A Multigrid Framework for Real-Time Simulation of ...vcg.isti.cnr.it/vriphys05/material/paper46/paper46.pdf · per vertex [MDM∗02], the global stiffness matrix has to be reassembled

Workshop On Virtual Reality Interaction and Physical Simulation (2005)F. Ganovelli and C. Mendoza (Editors)

A Multigrid Frameworkfor Real-Time Simulation of Deformable Volumes

Joachim Georgii and Rüdiger Westermann

ComputerGraphics & Visualization Group, Technische Universität München†

Abstract

In this paper, we present a multigrid framework for constructing implicit, yetinteractive solvers for the govern-ing equations of motion of deformable volumetric bodies. We have integratedlinearized, corotational linearizedand non-linear Green strain into this framework. Based on a 3D finite element hierarchy, this approach enablesrealistic simulation of objects exhibiting an elastic modulus with a dynamic range of several orders of magnitude.Using the linearized strain measure, we can simulate 50 thousand tetrahedral elements with 20 fps on a singleprocessor CPU. By using corotational linearized and non-linear Greenstrain, we can still simulate five thousandand two thousand elements, respectively, at the same rates.

Categories and Subject Descriptors(according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling I.3.7 [Computer Graphics]: Three-DimensionalGraphics and Realism

1. Introduction

Over the last decade, interactive, yet physics-based simula-tion of deformable volumetric bodies has received increasingattention in a number of applications. Especially in medicalapplications there is an ongoing demand for ever realisticsimulations of such objects. Popular examples include plas-tic and reconstructive surgery, breast augmentation or virtualtraining simulators. In these scenarios, physical correctnessis often sacrificed for efficiency, resulting in approximate be-havioral simulations or in the restriction to non-realistic ma-terial properties and small deformations.

The reason why interactive and physics-based simula-tion of reasonably sized volumetric bodies is still difficultto achieve is twofold: First, the most efficient techniquesknown so far require ever smaller simulation time stepswith increasing material stiffness. For the simulation of real-world volumetric bodies, which exhibit an elastic moduluswith a dynamic range of several orders of magnitude, thesetechniques in general can not fulfill both the requirements onframe rate and numerical stability. Second, even for simpleabstractions, calculations involved in stable techniques are

† [email protected], [email protected]

usually too expensive as to allow for real-time simulationsof such bodies.

In this work, we present a multigrid framework for con-structing implicit and stable solvers for the governing equa-tions of motion of deformable volumetric bodies. Thisframework provides an interactive means for simulating thedynamic behavior of an elastic solid under external forces,and it is open to a variety of different formulations of strain(see Figure 1). Independent of the formulation used, by tak-

Figure 1: Deformations of a tetrahedral model under thesame external forces are shown. From left to right, lin-ear Cauchy strain, corotational linear Cauchy strain, andnon-linear Green strain is simulated. Because the Cauchystrain is not invariant under rotations, it introduces artificialforces. Very close results are obtained using the corotationaland the non-linear formulation of strain.

c© The Eurographics Association 2005.

J. Georgii & R. Westermann / A Multigrid Framework for Real-Time Simulation of Deformable Volumes

ing advantage of a finite element hierarchy the proposedsolvers are significantly faster than previous solution meth-ods. If the linear strain measure is used, about 50 thousandtetrahedral elements can be simulated at 20 fps on a sin-gle processor CPU. At the same rate, up to five thousandelements can be deformed using the corotational multigridsimulation of the linear Cauchy strain. Even the simula-tion of non-linear Green strain can be performed interac-tively for a few thousand elements. The latter alternative isof particular interest in applications where internal stress be-comes large and the corotational formulation of strain tendsto produce artificial forces, e.g. if immediate or abrupt forcesare applied. Due to the linear asymptotic complexity of themultigrid solvers proposed, with increasing problem size anever increasing performance gain compared to previous ap-proaches is achieved.

The implicit nature of the presented framework makes itamenable to the simulation of heterogeneous volumes ex-hibiting a wide range of stiffness characteristics. An exampleis shown in Figure 2.

