# CS计算机代考程序代写 assembly scheme algorithm Interactive Virtual Materials

Interactive Virtual Materials

Matthias Mu ̈ller Markus Gross ETH Zu ̈rich

Figure 1: The pitbull with its inflated head (left) shows the artifact of linear FEM under large rotational deformations. The correct deformation is shown on the right.

Abstract

In this paper we present a fast and robust approach for simulating elasto-plastic materials and fracture in real time. Our method extends the warped stiffness finite ele- ment approach for linear elasticity and combines it with a strain-state-based plasticity model. The internal principal stress components provided by the finite element compu- tation are used to determine fracture locations and orien- tations. We also present a method to consistently animate and fracture a detailed surface mesh along with the under- lying volumetric tetrahedral mesh. This multi-resolution strategy produces realistic animations of a wide spectrum of materials at interactive rates that have typically been simulated off-line thus far.

Key words: Physically Based Animation, Finite Element Method, Large Deformations, Stiffness Warping, Elastic- ity, Plasticity, Fracture.

1 Introduction

Today’s interactive graphics applications, such as com- puter games or simulators, demand a continuously grow- ing degree of visual realism and technical sophistication. This demand poses great challenges for the underlying real-time graphics algorithms including rendering and an- imation. In addition to the display quality, it is espe- cially the way in which the physical behavior of char- acters, objects, or entire scenes are simulated that even- tually determines the degree of realism experienced by the user. Using animation sequences predefined by an artist is a viable way to simulate physical effects in in- teractive systems. However, this method exhibits clear

limitations when it comes to the realistic simulation of material behavior, such as elasticity, plasticity, melting, or fracture. Therefore, methods that control the object by physical laws have received increasing attention.

In the past decades, computational scientists have de- vised various methods for the simulation of material be- havior with a high level of accuracy. The main goal of such simulations has been the authentic reproduction of the real world, whereas interactivity has not been a pri- mary focus.

In contrast, in interactive graphics it is often not neces- sary to simulate the physical behavior with such numer- ical accuracy. Approximations are acceptable as long as they yield plausible and visually realistic results. Con- siderable research has been devoted to the development of such approximate simulation techniques. Going from off-line computations to interactive simulations, however, is not just a matter of making things faster. In order to al- low for random interaction with the system, stability, vi- sual quality and speed are the crucial requirements such methods have to satisfy.

In this paper, we present an unconditionally stable method to simulate a wide range of materials including elasticity, plasticity, melting, tearing and fracture in real time, while retaining geometric detail on the object’s sur- face. Our method is primarily designed for applications in computer games, virtual surgery and related fields.

1.1 Related Work

During the last two decades, physically-based simulation methods tailored to the field of computer graphics have been developed. In the late eighties, Terzopoulos et al. proposed the use of physical models for animating elastic and plastic objects [21] and fracture effects [20] which pi- oneered the use of physically-based models in the field of computer graphics. They used surfaces to represent de- formable objects and solved the governing partial differ- ential equations using finite difference schemes. While their work focused on the correct modeling of physical effects, the speed of the simulation was of secondary im- portance and, at that time, computations were done off- line. In the following years, a variety of new models for linear and non-linear elastic objects were proposed in computer graphics (summarized in [8] and [22]) while

little attention was given to plasticity and fracture effects.

Recently, O’Brien et al. presented a finite-element- based technique for simulating brittle [16] and ductile [15] fracture in connection with elasto-plastic materials. Their method produces visually convincing results, but it is not designed for interactive or real-time use.

ArtDefo [10] designed by James and Pai was among the first systems to simulate elastic objects at interactive rates. They use a linear boundary element model with pre-computed modes for a fixed geometry. Debunne [4] and Wu [24] use a hierarchy of volumetric meshes and Grinspun [9] and James [11] a hierarchy of basis func- tions which speed up the simulation but make changes in a mesh caused by fracture difficult to handle. On the other hand, interactive fracture methods such as the ones proposed by Smith [19] and Mu ̈ller [13] are tailored for rigid or brittle objects.

