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SIGGRAPH 2005 Course #6: Advanced Topics on Clothing Simulation and Animation Organizers: Hyeong-Seok Ko Kwang-Jin Choi Lecturers: Robert Bridson Dongliang Zhang A Brief History of the Course SIGGRAPH 1998 “Cloth and Clothing in Computer Graphics” Organized by House SIGGRAPH 2003 “Clothing Simulation and Animation” Organized by Breen and Ko SIGGRAPH 2005 “Advanced Topics on Clothing Simulation and Animation” Organized by Choi and Ko Course Schedule 1st Session 1:45~2:00 Introduction (Ko) 2:00~3:00 Physical Model of Cloth I (Ko) 3:00~3:30 Physical Model of Cloth II (Choi) 2nd Session 3:45~4:30 Collision Handling (Bridson) 4:30~5:00 Cloth Design and Applications (Zhang) 5:00~5:30 Current State-of-the Art / Challenges Ahead (Ko) Introduction How Cloth Simulation Works? Hyeong-Seok Ko Seoul National Univ. Graphics & Media Lab First of All, How to Represent Cloth? Particles interconnected by springs Particle-based Cloth Simulation Repeat the following: 1. Find the new position of the particles 2. Draw the surface from the particles How Does Cloth Simulation Work? Finding Governing Equation Solving the Equation Collision Handling Finding the Governing Equation Physically-based Simulation −1∂E && = M (− x + F) ⎡ x1 ⎤ ∂x ⎢M ⎥ x=⎢ ⎥ (m1, x1) (m2, x2) (m3, x3) ⎢ xN ⎥ ⎣ ⎦ (mi, xi) ⎡mi 0 0⎤ ⎢0 Mi = ⎢ mi 0⎥ ⎥ ⎢0 ⎣ 0 mi ⎥ ⎦ Physically-based Simulation −1 ∂E && = M (− x + F) ∂x Solving the Equation Now, numerically solve −1 ∂E &&(t ) = M (− x + F )(t ) ∂x while(1) { compute the force part at t; compute accel, vel, pos; t = t + ∆t; } We want to use large ∆t Animators love large ∆t Large ∆t can cause inaccuracy Large ∆t can cause a more serious problem We must have a stable algorithm Course Schedule Finding the Governing Equation 1st Session Solving the Equation 1:45~2:00 Introduction (Ko) 2:00~3:00 Physical Model of Cloth I (Ko) 3:00~3:30 Physical Model of Cloth II (Choi) 2nd Session 3:45~4:30 Collision Handling (Bridson) 4:30~5:00 Cloth Design and Applications (Zhang) 5:00~5:30 Current State-of-the Art / Challenges Ahead (Ko) Collision Handling Collision Handling Consists of Detection/Resolution Undetected collision can cause a lot of trouble. Cloth simulation is abundant of challenging cases for collision detection. A difficult problem: Multiple simultaneous collisions. Try this: simulate a person wearing multiple pieces of clothes. “Collision Handling” by Bridson (Session 2) Additional Parts 1st Session 1:45~2:00 Introduction (Ko) 2:00~3:00 Physical Model of Cloth I (Ko) 3:00~3:30 Physical Model of Cloth II (Choi) 2nd Session 3:45~4:30 Collision Handling (Bridson) 4:30~5:00 Cloth Design and Applications (Zhang) 5:00~5:30 Current State-of-the Art / Challenges Ahead (Ko) Physical Model of Cloth I Hyeong-Seok Ko Seoul National Univ. Graphics & Media Lab Physical Model of Cloth I 1. Understanding the problem 2. Immediate buckling model (IBM) 3. Stability analysis of the IBM 4. Damping analysis in clothing simulation Understanding the Problem What are the nontrivial parts in... −1 ∂E && = M (− x + F) ∂x E What would you use for E? ∂E −1 −1 && = M (− x + F ) ≡ M f ( x, x ) & ∂x −1 ≡ M f ( x, v ) Cloth is subject to stretch, shear, bending E = Estretch + Eshear + Ebending What are Estretch, Eshear, Ebending? They should be monotonic xi xj For example, Estretch = {||xi−xj||−L}2 L E should accurately represent the property of the cloth E should not cause problems in numerical integration Integration of ⎛ ∂E −1 ⎞ −1 && = M ⎜ − x + F ⎟ = M f ( x, v ) ⎝ ∂x ⎠ d ⎡ x⎤ ⎡ v ⎤ ⎢ v ⎥ = ⎢ M −1f ( x, v)⎥ dt ⎣ ⎦ ⎣ ⎦ ⎡∆x ⎤ ⎡ v 0 ⎤ ⎡ v ⎤ 0 ⎢∆v ⎥ = ∆t ⎢ −1 0 0 ⎥ ≡ ∆t ⎢ −1 0 ⎥ ⎣ ⎦ ⎣ M f ( x , v )⎦ ⎣M f ⎦ Explicit Integration We want Implicit Integration ⎛ ∂E −1 ⎞ −1 && = M ⎜ − x + F ⎟ = M f ( x, v ) ⎝ ∂x ⎠ d ⎡ x⎤ ⎡ v ⎤ ⎢ v ⎥ = ⎢ M −1f ( x, v)⎥ dt ⎣ ⎦ ⎣ ⎦ ⎡∆x ⎤ ⎡ v ⎤ ⎡ v ⎤ 0 0 ⎢∆v ⎥ = ∆t ⎢ −1 0 0 ⎥ ≡ ∆t ⎢ −1 0 ⎥ ⎣ ⎦ ⎣ M f ( x , v )⎦ ⎣M f ⎦ Explicit Integration Baraff and Witkin, Large Steps in Cloth Simulation, SIGGRAPH ‘98 We want Implicit Integration ⎛ ∂E −1 ⎞ −1 && = M ⎜ − x + F ⎟ = M f ( x, v ) ⎝ ∂x ⎠ d ⎡ x⎤ ⎡ v ⎤ ⎢ v ⎥ = ⎢ M −1f ( x, v)⎥ dt ⎣ ⎦ ⎣ ⎦ ⎡∆x ⎤ ⎡ v + ∆v 0 ⎤ ⎢∆v ⎥ = ∆t ⎢ −1 0 ⎥ ⎣ M f ( x + ∆x, v + ∆v)⎦ 0 ⎣ ⎦ ⎛ −1 ∂f −1 ∂f ⎞ −1 ⎛ 0 ∂f 0 ⎞ ⎜ I − ∆tM − ∆t M ⎟∆v = ∆tM ⎜ f + ∆t v ⎟ 2 ⎝ ∂v ∂x ⎠ ⎝ ∂x ⎠ HW 1: Derive this Into a Schematic Equation ⎛M ∂f ∂f ⎞ ⎛ 0 ∂f 0 ⎞ ⎜ 2− − ⎟∆v∆t = ⎜ f + ∆t v ⎟ ⎝ ∆t ∆t∂v ∂x ⎠ ⎝ ∂x ⎠ ⎛ ∂f 2 ∂f ⎞ ⎛ 0 ∂f 0 ⎞ ⎜ M − ∆t − ∆t ⎟∆v = ∆t ⎜ f + ∆t v ⎟ ⎝ ∂v ∂x ⎠ ⎝ ∂x ⎠ ⎛ −1 ∂f −1 ∂f ⎞ −1 ⎛ 0 ∂f 0 ⎞ ⎜ I − ∆tM − ∆t M ⎟∆v = ∆tM ⎜ f + ∆t v ⎟ 2 ⎝ ∂v ∂x ⎠ ⎝ ∂x ⎠ ⎛M ⎞ n n −1 ⎜ 2 + K ⎟∆x = f ⎝ ∆t ⎠ E can cause problems in Num. Int. ∂E −1 && = M (− x + F) ∂x ⎛ −1 ∂f −1 ∂f ⎞ −1 ⎛ 0 ∂f 0 ⎞ ⎜ I − ∆tM − ∆t M ⎟∆v = ∆tM ⎜ f + ∆t v ⎟ 2 ⎝ ∂v ∂x ⎠ ⎝ ∂x ⎠ ⎛M ⎞ n n −1 ⎜ 2 + K ⎟∆x = f ⎝ ∆t ⎠ Ill-conditioned or indefinite system matrix Divergence! Quest for a New Physical Model Find a Physical Model that makes K always positive definite that produces realistic movements of cloth ⎛M ⎞ n n −1 ⎜ 2 + K ⎟∆x = f ⎝ ∆t ⎠ Source of the Problem In simulating hookean model, (compress. stiffness ≈ tens. stiffness) σ = Eε , f = k ( x − l ) Instability occurs when cloth is bended or compressed rather than when it is stretched It appears that instability occurs when wrinkles are formed Compression/Bending, are they Unstable Phenomena? Yes! “Buckling” P P P Force : P Compressed