Figure 2: Both images show a tetrahedral horse model un-der the influence of gravity. The model on the left is simu-lated using an elastic modulus of5 ·1010N/m2 (aluminium)for all elements. On the right, the abdomen of the horse wassoften using an elastic modulus of104N/m2 (organic mat-ter), while all other parts of the model remain stiff.

This property makes the framework amenable to the de-formation of real-world objects, in which an elastic moduluswith a dynamic range of several orders of magnitude is notunusual to be found. These parameters considerably affectthe objects dynamic behavior, which makes them importantin a number of applications ranging from surgery simulationand image registration to solid mechanics. Besides physicalrealism, such parameters also provide a plausible means forcontrolling the behavior of arbitrary bodies as they allow fora flexible and realistic setting of deformation characteristics.

2. Related Work

To study the motion of a mechanical system caused by exter-nal forces, physics-based simulation is needed. For a set ofconnected rigid or flexible parts exhibiting material depen-dent properties the equations of motion can be formulatedand solved to predict the dynamic behavior of such systems[Bra01]. In computer graphics and medical applications, a

variety of interactive approaches for simulating such sys-tems have been derived. From a large scale perspective, thesetechniques can be classified according to the underlying ob-ject discretization, the object’s intrinsic deformation behav-ior, i.e. strain measure, and the method employed to integratethe equations of motion over time (see [GM97, NMK∗05]for thorough overviews of the state of the art in this field).

In this work, we employ finite element methods [Bat02]to derive numerical solvers for the governing equations ofmotion of deformable volumetric bodies. Based on a dis-cretization of the body into a set of elements, e.g. lineartetrahedral elements, boundary elements [JP99] or finite vol-umes [TBHF03], the solution of the equations to be solvedon the domain is then characterized by parameters of theseelements.

Implicit solution methods require the assembling ofall element equations into a large system of algebraicequations, which can then be solved using matrix pre-inversion [BNC96] or the conjugate gradient method[MDM ∗02, EKS03, HS04, MG04]. An acceleration methodwas proposed in [CDA99], where a precomputed linear elas-tic model is interpolated at run-time. Besides the use of im-plicit methods in finite element simulations, they have alsobeen employed in finite difference and mass-spring models[TPBF87, LTW95, BW98, DSB99] to enable stable simula-tions even for large time steps.

While the mentioned approaches consider the linear strainmeasure, i.e. the Cauchy strain, in [MG04, EKS03] a corota-tional formulation of the linear strain was used, which elim-inates artifacts typically introduced by the Cauchy strain.In this method, the rotational part of the deformation is ex-tracted for each finite element and the forces are computedwith respect to the initial reference frame. In this way, sta-ble and fast simulation results can be obtained. In contrast toan earlier approach, where the rotational part was extractedper vertex [MDM∗02], the global stiffness matrix has to bereassembled at every time step.

Explicit finite element methods avoid the construc-tion and solution of a large system of equations. There-fore, the non-linear Green strain can be integrated muchmore efficiently into these methods. Interactive simula-tion techniques using this measure have been presentedin [ZC99, WDGT01, PDA01, DDBC01, ML03]. However,methods based on explicit time integration are limited dueto the Courant condition, which significantly restricts thelargest possible time step for very stiff materials.

To accelerate finite element methods, multiresolutiontechniques based on adaptive refinements have been pro-posed [CGC∗02, DDBC01, GKS02]. An explicit multigridscheme for the simulation of surface deformations was pre-sented in [WT04]. To the best of our knowledge, an implicityet interactive multigrid solver for volumetric bodies has notyet been developed.