Since linear elasticity models are both stable and com- putationally cheap, they have become popular in the field of real-time animation. Linear models are, however, not suitable for large rotational deformations, because the non-linear effects create displeasing distortions of the ge- ometry (see Fig. 1). To solve this problem Capell et al. [1] suggest manual division of an object into small parts based on its skeleton. Each part uses its local (rotated) co- ordinate frame to compute the linear elastic forces. The primary disadvantage of this approach is the discontinu- ity at the boundary between two parts.

An alternative to Capell’s method is the warped stiff- ness approach proposed by Mu ̈ller [12] which is a co- rotational formulation [6] and uses a local coordinate frame for each vertex to compute the linear force. The resulting discontinuities are smaller because individual rotations are computed per vertex. However, small lo- cal errors sum up and can cause ghost forces. For FEM- based cloth modeling, Etzmuss [5] improved the warped stiffness approach by using element-based rotation ex- traction.

1.2 Our Contribution

In this paper, we present a unified algorithmic framework for the real-time simulation of a variety of physical phe- nomena for geometrically complex objects. We propose a way to efficiently and robustly compute the elastic forces avoiding linear distortions and ghost forces. Both a plas- ticity and a fracture model are integrated into the pseudo- linear elastic force computation. We also propose an al- gorithm to consistently animate and fracture a high reso- lution surface mesh based on a coarser underlying volu- metric mesh. This multi-resolution approach enables us to simulate geometrically complex objects in real time while simultaneously retaining their surface details.

2 Elasticity Model

Our elasticity model is based on the linear continuum elasticity theory [2]. We use the Finite Element Method (FEM) [3] with linear displacement tetrahedra to solve the governing partial differential equations. Here we summarize the steps essential to understand our method. We also include details of how the stiffness matrix is computed because some intermediate quantities are used in sections 3 and 4 to model plasticity and fracture.

2.1 The Stiffness Matrix

In continuum elasticity theory the deformation of an object is described by a vector field u(x) = [u(x), v(x), w(x)]T meaning that every point x = [x, y, z]T in the undeformed body corresponds to point x + u(x) in the deformed body. The first step in a finite element approach is to replace the continuous displace- ment field u(x) with a discrete set of displacement vec- tors defined only at the vertices of a mesh – in our case a tetrahedral mesh. Within each tetrahedral element e, the displacement field is linearly interpolated as

u(x) = He(x) · uˆ, (1)

where He(x) is a 3 × 12 matrix that contains the shape functions of the tetrahedral element and uˆ = [u1, v1, w1, . . . , u4, v4, w4]T is the collection of the dis- placement vectors at the four vertices of the tetrahedron. Using Cauchy’s linear strain tensor [3], for the strain within the tetrahedron we get

ε = Be · uˆ, (2)

where Be a constant 6 × 12 matrix, which can be pre- computed for every tetrahedron. Using Hooke’s law, for the stress within an element we get

σ = E · ε = EBeuˆ, (3)

where E is a 6 × 6 matrix which – for isotropic materi- als – only depends on two scalars, Young’s modulus and Poisson’s ratio [3]. The elastic forces fe acting on the nodes of an element are derived from the strain energy which, in turn, depends on the strains and stresses within the element. Using Cauchy strain, the forces turn out to be linearly dependent on the vertex displacements uˆ:

fe = Keuˆ (4)

where the 12×12 matrix Ke = VeBTe EBe is the stiffness matrix and Ve the volume of the element. Finally, the 3n×3n dimensional stiffness matrix K of the entire mesh is an assembly of the individual Ke of all the elements.

2.2 Dynamic Deformation

To simulate the dynamic behavior of an object, the co- ordinate vector x is made a function of time, x(t). The following governing equation for x(t) in Lagrange’s form describes the dynamics of the system:

where x contains the positions of the four vertices and f0e contains force offsets. Now let us assume that we know the rotational part Re of the deformation of the tetrahe- dron. Then, using the warped stiffness concept adopted to elements, we compute the forces as

Mx ̈+Cx ̇+K(x−x)=f , 0 ext

(5)

f

e

=RK ·(R−1x−x) e e −1e 0

= ReKeRe x − ReKex0 (7) = K ′e x + f 0′ e ,

where x ̇ and x ̈ are the first and second derivatives of x with respect to time, M is the mass matrix, C the damp- ing matrix and fext a vector of external forces. Eqn. (5) defines a coupled system of 3n linear ordinary differen- tial equations for the n position vectors contained in x.

