely separate these two components. With theexplicitly defined linear rotation and deformation, theapproximation error does not accumulate or cause anystability issues.

Figure 2. A hybrid bar model consisting of 323 tetrahedra is twisted: (a) The rest shape, (b) Equation (10) is used as boundaryconstraint, (c) all eight boundary nodes are constrained (over-constrained), and (d) Only two nodes on the boundary are constrained

(over-constraint resolved).


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Figure 3. Boundary deviation comparisons with differentnumber of modes and boundary constraint nodes.

For every constrained boundary node, the number ofvanished DoF raises from 3 (using Equation (10)) to 6(using Equation (11)), indicating that twice as many con-straints are induced. Importing more constraints into thesystem, however, brings overhead: the increased num-ber of eliminated DoF due to boundary constraints couldturn out to degenerate the system, i.e., there may notbe enough available DoF to allow the body to deformproperly. This causes the system to be over-constrained asshown in Figure 2(c). One solution could be to simply in-crease the number of spectral bases used. However, thiswill weaken the major benefit of spectral simulation. Al-ternatively, we resolve this problem by capping the num-ber of constrained nodes at the boundary as shown in Fig-ure 2(d) where we only constrain two boundary nodesand the bar behaves naturally. The reader may wonderwhether this partially constrained boundary still bondstwo neighboring regions with high quality. The answer isyes, because our size-reduced spectral simulator implic-itly neutralizes the demand of the number of boundaryconstraints. To be more specific, the deformable regionsin the hybrid solid are actually becoming ‘stiffer’ dueto the absence of high frequency vibrations. As a side-effect, the stiffened boundary requires fewer boundaryconstraints. An extreme example may explain this moreclearly: if the deformable region degenerates to a sub-system of only six DoF, it actually becomes a purely rigidone; as a result, a single boundary node will suffice toconnect the this subsystem with its rigid neighbor. Ofcourse, the boundary nodes that are not constrained canstill experience slight deviations. Figure 3 quantifies sucha situation. The curves are generated from a bar modelwith 323 tetrahedra (same model as in Figure 2). One endof the bar is fixed and the other end drops freely under

gravity. The deviation is the boundary nodal displace-ment difference between the deformable region and therigid region. This quantity evaluates how well two ma-terials are physically glued by the boundary constraint.In order to give the reader a better understanding of thevalues of deviations in the figure, the bar model is nor-malized into a 1 × 1 × 1 cube. As shown in the figure,when all the boundary nodes are constrained (square dotcurve) the deviation is very close to its analytical value, 0.However if the constrained boundary nodes are reducedto two, a system with 50 modes (triangle dot curve) sat-isfies the boundary condition much better than a systemwith 300 modes (round dot curve) does. As we alwayswant to use smaller number of spectral bases to reducethe size of the system, 50 modes are always preferred over300 modes. Such tiny errors at the boundary (< 0.2% ofmodel dimension) are negligible during simulation anddo not cause force feedback (the interactivity betweensubsystems are affected only by constrained boundarynodes) and thus do not affect the system’s stability.

The selection of constrained boundary nodes is not ar-bitrary, as we always want the constrained nodes to beevenly scattered over the boundary. In other words, theset of chosen constrained nodes C is a subset of bound-ary nodes B such that the distance between C and B − C

is minimized, assuming each edge on the boundary isequally weighted. We employ a fast, simple algorithm toselect C from B as follows:

� An undirected connected graph G(V, E) is constructedfrom the topology of the boundary. Each node on theboundary corresponds to a vertex in V and every edgeconnecting two nodes on the boundary corresponds toan undirected edge, e ∈ E.

� An adjacency matrix D0 from G is computed such that

∀di,j ∈ D0, di,j ={

1, if e(i, j) ∈ E

0, otherwise.and D |B|

2 is com-

puted using the Floyd–Warshall algorithm.†� The first node is chosen as the one with the largest

number of one-ring neighbors (cover the largest area).The next candidate is b ∈ B − C such that ∀c, c ∈ C,min{Distance(c, b)} is maximized, where Distance(c, b)denotes the number of edges on the shortest patch con-necting c and b in G. This value can be found in D |B|

2 .

†The degree of the adjacency matrix is |B|2 because G is a bicon-

nected graph: the removal of any vertex on the graph does notaffect the connectivity of the resulting graph. |B|

2 guarantees thatthe length of all pair shortest path is evaluated correctly.


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Figure 4. Results of boundary node selection. Selected nodes are shown in orange.

Figure 4 shows the resulting selection of 15th, 9th, and6th nodes out of the 37 boundary nodes.

Time Integration andConstraint Manipulation

A coupled ODE must be solved at each time step in ev-ery subsystem. The Newmark family is the most widelyused family of direct methods for solving systems likeMu + Cu + Ku = f . Here we use an implicit familymember, average acceleration method to linear the ODEas Au = b, where A = M + γhC + βh2K, b = f n+1 −C ˜u

n+1 − Kun+1, β = 14 and γ = 1

2 . un+1 and ˜un+1

are twopredictors defined as:

{un+1 = un + hun + h2

2 (1 − 2β)un,

˜un+1 = un + (1 − γ)hun

(12)

The unknown displacements and velocities can be com-puted through:

{un+1 = un+1 + βh2un+1,

un+1 = ˜un+1 + γhun+1

(13)

In our implementation, the variables are the spectralcounterparts, i.e. q. Constraints expressed as linear equa-tions are integrated using the Lagrange Multiplier Method.In addition to the boundary constraint (Equation (11)),other users’ manipulation constraints including eitherposition or orientation constraints are necessary as well.These constraints can be formulated in a symmetric fash-ion. Then we can build a generalized constraint matrixdemoted by J such that:

J

[qd

qr

]=

[0

c

](14)

where

J =

Ebd�d −Eb

r�r

Ebd�d −Eb

r�r

Ep

dRd�d 0

EodRd�d 0

0 Epr Rr�r

0 EorRr�r

(15)

Ep andEo play roles similar toEb, i.e., picking out the cor-responding constrained nodal values of each subsystem.Vector c is assembled by the desired constraint values. Asin Reference [20], the damped second-order constraintform is adopted:

[Ad 0

0 Ar

]J�

J 0

qd

qr

βh2λ

=

bd

br

0[R�

d 0

0 R�r

]c

− 2p1C − p2

2C

(16)

where λ denotes the Lagrange Multipliers, and p1 and p2

are stabilization factors. The rotation terms are movedto the right side of the equation so the matrix on theleft-hand side can be pre-computed and J turns to ˜(J).From the computed qd and qr we can get qd and qr asin Equation (13). The spatial displacements for updatinggeometry can be computed using Equation (7).


DOI: 10.1002/cav


Figure 5. Computational flow chart.

Implementation andExperimental Results

Our framework is implemented using Microsoft VisualC++ 2005 on a Windows XP PC with Intel Core2 Duo2.93GHz CPU, 2GB DDR2 RAM, and NVIDIA GeForceGTX 280 GPU with 1GB DDR3 VRAM.

Figure 5 is a general flow chart illustrating the com-putational procedure of this framework. All the compu-tation is partitioned into offline computation and onlinecomputation. The spatial geometry update is executedby the GPU. The original hybrid solid is subdivided intoseveral regions as defined by the user. Once the linear

Figure 6. Simulate leg behavior of Armadillo model.

system in Equation (16) is resolved, the framework pro-ceeds to online computation.

The most time consuming part in online computationis updating the geometry of the hybrid solid at each timestep as in Equation (7). Fortunately, this part can be par-allelized with the GPU in our framework. In order to dothis, �d or �r for all the subsystems are assembled intoone single global matrix � and � called global displace-ment matrix and global rotation matrix, respectively. Simi-larly, we also build a global spectral displacement vector q

and global spectral rotation vector w. Every three consecu-tive rows correspond to the spatial position or rotation ofone node and are updated in parallel with vertex programsexecuting concurrently on multiple nodes.

We have conducted extensive experiments and tests onvarious 3D tetrahedral meshes. Some of the models areof very large size and not commonly seen in other real-time physically-based animation applications. Detailedbenchmarks of each computational stage of our frame-work are shown in Table 1 where the usage of the GPUmakes simulation about three times faster.

Hybrid solids provide more natural solutions to simu-late real world objects. In Figure 6, the leg motion of theArmadillo is simulated where two rigid interior regionsmarked in dark red function as bones. Figure 7 containsseveral snapshots of our implementation where the userinteractively defines the regions’ properties and manip-ulates the models in real-time. Most experiments men-tioned can be found in the accompanying video demowhich was produced by real-time screen capture of ourimplementation.

Further Work

This work provides a spectral solution of real-time simu-lation of hybrid objects. However, currently we have notintegrated collision or contact handling into our frame-work. Though we believe due to the uniformity of this


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Y. YANG ET AL............................................................................................

Figure 7. (a) Deformable bar, (b) hybrid bar (5372 tetrahedra), and (c) the dragon model of 32 959 tetrahedra opens its mouthwith two rigid jaws (purple).

framework, collision, and contact could be incorporatedin a straightforward manner. In addition, the rapid de-velopment of graphical hardware raises some more ad-vanced parallel computation languages such as CUDA.As this work is heavily involved with matrices and vec-tors based computation, these new-emerging program-ming tools can definitely contribute a faster simulationand benefit the final performance. On the other hand,the rigid region discretized by the 3D tetrahedral mesh,in many situations may not be the best candidate for ef-ficiently describing some natural constraints. One exam-ple is the human body, where the rigid subsystem couldbe more properly expressed using only line-based rigidconstraints.5 In the future, we will investigate differenttypes of rigid subsystems as well as some joint hierarchiesmodels21 and attempt to integrate them into the spectralhybrid framework to simulate more realistic behaviorsof natural creatures in the real world.

ACKNOWLEDGMENTS

Authors thank the anonymous reviewer’s diligent efforts andbeneficial comments on the paper. This work is supported bythe National Science Foundation under Grant No. CCF-0727098.

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Authors’ biographies:

Yin Yang is currently a Ph.D student at Department ofComputer Science, University of Texas at Dallas. His re-search interests include computer graphics and anima-tion, visualization and medical imaging, especially inphysically based simulation and animation.

Guodong Rong received his B.Eng. and M.Eng. degreesboth in computer science from Shandong University in2000 and 2003 respectively, and his Ph.D. degree in com-puter science from National University of Singapore in2007. He is currently a research scholar at Departmentof Computer Science, University of Texas at Dallas. Hisresearch interests include computer graphics, computa-tional geometry, visualization and image processing, es-pecially on using the GPU to accelerate geometry-relatedproblems.

Luis Torres is a undergraduate student at Departmentof Computer Science, University of Texas at Dallas. Hisresearch focuses on computer graphics.

Xiaohu Guo is an assistant professor of computer scienceat the University of Texas at Dallas. He received the PhDdegree in computer science from the State University ofNew York at Stony Brook in 2006. His research interestsinclude computer graphics, animation and visualiza-tion, with an emphasis on geometric and physics-basedmodeling. His current researches at UT-Dallas include:spectral geometric analysis, deformable models, cen-troidal Voronoi tessellation, GPU algorithms, 3D and4D medical image analysis, etc. For more information,please visit http://www.utdallas.edu/∼xguo.


DOI: 10.1002/cav

Real-time hybrid solid simulation: spectral unification of deformable and rigid materials

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