3. Elasticity Theory

The motion of a deforming volumetric object can be simu-lated by a displacement field in an elastic solid. Given sucha solid in the reference configurationx ∈ Ω, the deformedsolid is modeled using a displacement functionu(x),u :R

3→ R

3. This function describes the displacement vec-tor at every point∈ Ω, yielding the deformed configurationx+u(x).

3.1. Lagrangian Equation of Motion

Driven by external forces, the dynamic behavior of the de-formed solid is governed by the Lagrangian equation of mo-tion

Mu+Cu+K(u) = f (1)

whereM, C, andK are respectively known as the system’smass, damping, and stiffness matrix.u consists of the lin-earized displacement vectors of all vertices andf is the lin-earized force vectors applied to these vertices.

By discretization ofu,u andu with respect to time, the dif-ferential equation can be transformed into a set of differenceequations. To avoid artificial damping typical to implicit Eu-ler integration, a second order accurate Newmark scheme isused for time integration:

ut+dt = ut +(

0.5ut +0.5ut+dt)

dt

ut+dt = ut + ut dt +(

0.25ut +0.25ut+dt)

dt2

By discretizingu as well as the partial derivatives ofu withrespect to time, and by replacing ˙ut+dt andut+dt in equation(1), the system of algebraic equationsK(ut+dt) = f t+dt isderived.

3.2. Finite Element Method

If a finite element method is used to model the system, sys-tem matrices are built by assembling all element matrices.Since each element in the finite element discretization onlyhas a very small number of neighbors, this system is verysparse. We are using tetrahedral elements with linear nodalbasis functions [Bat02]. Displacements are expanded in a ba-sis of shape functionsΦ as

u(x) = Φ(x)ue, (2)

whereue = (uT1 , . . . ,uT

4 )T contains the single node displace-ments. The matricesM andC are derived from simple masslumping and Rayleigh damping.

The stiffness matrixK accounts for the strain energy as-sociated with the displacement field, and it is thus depen-dent on the elastic energy stored in the solid and on the workdone by body forces and tractions applied through the dis-placement fieldu. The Green strain tensorE describes the

non-linear relation between deformation and displacement:

Ei j =12

(

∂ui

∂x j+

∂u j

∂xi

)

+12

3

∑k=1

∂ui

∂xk

∂u j

∂xk(3)

In an isotropic and fully elastic body, stress (S) and straintensors are coupled through Hooke’s law (linear materiallaw)

S = λ

(

3

∑i=1

Eii

)

· I33+2µE , (4)

with the Lamé coefficientsλ andµ.

Given the nodal basis functions as well as per-node strainand stress, the potential energy

V =12

Z

Ω∑i, j

Ei jSi j dx

of every element can be computed. For a solid element tobe in equilibrium, the first variation ofV with respect tothe per-node displacements has to vanish. The resulting sin-gle element equations are finally assembled into a system ofnon-linear equations.

3.3. Corotational Linear Strain

A common simplification is to neglect the quadratic termsin the definition of the strain tensor (3), yielding the (lin-earized) Cauchy strain tensor. Then, the assembling processresults in a system of linear equations, where the matrixdoes not change during the simulation. While this approx-imation is appropriate for small deformations, for large de-formations it leads to non-realistic displacements. Further-more, as the Cauchy tensor is not invariant under rotations,incorrect forces are very likely to occur in the linear setting(see Figure 1 and 6).

A rotational invariant formulation of the Cauchy straintensor is obtained using the so-called corotational strainof linear elasticity [MG04]. In this formulation finite el-ements are first rotated into their initial configuration be-fore the strain is computed. In this way, although strain isstill approximated linearly, artificial forces as they are com-puted using the Cauchy strain are significantly reduced. Ro-tations are calculated per element using a polar decomposi-tion of the deformation gradient∇(x+u(x)) as proposed in[EKS03, HS04]. Once the rotationOe of all finite elementsare calculated, the element stiffness matrixKe is replaced byOeKe(Oe)T , and the global stiffness matrix is reassembled.A solution to equation (1) is then found by solving a systemof linear equations with the updated system matrixK.

In the following we present a multigrid framework thatsignificantly speeds up the simulation of all three differentstrain measures. In particular we show, that the solution ofthe system of linear equations with fixed and changing sys-tem matrix as in the linear and corotational setting, as wellas the solution of a system of non-linear equations as in thenon-linear setting take advantage of such a framework.