The advantages of using linearized elastic forces are, first, that the stiffness matrix K is constant and can be pre-computed before the simulation starts. Second, when an implicit scheme is used to solve Eqn. (5), a purely linear system has to be solved at every time step. We choose implicit Euler integration because unlike explicit schemes it is unconditionally stable [23, 18]. Uncondi- tional stability is essential in an interactive system. Third, if the tetrahedra in the original mesh are well shaped, the system matrix is well conditioned throughout the sim- ulation. In contrast, when non-linear elastic forces are used, a stiffness matrix has to be computed at every time step and the system matrix can become arbitrarily ill- conditioned because it depends on the deformed model.

Linearized elastic forces are, however, only valid close to the equilibrium configuration. Under large rotational deformations, they cause unrealistic growth in volume as demonstrated by the pitbull model in Figure 1.

2.3 Element-Based Warped Stiffness

In [12] Mu ̈ller et al.propose a method they call Stiffness Warping to remove the artifacts that linear elastic forces show while keeping the governing equation linear. They compute the elastic forces for every vertex in a local un- rotated coordinate frame. However, the proposed way of extracting the rotational components of the deformation has two main disadvantages. First, the rotation Ri of ver- tex i has to be computed from the locations of its adja- cent vertices which is an ambiguous problem. Second, the elastic forces are not guaranteed to sum up to zero resulting in possible ghost forces.

Following Etzmuss [5], we solve these problems by ex- tracting rotations of elements rather than rotations of ver- tices. Contrary to the vertex-based approach, our method also allows the integration of the plasticity model de- scribed in the next section. For a single tetrahedral el- ement with stiffness matrix Ke, the forces fe acting at its four vertices are

fe =Ke ·(x−x0)=Ke ·x+f0e, (6)

where Re is a 12 × 12 matrix that contains four copies of the 3 × 3 rotation matrix along its diagonal. By do- ing so, we reach the exact same forces as though we had computed the regular linear elastic forces in a rotated co- ordinate frame (see Fig. 2). The forces in fe are, thus, guaranteed to sum to zero. Now, for the elastic forces of the entire mesh we get

f =K′x+f0′, (8)

where the global stiffness matrix K′ and force offset vec-

tor f0′ are the sums of the element’s rotated stiffness ma-

tricesK′ =RKR−1 andforceoffsetsf′ =Rf

e eee 0e e0e

with global vertex numbers. Our method has the follow- ing main features:

• No ghost forces: Since the individual fe all sum to zero, the forces in f sum to zero, too, which means the method does not produce ghost forces.

• Stability: As in non-linear FEM, the global system matrix derived from Eqn. (5) changes for every time step. However, in non-linear FEM it can become arbitrarily ill-conditioned because it depends on the shape of the deformed tetrahedra. With our method, the local system matrices of elements are only ro- tated which does not change their condition. Since the global system matrix depends on the sum of dif- ferently rotated element matrices, its condition num- ber might still change slightly. However, in all of our examples presented in section 7, the global con- dition number never changed by more than 25 per- cent. If the tetrahedra in the original mesh have good aspect ratios, the simulation remains stable for arbi- trary deformations.

•

Speed: In contrast to non-linear FEM, the ele- ment’s stiffness matrices Ke can be pre-computed and reused saving a considerable amount of com- putation time [12]. However, unlike in the vertex- based technique, these matrices are rotated before the summation. This means that, while the Ke can still be pre-computed, the global stiffness matrix K′ has to be summed up whenever the rotations change, i.e. at every time step. Fortunately, the time for the

R −1x x0 e

x0 are multiplied with the stiffness matrix yielding the forces Ke(R−1x − x0) that are finally rotated back to the frame of

e

the deformed tetrahedron by multiplying them with Re.

2.4

Rotation of a Tetrahedron

summation is linear in the number of elements and according to our experiments (Fig. 7), small in com- parison to the time it takes to solve the linear system.

