2.7 Cloth Animation. Jacobs University Visualization and Computer Graphics Lab : Advanced Graphics - Chapter 2 123

Transcription

1 2.7 Cloth Animation : Advanced Graphics - Chapter 2 123

2 Example: Cloth draping Image Michael Kass : Advanced Graphics - Chapter 2 124

3 Cloth using mass-spring model Network of masses and springs Structural springs: link (i j) and (i+1, j); and (i, j) and (i, j +1) Shear springs (i j) and (i+1, j+1) Flexion springs (i,j) and (i+2,j) (i,j) and (i,j+2) : Advanced Graphics - Chapter 2 125

4 External forces Gravity Viscous damping Wind etc : Advanced Graphics - Chapter 2 126

5 Cloth simulation Then, the main trick is to set the stiffness of all springs to get realistic motion! Remember that forces depend on other particles (coupled system) But it is sparse (only neighbors) : Advanced Graphics - Chapter 2 127

6 Contact forces Hanging curtain: 2 contact points stay fixed What does it mean? Sum of the forces is zero How so? Because those point undergo an external force that balances the system What is the force at the contact? Forces from other particles, gravity Depends on all other forces in the system Gravity, wind, etc. Reaction force : Advanced Graphics - Chapter 2 128

7 Contact forces How can we compute the external contact force? Inverse dynamics! Sum all other forces applied to point Take negative Do we really need to compute this force? Not really, just ignore the other forces applied to this point! : Advanced Graphics - Chapter 2 129

8 Example : Advanced Graphics - Chapter 2 130

9 Example Excessive deformation: the strings are not stiff enough Initial position After 200 iterations : Advanced Graphics - Chapter 2 131

10 The stiffness issue We use springs while we mean constraint Structural springs should be super stiff, which requires tiny Δt remember x =-kx system Even though clothes are a little elastic, they usually don t deform more than 10% Many numerical solutions Reduce Δt Actually use constraints Implicit integration scheme : Advanced Graphics - Chapter 2 132

11 One solution Constrain length to increase by less than 10% Simple mass-spring system Improved solution : Advanced Graphics - Chapter 2 133

12 Spatial discretization issue What happens if we discretize our cloth more finely? Do we get the same behavior? Usually not! It takes a lot of effort to design a scheme that does not depend on the discretization : Advanced Graphics - Chapter 2 134

13 The collision problem A cloth has many points of contact Stays in contact Requires Efficient collision detection Efficient numerical treatment (stability) : Advanced Graphics - Chapter 2 135

14 2.8 Finite Element Method : Advanced Graphics - Chapter 2 136

15 Finite element method Common technique in computational sciences to solve PDEs In terms of computer graphics, deformable objects are regarded as continuous connected volumes. Continuum mechanics provide PDEs that are to be solved for the object : Advanced Graphics - Chapter 2 137

16 Continuum mechanics PDE governing dynamic elastic materials where ρ is the density, f is the combination of all external forces, σ is the stress (= force / area), and the divergence operator turns the 3x3 stress tensor back into a vector by : Advanced Graphics - Chapter 2 138

17 Finite element method The finite element method brings the PDEs in an algebraic form that is solved numerically. The continous volumetric domain is discretized into a finite number of disjoint elements in form of a mesh. Typically, tetrahedral meshes are used. Instead of solving for a continuous function x, one only solves for some positions x i. The positions x i are multiplied with basis functions such that the sum of the products approximates x : Advanced Graphics - Chapter 2 139

18 Displacement The displacement u(m) of an object m is given in form of u(m) = x(m) m, where x(m) is the new position. The finite element method provides a linearization of u : Advanced Graphics - Chapter 2 140

19 Displacement The relationship between nodal forces and nodal positions (displacement) for an element e connecting n e nodes can be expressed as f e = K e u e, where K e is the stiffness matrix of e. For the entire mesh we get that where the stiffness entries are zero for nonadjacent nodes : Advanced Graphics - Chapter 2 141

20 Elastic forces When assuming elastic forces, we obtain the equation of motion for the entire mesh as where M is the mass matrix and D is the damping matrix. M and D are frequently diagonal matrices. The equation is linear and, thus, can be solved using standard numerical methods : Advanced Graphics - Chapter 2 142

21 Examples : Advanced Graphics - Chapter 2 143

22 Examples : Advanced Graphics - Chapter 2 144

23 2.7 Particle Systems : Advanced Graphics - Chapter 2 145

24 Euler vs. Lagrange Among all these physically-based animation methods, one distinguishes between the following categories: Euler methods: Material properties are computed at stationary discrete points, i.e., the points are not moving. The points are typically connected to form a (regular hexahedral) grid. Lagrange methods: Material properties are computed at nonstationary discrete points, i.e., the points are moving. The points may be connected to form a grid/mesh or may be freely moving and, thus, changing the neighborhoods during animation : Advanced Graphics - Chapter 2 146

25 Examples Mass-spring method Lagrange method mesh-based Finite element method Lagrange method mesh-based : Advanced Graphics - Chapter 2 147

26 Euler methods Euler methods are common to compute the motion of fluids or gases. The motions are, typically, governed by the Navier- Stokes equations, which represent the convervation of mass and momentum for an incompressible fluid: where u is the fluid s velocity, u t its time derivative, p is pressure, and ν the kinematic viscosity : Advanced Graphics - Chapter 2 148

27 Euler methods Euler methods solve the PDEs for all cells. There are different ways to solve Navier-Stokes. There are different forces f that can be applied. Results are given by rendering isosurfaces. Example: Dripping viscoelastic fluid: : Advanced Graphics - Chapter 2 149

28 Particle Systems Lagrangian method not mesh-based set of particles to model time-dependent phenomena such as snow fire smoke : Advanced Graphics - Chapter 2 150

29 Particle systems particles are characterized by mass, position, and velocity. forces determine the dynamic behavior particles can carry attributes for rendering like shape, color, transparency, : Advanced Graphics - Chapter 2 151

30 Particle quantities : Advanced Graphics - Chapter 2 152

31 Particle motion : Advanced Graphics - Chapter 2 153

32 Governing equations : Advanced Graphics - Chapter 2 154

33 Initial value problem : Advanced Graphics - Chapter 2 155

34 Initial value problem : Advanced Graphics - Chapter 2 156

35 Finite differences : Advanced Graphics - Chapter 2 157

36 Euler method : Advanced Graphics - Chapter 2 158

37 Accuracy and stability : Advanced Graphics - Chapter 2 159

38 Higher-order integration scheme : Advanced Graphics - Chapter 2 160

39 Video 750,000 particles : Advanced Graphics - Chapter 2 161

40 Particle interactions Up to now, particles did not interact with each other. Still, certain phenomena can be modeled quite realistically. If particle interaction is desired, one can apply the methods for spring-mass models, where the attracting and repelling forces between particles can, for example, be modeled using the Lennard-Jones function : Advanced Graphics - Chapter 2 162