4. Multigrid Method

Multigrid methods provide a general means for construct-ing scalable linear solvers. Such methods exploit the factthat a problem can be solved on different scales of resolu-tion. At the core of such approaches two observations areof major importance. First, a basic property of many itera-tive solvers for linear or non-linear systems of equations issmoothing. Many relaxation methods like Gauss-Seidel re-duce high frequencies in the error very quickly, while lowfrequencies are damped rather slowly. Second, the remaininglow-frequency errors can be accurately and efficiently solvedfor on a coarser grid. Both observations can be combinedinto a multigrid strategy that enables improved convergenceat the same time avoiding any loss in accuracy, because onlythe smoothed error is transferred to the coarser grid. Recur-sive application of this basic idea to each consecutive systemon a hierarchy of grid levels leads to a multigrid V-cycle.

For the efficient simulation of an elastic deformable solidwe have developed a geometric multigrid method. In partic-ular, this method includes geometry-specific relaxation, re-striction, and interpolation operators. These operators formthe essential multigrid components, as they are used to trans-fer quantities along the object hierarchy.

In this work, we define an appropriate finite element hi-erarchy, which allows for an efficient implementation ofmultigrid components. The result is a method that uniformlydamps all error frequencies with a computational cost thatdepends only linearly on the problem size.

4.1. Unstructured Hierarchy

The geometric multigrid method requires a mesh hierarchythat represents the deformable object at different levels ofresolution. On this hierarchy, appropriate transfer operatorsto map quantities between different levels have to be de-signed. Starting with a finite element mesh at the coars-est resolution level, a common way to construct the hier-archy in a top-down approach is to split each tetrahedronas shown in Figure 3. The octahedron is subsequently splitinto four tetrahedra, such that eight children are generatedoverall. This approach results in a nested hierarchy, whichallows the transfer operators to be defined in a straight for-ward way, but it requires the initial mesh to be fine enoughto achieve a proper representation of the object’s boundaryat ever finer resolution levels. Furthermore, subsequent sub-divisions might lead to a fine mesh which contains ill-shapedtetrahedra that are not suited for finite element simulation.

To avoid these drawbacks, we propose linear transfer op-erators that do not require a nested hierarchy and can be inte-grated efficiently into the multigrid scheme. These operatorsestablish relations in a multilevel hierarchy of unstructuredand unrelated meshes by means of barycentric interpolationas illustrated in Figure 3.

Initially, we start with a coarse meshH and a fine mesh

Figure 3: Left: tetrahedral subdivision lends itself directlyto a nested hierarchy. Right: geometric relations between el-ements in the non-nested hierarchy are illustrated (the 2Dcase is shown for simplicity). Dotted and solid lines indicatethe coarse and the fine mesh respectively. Barycentric inter-polation weights are highlighted by dotted red lines.

h. For every tetrahedron inH, all vertices ofh inside thistetrahedron are determined. The barycentric coordinates ofthese vertices with respect to the circumscribed element arecalculated, and they are used as interpolation weights to mapvalues from the coarse grid to vertices on the fine grid via theinterpolation operatorRT

h . Each fine grid vertex stores therespective coarse grid vertices and corresponding weights.For vertices inh that lie outside the coarse mesh, barycen-tric coordinates to the closest tetrahedron in the coarse meshare computed. The restriction operator that is required in themultigrid method to gather values from a finer resolutionlevel is realized by inverse interpolationRh.

We should also note that during the construction of thematrix hierarchy the Galerkin property [BHM00] is en-forced. In particular, for all but the finest hierarchy level thesystem matrices are computed as

KH = RhKhRTh .

The Galerkin property guarantees a consistent calculation ondifferent levels of resolution, and it assures optimal conver-gence of the multigrid scheme.

It should be evident that the multigrid method computesthe correct FEM solution on the entire mesh at the finestlevel. This is in contrast to other multiresolution techniques,e.g. [DDBC01], where the solution is computed adaptivelyfor sub-meshes at different resolutions. This can lead to in-consistent deformations on different hierarchy levels, whichis entirely avoided by the multigrid approach.