• Quality: The error of the approximation is the same as in linear FEM if the rotation matrices Re = I for the entire simulation. If all the Re are the same but follow the rigid body transformation of the entire mesh, our method outperforms linear FEM because it produces the same forces as if the linear model was re-computed in the rotated frame for every time step and, thus, correctly produces zero forces for pure rigid body transformations. When individual Re’s are computed for every element, the approxi- mation gets even better because it can then adopt to local rotations.

There is a simple way to compute rotations Re of tetrahe- dral elements. Let us look at a tetrahedron whose vertices have coordinates p1 , . . . , p4 in the undeformed state and q1, . . . , q4 in the deformed state. The barycentric coor- dinates β1, . . . , β4 of a point p with respect to the unde- formed tetrahedron satisfy

p1x p2x p3x p4x β1 px

p1y p2y p3y p4y β2 py

p p p p·β=p(9)

R K (R −1x−x ) eee0

and, thus, combining Eqn. 9 and 10 we get

q = Qβ = QP−1p = Ap. (11)

(x−x0)

xRe −1

The unique matrix A = QP that describes the trans-

formation of the tetrahedron has the form

A= B t , (12)

0001

where t contains the translational part of the transforma- tion and B the rotation and stretching parts. The 3 by 3 matrix B is independent of translations of both P and Q. Finally, the rotational part of the transformation can be extracted by a polar decomposition of B as proposed in [5].

3 Plasticity Model

So far, our model is perfectly elastic, i.e. when exter- nal forces are removed, the model returns to its origi- nal shape. In contrast, an elasto-plastic material stores part of the deformation and returns to a configuration be- tween the deformed and undeformed state when the ex- ternal forces are removed. In [15] O’Brien describes a method for modelling plasticity. The method is used in the context of non-linear FEM and explicit integration. Here we show how it can be adopted to our framework, namely linear, warped FEM in connection with implicit integration. We also suggest a way to simulate plastic creep, an effect not present in O’Brien’s model.

According to Eqn. (2) and our warped stiffness ap- proach, a deformed tetrahedon is under a total strain of

ε =B ·uˆ=B(R−1x−x). (13) total e e e 0

A plastic element stores plastic strain in a state variable εplastic and only the difference between the total strain and the plastic strain – the elastic strain – causes internal elas- tic forces (see Fig. 3):

εelastic = εtotal − εplastic (14)

Note that these strains are six-dimensional vectors. The state variable εplastic is initialized with 0. At every time step, it is updated as follows:

(R−1x−x) e0

−1 Ke(Re x−x0)

Figure 2: To compute the elastic forces acting at the vertices

of a tetrahedron, its deformed coordinates x are rotated back to

an unrotated frame R−1x. There the displacements R−1x − ee

1z 2z 3z 4z 3

1 1 1 1 β4 1 total e e 0

or Pβ = p. The point q in the deformed tetrahedron that corresponds to p has the same barycentric coordinates β but with respect to the deformed tetrahedron

Qβ = q (10)

ε ←B (R−1x−x ) εelastic ← εtotal − εplastic

if ||εelastic||2 > cyield : εplastic += ∆t · ccreep · εelastic if ||εplastic||2 > cmax : εplastic *= cmax/||εplastic||2

First, the total and elastic strains are updated. The plas- ticity model has three scalar parameters cyield , ccreep and

z

fplastic

ftotal

original state

plastic rest state

-fplastic felastic

deformed state

for every tetrahedral element and to fracture the model

normal to nmax if σmax exceeds the threshold of the ma-

terial. Thus, at every time step we compute, for each

tetrahedron, the stress tensor σ = EB (R−1x − x ), its ee0

largest eigenvalue σmax and its corresponding eigenvec- tor nmax. If σmax exceeds the fracture threshold of the material, we split the mesh as follows (see Fig. 4): First, we select randomly one of the tetrahedron’s vertices that is marked as a crack tip. If none of them are marked, we randomly make a choice amongst all four vertices. We do this for two reasons. First, because it is more likely that an existing crack propagates than that a new one is formed and second, because in real materials, randomly located microscopic imperfections are the points where cracks are initiated [7].