Length “Buckling” P P P Force : P Post-buckling Instability Compression/Bending, are they Unstable Phenomena? Apparently, No Post-buckling instability passes instantly, and reaches a static equilibrium. Why was Compression/Bending Unstable in Clothing Simulation? Hookean model causes the system matrix indefinite when compression/bending occur ⎛ −1 ∂f −1 ∂f ⎞ −1 ⎛ ∂f ⎞ ⎜ I − ∆tM − ∆t M ⎟∆v = ∆tM ⎜ f 0 + ∆t v 0 ⎟ 2 ⎝ ∂v ∂x ⎠ ⎝ ∂x ⎠ Suggestion for a Stable Solution Hookean model causes the system matrix indefinite when compression/bending occur The model works well for stretch Suggested solution: use different physical models for stretch and compression Type 1 for stretching Type 2 for compression/bending How to Connect Springs? Red Links = Type 1 (for stretch) Blue Links =Type 2 (for compression and bend) Immediate Buckling Model How to realize... Post-buckling instability passes instantly, and reaches a static equilibrium. Compression causes bending rather than shortening Immediate Buckling Assumption Experiment # of particles = 5608, ∆t=0.011s, CPU=0.51s How to implement this idea? Compression causes bending rather than shortening Immediate Buckling Assumption The Steps We Took 1. Predict the shape at the static equilibrium after buckling 2. Formulate the energy function from the deformed shape 3. Derive the force, Jacobian, etc. Predicting the Static Equilibrium Predicting the Static Equilibrium Momentum Equilibrium Equation kbκ + Py = 0 P/kb = 23.5 P/kb = 35.25 Equilibrium shape is similar to a circular arc. Simplifying Assumptions Equilibrium shape = circular arc Arc length remains constant L L X Equilibrium shape is completely characterized when xi and xj are given. Energy Function in terms of xij=xj-xi 1 L The energy function E = ∫ Mκ dx 2 0 1 Since M∝κ & curvature is constant E = k b Lκ 2 2 Since arc length is constant ⎛ xij ⎞ κ = sinc -1 ⎜ ⎟ 2 L ⎜ L ⎟ ⎝ ⎠ 2 ⎡2 ⎛ xij ⎞ ⎤ E = kb L ⎢ sinc -1 ⎜ ⎟ ⎥ 1 2 ⎢L ⎜ L ⎟ ⎥ ⎣ ⎝ ⎠ ⎦ The Steps We Took 1. Predict the shape at the static equilibrium after buckling 2. Formulate the energy function from the deformed shape 3. Derive the force, Jacobian, etc. ⎛ −1 ∂f −1 ∂f ⎞ −1 ⎛ 0 ∂f 0 ⎞ ⎜ I − ∆tM − ∆t M ⎟∆v = ∆tM ⎜ f + ∆t v ⎟ 2 ⎝ ∂v ∂x ⎠ ⎝ ∂x ⎠ 2 ⎡2 ⎛ xij ⎞ ⎤ E = kb L ⎢ sinc -1 ⎜ ⎟ ⎥ 1 Force Derivation 2 ⎢L ⎜ L ⎟ ⎥ ⎣ ⎝ ⎠ ⎦ ∑ f (x ) x ∂E xij fi = − = , ∂x i b ij j∈Ν ( i ) ij The Force-Deflection Curve Finally Used * ( ) ( ⎧cb xij − L : f b < cb xij − L f =⎨ ) b ⎩ f b : otherwise New Force-Displacement Curve Force : P Compressed Length ∂f i Meaning of the Force Jacobian ∂x j ∂ ∑ f ik ∂f i ∂f ij = k∈Ν = HW 2: Why? ∂x j ∂x j ∂x j The rate of change of fi as xj changes ∂f i ∂fij = −∑ HW 3: Why? ∂x i j ≠ i ∂x j