4.2. Linear Elasticity Multigrid

Given the linear transfer operatorRh, as well as an initialapproximation ˆuh of the displacement values on the fine gridh of the deformable solid, a new approximationuh can becomputed as follows (eh andeH denote the error):

① compute residual rh = f h−Khuh

② restrict residual to coarse grid rH = Rhrh

③ solution on coarse grid KHeH = rH

④ transfer correction eh = RTh eH

⑤ correction uh = uh +eh



The algorithm extends to a complete 2-grid approach byrelaxation of ˆuh prior to stage① to avoid the transfer of quan-tities from theh-grid that can not be reduced on the coarseH-grid, and by relaxation of the resultuh after stage⑤ toavoid high frequencies introduced by numerical inaccura-cies. In general, 1− 2 Gauss-Seidel steps are sufficient forboth pre- and post-smoothing. By recursive application ofthe coarse grid correction to stage③ and by using a pre-conditioned conjugate gradient method to compute the solu-tion on the coarsest grid, a full multigrid V-cycle is derived.

4.3. Corotational Elasticity Multigrid

Because the integration of corotational strain leads to a sys-tem of linear equations, the multigrid method is essentiallythe same as for the Cauchy strain tensor. However, as the sys-tem matrixK on the finest level changes in every time step,the coarse grid matrices have to be rebuilt, too. Fortunately,asK is very sparse, a row-based index data structure can beutilized to significantly speed up this process. In addition,such a data structure reduces the number of numerical andmemory access operations to ensure the Galerkin propertyon every hierarchy level.

4.4. Non-linear Elasticity Multigrid

The simulation of deformations based on the Green straintensor using an implicit time integration scheme requires anon-linear equation system to be solved. To calculate a solu-tion to this system we employ the Newton method, which isbased on the first order Taylor approximation of the systemof equations:

K(u+e) ≈ K(u)+K′(u)e

Here,K′ is the Jacobian matrix ofK. Given an initial solu-tion u, this solution can be corrected by solving the equationfor e, which only requires a system of linear equations basedon the Jacobian matrix to be solved:

K′(u)e= f −K(u),

For this system, the multigrid approach as described in chap-ter 4.2 is used.

To construct the non-linear system of equationsK(u),we utilize symbolic algebra operations in the preprocessingstep. The set of all non-linear element stiffness equations isassembled symbolically into a system of non-linear equa-tions. From equation (3) one can see, thatSi jEi j can be ex-pressed in terms of nodal basis functionsΦ(x). By first ap-plying the material law to expressS in terms ofE , E canthen be expressed by the partial derivatives ofu, whereu isinterpolated as shown in equation (2). Symbolic calculationof Si j Ei j results in a polynomial in the unknown variablesui .These polynomials, which share a large number of monomi-als, can then be assembled into a global system of symbolicequations. The number of monomials to be evaluated in this

system is significantly smaller (about a factor of 3) than thenumber of monomials contained in the set of element equa-tions. Consequently, multiple evaluations of monomials canbe avoided at run-time.

In the same way, the Jacobian matrix can be expressedsymbolically, and then calculated using the current parame-ter values. The multigrid solver then updates the hierarchytaking into account the current evaluation of the Jacobianmatrix. Although building the matrix hierarchy in every stepis expensive both in terms of numerical and memory accessoperations, the multigrid solution is still up to 10 times asfast as a preconditioned conjugate gradient method.

To further improve the evaluation of polynomials at run-time, all required monomials∏i, j u j

i are computed from thecurrent displacement vectoru. As these monomials occur inseveral equations of the system as well as in the Jacobian ma-trix, an additional speed-up of about a factor of 2 is achieved.