Once a vertex v is selected to be split, we virtually put a plane α through v perpendicular to nmax. Then v is split into two new vertices v+ and v−. For all tetrahedra adjacent to v, their vertex v is replaced by either v+ or v− depending on whether their center lies on the positive or negative side of α. For all pairs of tetrahedra lying on opposite sides of α who shared a common face (v, v1, v2) before the split, two sides (v+, v1, v2) and (v−, v1, v2) get exposed after the split and v1 and v2 are marked as crack tips.

With this method the cracks can only go along the tetrahedral boundaries which can result in visual artifacts. A way to reduce them is to use a non-regular tetrahedral mesh. We use this simple, fracture procedure for two main reasons. First, it is very fast because the manip- ulations in the mesh are minimal. Second, the stiffness matrices Ke of the elements do not change. The way they are added to the global stiffness matrix does change but the summation is done at every time step anyway. Frac- ture, thus, causes no computational overhead other than some updates in the adjacency lists of the mesh. Also, as with the plasticity modifications, fracture does not intro- duce any instability to the linear system to be solved.

5 Surface Mesh Animation

Even though our method is fast enough to animate a few thousand volumetric elements at interactive rates (Fig. 7), a tetrahedral mesh with a resolution of this order cannot represent a surface of appealing quality. A natural way to solve this problem is to work with two different represen- tations for the same object, a low resolution volumetric mesh for the FEM simulation and a high resolution sur- face mesh for rendering. This is a reasonable solution because firstly, surface detail does not significantly af- fect the physical behavior of an object and secondly, the deformation field stored in the low resolution volumetric mesh provides enough information to animate a detailed

Figure 3: Relative to the original state the deformation of the tetrahedron yields a force of ftotal. In a plastic material only the offset force ftotal − fplastic relative to the plastic rest state acts as elastic force felastic. Here, the stiffness is chosen such that displacement equals force.

cmax. If the 2-norm of the elastic strain exceeds the threshold cyield, the plastic strain absorbs part of it. If ccreep ∈ [0 . . . 1/∆t] is 1/∆t, the elastic strain is imme- diately and completely absorbed. Small values for ccreep yield slow plastic flow in the material. The parameter cmax defines the maximum plastic strain an element can store. If the 2-norm of the plastic strain exceeds cmax, the plastic strain is scaled down accordingly.

We integrate the effect of the plastic strain into Eqn. (5) via corresponding plastic forces. This way the stiff- ness matrices Ke of the elements do not have to be re- computed. We use the definition of the stiffness matrix from Eqn. (4) to compute the plastic forces fplastic that correspond to εplastic:

fplastic

The plasticity matrix Pe = VeBTe E that maps the plas- tic strain to plastic forces is constant and can be pre- computed for each element. At every time step we com- pute the sum of the plastic forces coming from all ele- ments and simply subtract them from Eqn. 5 to get

Mx ̈ + Cx ̇ + K′x + f0′ − fplastic = fext. (16)

Since we only add a force vector fplastic to the equation of motion at every time step, plasticity does not change the condition of the linear system which needs to be solved.

4 Fracture Model

To complete our model, we propose a method, similar to the ones described in [13] and [16], for animating mate- rial tearing and fracture if internal stresses exceed a frac- ture threshold.

The concept behind our fracture algorithm is to find the maximum tensile stress σmax and its direction nmax

= ReKeuˆplastic = R K B−1 · ε

e e e plastic

=R (V BTEB )B−1 ·ε (15)

eee ee plastic = ReVeBTe E · εplastic

= RePe · εplastic

cracktip α σmax +-

– v + v−

–

cracktip

Figure 4: Planar sketch of a vertex split in 3-d. To fracture a tetrahedron, an adjacent vertex v is chosen. Plane α is placed perpendicular to the maximum stress component σmax through v. Then v is split into v+ and v−. Tetrahedra adjacent to v that lie on the positive side of α are linked to v+, the others to v−.

Re

Figure 5: Vertices of the surface mesh (shown in red) are dis- placed according to the displacement field of the tetrahedron in which they lie. The rotation of the tetrahedron Re is used to rotate the normals.

surface mesh. Pentland et al. [17] made use of this gen- eral idea by approximating deformation modes with poly- nomial mappings. The approach we propose here is more closely related to the work of Capell et al. [1] where a coarse control mesh is used to compute dynamic defor- mations of detailed surfaces.

5.1 Mesh Coupling

To animate a surface mesh consistently with a volumetric mesh, we first link every vertex of the surface mesh to the closest tetrahedron in the volumetric mesh and store its barycentric coordinates with respect to that tetrahe- dron. During the simulation, the position of each vertex of the surface mesh is interpolated from the positions of the linked tetrahedron using the stored barycentric coor- dinates. An advantage of the warped stiffness algorithm is that it computes the rotation Re of every tetrahedral element which can directly be used to transform the nor- mals stored in the undeformed surface mesh (see Fig. 5). This simple algorithm works very well with elastic and elasto-plastic deformation (see Figures 1 and 8).

5.2 Consistent Watertight Fracturing

When the volumetric mesh fractures as described in sec- tion 4, the mesh coupling becomes more complicated.

There are two main problems that have to be solved:

α

v+ +

•

•

The vertices of the surface mesh and those of the volumetric mesh are not aligned. The three vertices of a surface triangle can be linked to different tetra- hedra that can get detached by the fracture proce- dure. Such a triangle has to be split to prevent the surface from getting stretched like rubber across the crack of the volumetric mesh.

If a two-dimensional surface mesh is fractured, un- acceptable holes will appear.

The method that we propose solves the aforemen- tioned problems and keeps an initially watertight surface mesh watertight throughout the entire simulation. The two requirement for the procedure to work correctly are, first, that the volumetric mesh totally contains the surface mesh and second, that the surface mesh is a manifold.

The basic event in the fracture procedure of the volu- metric mesh that triggers surface updates is the discon- nection of two tetrahedra that share a face (v,v1,v2) when vertex v is split. In this case two new faces (v+, v1, v2) and (v−, v1, v2) with identical coordinates are created in the volumetric mesh. The surface mesh is updated in two steps (see Fig. 6).

First, it is cut along triangle Tv = (v, v1, v2), the com- mon face along which the two tetrahedra are separated. If Tv does not intersect the surface, this step is skipped.

The cut operation can be done using the undeformed coordinates of both the volumetric and the surface mesh. We use a uniform spatial grid to quickly find the triangles of the surface mesh that intersect triangle Tv . By working with undeformed coordinates, this grid does not need to be updated when the object deforms.

In a second step, the holes in the surface are closed. Therefore a new surface triangle at the location of Tv is generated. This triangle is linked to the first tetrahedron and a copy of it to the second one. These two triangles close the hole. In the event that the new triangles reside completely inside the surface, the second step is finished. However, if Tv intersects the surface, the new triangles are subdivided along the surface mesh resulting in a set of smaller triangles of which those who lie outside the surface are deleted.

6 The Algorithm

Now we are ready to put everything together and sum- marize the entire simulation algorithm:

200

150

100

50

0

0 1000 2000

3000 4000 5000

number of tetrahedra

warped stiffness

linear

Tetra 1 Tetra 2

Tv

Ts

Figure 6: The surface mesh needs to be updated when two tetrahedra that share side Tv are detached. First, the surface mesh is cut along Tv. Then, a new surface triangle congruent with Tv is subdivided along the surface mesh and the triangles that are outside the surface are removed. The ones that are in- side are linked to the first tetrahedron and identical copies to the second one.

forall elements e compute Be, Pe, Ke from x0 initialize x0, v0

i←0

loop

forall elements e compute Re based on xi

e

inside

outside

new surface

forall elements e update εplastic,e

Figure 7: Computation time [ms] per time step against size of the mesh for warped stiffness and linear FEM.

Figure 8: The snake on the left demonstrates large rotational elastic deformations while the cow made of warm wax melts plastically under gravity.

parison to the task of solving the linear system. We use a conjugate gradients solver with a fixed number of itera- tions (20 for our demos).