5. Results

In the following, we give several examples that demonstratethe efficiency of the proposed multigrid method. The mod-els used in these examples are shown in the Figures below.All experiments were run on a Pentium4 3.0 GHz processorequipped with 1GB RAM. As shown in Table 1, the multi-grid method scales linearly with the number of elements, andit achieves excellent performance rates even for large mod-els. Tetrahedral meshes with about 50k elements can be sim-ulated interactively using the linear strain measure. Particu-larly in the last example, where a larger hierarchy allows themultigrid approach to deploy it’s full potential, for the num-ber of elements used the method is considerably faster thanimplicit approaches utilizing the conjugate gradient method.The star∗ in all tables denotes the use of a non-nested gridhierarchy.

tps tpsModel # level # tet # vert mgrid CG

Liver 2* 1467 464 720 30Bridge 3 3072 825 460 2.4

Liver 3* 8078 1915 140 4.3Breast 3* 10437 2542 120 1.5Horse 4* 49735 12233 18 -

Table 1: Timing results in time steps per second (tps) for dif-ferent models using the linearized Cauchy strain measure.The multigrid solver is compared to a preconditioned conju-gate gradient solver (CG).

Compared to previous approaches, the implicit multigridsolvers enable much larger integration time steps of up toone second. Even more importantly, the time step does notdepend on material stiffness. This property enables stablesimulations of heterogeneous bodies with an elastic modulus



varying from 103 N/m2 to 1011 N/m2. Figure 5 shows theinfluence of wind forces and gravity to bars of different den-sity and stiffness. As the force is constant everywhere, softerbars are deformed much more significantly than stiffer ones.

As it is shown in Figure 6, the artifacts introduced by thelinear strain measure can almost entirely be avoided by usingthe corotational formulation of linear strain. However, theperformance gain of the multigrid approach is not as high asin the linear setting. This is due to the extra cost involved inupdating the system matrices. As updating these matrices fora non-nested hierarchy (marked by a∗ in the Tables) is muchmore costly than for a nested hierarchy, in case of a non-nested hierarchy only for larger meshes a significant speedup compared to [MG04] is achieved (see Table 2 and Figure4). However, for stiffer materials the performance gain getsever better as illustrated on the right of Figure 4. If a nestedhierarchy is employed, however, matrix updates only requirelinear interpolation along element edges. As a matter of fact,a performance gain of up to 10 compared to the conjugategradient method is achieved. Table 4 shows the time it takesto reassemble the system matrix in the corotational settingand to built up the matrix hierarchy used by the multigridapproach.


Bridge 2 384 153 150 110Liver 2* 1467 464 21 18

Bridge 3 3072 825 19 1.9Liver 3* 8078 1915 3.6 1.2

Breast 3* 10437 2542 2.8 0.7Breast 3 10432 2541 6 0.7

Table 2: Timings statistics for different models using thecorotational simulation of linear strain. The multigrid solveris compared to a preconditioned conjugate gradient solver(CG)

Compared to the linear setting, in the non-linear strainsetting real-time can only be achieved if the number of el-ements is significantly reduced. However, compared to thecorotational setting the performance is only about a factor of2 lower (see Table 3). Explicit timings of the reassemblingstep in the non-linear setting are given in Table 4.

The efficiency and effectiveness of the non-linear solver isalso demonstrated in Figure 7, where large and global defor-mations of about 3K tetrahedral elements including collisiondetection and response to static objects is performed in real-time. Figure 8 shows the application of the proposed simu-lation engine for breast augmentation. Deformations due togravity as well as additional implants that are inserted intothe breast can be simulated very realistically in real-time. InFigure 9 it is shown, that even the deformation of very largetetrahedral meshes is feasible on consumer class hardwareby means of the proposed multigrid approach.

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000 10000 100000

tim

e [

sec]

#tetrahedra

0.01

0.1

1

1000 100000 1e+07 1e+09 1e+11

tim

e [

sec]

elastic modulus [N/m2]

Figure 4: Left: on a double logarithmic scale, timings are shownfor the corotational simulation using the multigrid methodinclud-ing matrix reassembling and matrix hierarchy update (solidlines)and the CG method (dashed lines). Timings were measured usingan increasingly refined tetrahedral cube model (elastic modulus =2 ·106N/m2, integration time step =0.02sec).Right: performance measures for a bridge model (3k tetrahedra).For ever stiffer materials, the CG method requires more steps to beperformed to achieve the same relative error of10−4 as the multi-grid method. The elastic modulus affects the performance ofthe CGmethod significantly, while it does not affect the performance of themultigrid method.


Bridge 2 384 153 95 60Liver 2* 1467 464 13 8

Bridge 3 3072 825 12 1Liver 3* 8078 1915 1.8 1.2

Breast 3 10432 2541 4.8 0.7

Table 3: Timings statistics for different models using thenon-linear Green strain measure. The multigrid solver iscompared to a preconditioned conjugate gradient solver(CG)

corot. non-linear rebuildModel # level reassemb. reassemb. mgrid

Bridge 2 4 9 1Liver 2* 22 35 30

Bridge 3 42 72 17Liver 3* 95 209 198

Breast 3 115 255 65

Table 4: Timings in milliseconds for the most time-consuming parts of the multigrid solvers using the corota-tional and the non-linear formulation of strain.

6. Conclusion

In this work, we have presented an implicit multigrid frame-work for interactive and physics-based simulation of de-formable volumetric bodies, which is open to a variety of dif-ferent strain measures. The proposed solvers effectively ben-efit from coarse grid correction in that they produce numer-ically stable results yet minimizing the number of iterationsto be performed until convergence. The proposed multigridsolvers allow for the simulation of heterogeneous materials,i.e.,, materials exhibiting varying stiffness and density, with-out sacrificing speed or quality. Therefore, they have the po-



tential to be integrated into real-time scenarios such as sur-gical simulators or virtual environments.

As the described multigrid framework benefits extremelyfrom a nested hierarchy, in the future we intend to integratemore advanced meshing strategies into this framework. Inparticular, our goal is to generate nested tetrahedral meshesthat are amenable to multiresolution finite element simu-lations at the same time adapting to the object boundaries[MBTF03].

To integrate the proposed simulation framework into prac-tical applications, collision detection as well as topologychanges, i.e cutting, has to be supported. Challenging ap-proaches to detect self-collisions between parts of a deform-ing object have been presented recently [TKZ∗04]. It willbe of particular interest in the future to evaluate the integra-tion of these methods into the multigrid framework. Topol-ogy changes, on the other hand, require the multigrid matri-ces to be recomputed. Although this is possible in general, itremains to be shown that these updates can still be done inreal-time.

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Figure 5: A visualization of the internal states (i.e.,vonMises stress) of three towers exhibiting different stiffness isshown. A standard approach for direct volume rendering isused to generate the images. A constant wind force is appliedto all models. Stress values are color coded ranging from red(high) to blue (low).

Figure 6: Comparison of the linear, corotational and non-linear strain measure. The deformation of the latter one areshown as reference in blue color. While the linear Cauchystrain (red color) fails to approximate the deformation prop-erly, only very small differences can be observed in case ofthe corotatational strain (green color).

Figure 7: A simple example demonstrates the potential of theproposed non-linear simulation engine. The bridge is dis-cretized into 3K tetrahedral elements. Simulation and colli-sion detection with static obstacles is run at 12 fps.

Figure 8: Breast augmentation as one potential applicationof the proposed multigrid simulation framework is demon-strated. The simulation of gravity, different material proper-ties as well as additional forces induced by implants can besimulated interactively at high physical accuracy.

Figure 9: Iso-surface visualization from a deformed CT dataset. A tetrahedral simulation mesh was adaptively refinedalong the surface and bone structures in the 3D medicaldata set. This mesh consists of 1.1M tetrahedra, and it wasdeformed using the linear strain measure at 0.5 fps.


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A Multigrid Framework for Real-Time Simulation of ...vcg.isti.cnr.it/vriphys05/material/paper46/paper46.pdf · per vertex [MDM∗02], the global stiffness matrix has to be reassembled