Figures 8 and 9 show the range of phenomena that can be simulated at interactive rates with the method de- scribed in this paper from large, purely elastic deforma- tions (snake model), plastic deformation of warm wax un- der gravity (cow), local plastic deformation caused by a heat source (dragon) to fracture effects caused by user applied forces. The parameters used for these animations are summarized in Table 1. We made the entire simula- tion framework with a variety of additional scenes and a video available from www.matthiasmueller.info/demos.

8 Conclusions

We have presented a novel method for the real-time sim- ulation of a wide range of material effects including elas- ticity, plasticity, melting and fracture. The key to our technique is a pseudo-linear elasticity model utilizing the warped stiffness approach, which was extended to avoid ghost forces and to make it amenable to plastic- ity and fracture modeling. To enhance the visual quality of our simulations, we developed algorithms to consis-

assembleK= R K R−1 eee

assemble f0 = − ReKex0 e

assemble fplastic = e RePeεplastic,e compute external forces fext

solve (M + ∆tC + ∆t2K)vi+1 =

Mvi − ∆t(Kxi + f0 + fplastic − fext)

for vi+1

update xi+1 = xi + ∆tvi+1 forall elements e

compute stress σ = EB (R−1x − x ) ee0

if maximum eigenvalue of σ > fracture threshold then fracture mesh

endfor

i←i+1 endloop

7 Results

All the animations described in this section were com- puted and rendered in real time on a 1.8 GHz Pentium IV PC with a GForce 4 graphics card. For implicit in- tegration we used a fixed time step of 0.01 seconds. To generate tetrahedral meshes from surface meshes we used the approach described in [14].

The computational cost of one time step dependent on the number of animated tetrahedra for the linear and warped stiffness approach is depicted in Figure 7. The curves show that the additional work of computing rota- tions and assembling the stiffness matrix is small in com-

ms per time step

Figure 9: A heat source changes the material properties locally (left). The surface mesh of the dragon fractures along with the volumetric mesh when the model is pulled apart (right).

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[21] D.Terzopoulos,J.Platt,A.Barr,andK.Fleischer.Elasticallyde- formable models. Proceedings of ACM SIGGRAPH, pages 205– 214, 1987.

[22] A. Witkin and D. Baraff. Physically based modeling: Principles and practice. Siggraph Course Notes, August 1997.

[23] A. Witkin and D. Baraff. Large steps in cloth simulation. Pro- ceedings of ACM SIGGRAPH, pages 43–54, 1998.

[24] X. Wu, M. S. Downes, T. Goktekin, and F. Tendick. Adap- tive nonlinear finite elements for deformable body simulation us- ing dynamic progressive meshes. Eurographics, pages 349–358, September 2001.

Model parameters

Snake

Cow

Dragon

Tetrahedra

440

970

834

Triangles

22,000

5,800

10,000

Elast. Mod. [1e3N/m2]

100

50

100

Poisson Ratio [1]

0.33

0.33

0.33

Density [kg/m3]

1000

1000

1000

Yield stress [1e3N/m2]

∞

∞

60

Plastic yield [1]

∞

0.01

0.10

Plastic creep [1/s]

0

100

1

Plastic max [1]

0

10

1

Table 1: Parameters used for the animations.

tently combine a high resolution skin mesh with a compu- tational volumetric mesh of lower resolution keeping the overall representation watertight at all times. Our method is easy to implement, stable and, as we demonstrate, per- forms robustly even in cases of extreme parameter set- tings. In terms of material stiffness, our method covers the gap between purely rigid objects – for which rigid body simulators are best suited – and fluids that change their topology because no connectivity is defined. In the future, we plan to extend our algorithms towards fluid flow simulation by including object representations that can handle changes in topology.

9 Acknowledgements

This project was funded by the Swiss National Commis- sion for Technology and Innovation (KTI) project no. 6310.1 KTS-ET.

References

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[2] T. J. Chung. Applied Continuum Mechanics. Cambridge Univ. Press, NY, 1996.

[3] R. D. Cook. Finite Element Modeling for Stress Analysis. John Wiley & Sons, NY, 1995.