The rate of change of fi as xi changes Stiffness! Derivation of Force Jacobian ∂f i * T * T df x x f x x = + (I − b ij ij b ij ij ) ∂x j d x ij x x T x ij ij ij x x T ij ij HW 4: Derive this We can Start Implicit Integration ⎛ −1 ∂f −1 ∂f ⎞ −1 ⎛ ∂f ⎞ ⎜ I − hM −h M ⎟∆v = hM ⎜ f 0 + h v 0 ⎟ 2 ⎝ ∂v ∂x ⎠ ⎝ ∂x ⎠ For now, forget about the damping term. ⎛ −1 ∂f ⎞ −1 ⎛ ∂f ⎞ ⎜I − h M ⎟∆v = hM ⎜ f 0 + h v 0 ⎟ 2 ⎝ ∂x ⎠ ⎝ ∂x ⎠ Stability Analysis of Immediate Buckling Model ∂f Property of the Force Jacobian ∂x ⎛ −1 ∂f ⎞ −1 ⎛ ∂f ⎞ ⎜ I − ∆t M ⎟∆v = ∆tM ⎜ f 0 + ∆t v 0 ⎟ 2 ⎝ ∂x ⎠ ⎝ ∂x ⎠ ∂f i * T * T df x x f x x = + (I − b ij ij b ij ij ) ∂x j d x ij x x T x ij ij ij x x T ij ij What is the property of this 3×3 matrix? What is the property of the global 3n×3n matrix? T x x The Properties of A= ij ij T x x ij ij T xij xij det( T − λI ) = λ2 (1 − λ ) HW 5: Prove it x xij ij T x x ∴ ij ij T is positive definite. x x ij ij T x xij ij T z produces the projection of z along xij x xij ij HW 6: Verify this T x x I− =I−A ij ij The Properties of T x x ij ij ⎛ T xij xij ⎞ ⎜I − ⎟ is positive definite. ⎜ T xij xij ⎟ ⎝ ⎠ HW 7: Why? ⎛ xij xij ⎞ T ⎜I − ⎟ z produces the projection of z ⎜ xij xij ⎟ T ⎝ ⎠ on a plane orthogonal to xij HW 8: Verify this They Work as Projection Operators ∂f i * T * T df x x f x x dx j = dx j + (I − b ij ij b ij ij )dx j ∂x j d x ij x x T x ij ij ij x x T ij ij new x j xi xj df b* f b* = in-plane stiffness = out-of-pl stiffness d xij x ij Property of the Jacobian ∂f i * T * T df x x f x x = + (I − b ij ij b ij ij ) ∂x j d x ij x x T x ij ij ij x x T ij ij df b* is positive. d x ij * f But x is negative. b ij So the sum is not definite If 2nd term is omitted, it becomes pos. def. Property of the Jacobian ∂f i * T df x x = b ij ij ∂f ⎞ ∂f ⎞ ∂x j d x ij x x T ij ij ⎛ ⎜ ⎛ I − ∆t 2 M −1 ⎟∆v = ∆tM −1 ⎜ f 0 + ∆t v 0 ⎟ ⎝ ∂x ⎠ ⎝ ∂x ⎠ df b* is positive. d x ij * f But x is negative. b ij So the sum is not definite If 2nd term is omitted, it becomes pos. def. What is the effect of it? Is the global matrix negative definite? Is the Global Matrix Neg. Def.? ⎡ ∂f1 ∂f1 ∂f1 ⎤ ⎡− ∂f1 ∂f1 ∂f1 ⎤ ⎥ ⎢ ∑ ∂x j L L ⎢ ∂x ∂x 2 ∂x n ∂x 2 ∂x n ⎥ ⎢ 1 ⎥ ⎢ j ≠1 ⎥ ⎢ ∂f 2 ∂f 2 ∂f 2 ⎥ ⎢ ∂f 2 ∂f 2 ∂f 2 ⎥ ∂f L −∑ L = ⎢ ∂x1 ∂x 2 ∂x n ⎥ = ⎢ ∂x1 j ≠ 2 ∂x j ∂x n ⎥ ∂x ⎢ M M O M ⎥ ⎢ ⎢ M M O M ⎥ ⎥ ⎢ ∂f ∂f n ∂f n ⎥ ⎢ ∂f n ∂f n ∂f n ⎥ ⎢ n L ⎥ L −∑ ⎢ ∂x1 ⎣ ∂x 2 ∂x n ⎥ ⎢ ∂x1 ⎦ ⎣ ∂x 2 j ≠ n ∂x j ⎥ ⎦ Yes HW 9: Prove it. What is the meaning of Omitting the Second Term? ∂f i * T df x x = b ij ij ∂x j d x ij x x T ij ij dx j not counted L counted Does it mean ignoring the orthogonal component? No. Remember, ∂f/∂x was introduced for implicit integration ∂f/∂x In Global Implicit Integration... ⎛ −1 ∂f ⎞ −1 ⎛ ∂f ⎞ ⎜I − h M ⎟∆v = hM ⎜ f 0 + h v 0 ⎟ 2 ⎝ ∂x ⎠ ⎝ ∂x ⎠ Looking at the i-th row ⎡M M ⎡ ∆v ⎤ M M M M M M M⎤ ⎢ 1 ⎥ ⎢ ∂f ∂f ∂f i ⎥ ∆v LHS = ⎢0 L − h 2 M i−1 i L I − h 2 M i−1 i L − h 2 M i−1 L 0⎥ ⎢ 2 ⎥ ⎢ ∂x j1 ∂x i ∂x jN ⎥⎢ M ⎥ ⎢M M ⎣ M M M M M M M ⎥ ⎢∆v n ⎥ ⎦⎣ ⎦ ⎛ −1 ∂f i ⎟ ⎞ −1 ∂f i = ⎜ I + ∑ h Mi 2 ∆v i − ∑ h M i 2 ∆v j ⎜ ∂x j ⎟ ∂x j ⎝ j ≠i ⎠ j ≠i ⎛ n ∂f i ⎞ RHS = hM i f 0i + ∑ h −1 ⎜ v0 j ⎟ ⎜ ∂x j ⎟ ⎝ j =1 ⎠ Comparison with Explicit Method ∆vi = hM i−1f i0 Explicit ⎛ ⎞ ⎛ ⎞ ⎜ I + ∑ h 2 M i−1 ∂f i ∂f i ∂fi n ⎟∆v i − ∑ h M i 2 −1 ∆v j = hM i f 0i + ∑ h −1 ⎜ v0 j ⎟ ⎜ ∂x j ⎟ ∂x j ⎜ ∂x j ⎟ ⎝ j ≠i ⎠ j ≠i ⎝ j =1 ⎠ Implicit In LHS, every term except ∆v i is introduced due to using implicit integration It produces implicit filtering effect Desbrun, M., Schröder, P., and Barr, A. 1999. Interactive animation of structured deformable objects. In Proceedings of the 1999 Conference on Graphics interface '99 If 2nd Term of ∂fi/∂xj is Omitted... ⎛ ⎞ −1 ∂f i ⎟ −1 ∂f i ⎜ I + ∑ h Mi ⎟ i ∑ LHS = 2 ∆v − h M i 2 ∆v j ⎜ ∂x j ⎠ ∂x j ⎝ j ≠i j ≠i ⎛ ⎞ −1 ∂f i ⎟ −1 ∂f i = ⎜ I + ∑ h Mi 2 (∆v i + ∆v i ) − ∑ h M i p o 2 (∆v p + ∆v oj ) ⎜ ⎟ ∂x j ⎠ ∂x j j ⎝ j ≠i j ≠i ⎛ ⎞ p −1 ∂f i ⎟ −1 ∂f i ⇒ ∆v i + ⎜ ∑ h Mi 2 ∆v i − ∑ h M i 2 ∆v p ⎜ j ≠i ∂x j ⎟ ∂x j j ⎝ ⎠ j ≠i ⎛ n ∂fi ⎞ RHS = hM i f 0i + ∑ h −1 ⎜ v0 j ⎟ ⎜ ∂x j ⎟ ⎝ j =1 ⎠ ⎛ n ∂f i ⎞ ⎛ n ∂f i p ⎞ = hM i f 0 i + ∑ h −1 ⎜ ( v 0 j + v 0 j ) ⎟ ⇒ hM i f 0 i + ∑ h p o −1 ⎜ v0 j ⎜ ∂x j ⎟ ⎜ ∂x j ⎝ j =1 ⎠ ⎝ j =1 ⎠ So, What is the Effect? The force in the orthogonal direction is not affected by implicit filtering. Thus, the force in the orthogonal direction is in fact integrated by explicit method What happens to Stability? By the omission, the system matrix is positive definite However, since the method is now not fully semi-implicit, stability is not guaranteed semi-implicit, If the orthogonal force is small compared to stretch force, there is little possibility system will diverge How large is the orthogonal force? When is the Ortho Force Generated? More Probable Scenario When a compressive force is applied to two consecutive links Each link will immediately start to bend However, at the center, an orthogonal force may be generated due to this compression More Probable Scenario When a compressive force is applied to two consecutive links Each link will immediately start to bend However, at the center, an orthogonal force may be generated due to this compression The force can be large when the mesh resolution is high (high res. is bad!) The force can be large when the material’s stiffness is material’s large Is there any other Scenario? We are not sure But based on our experiments, previous case is probably the most probable scenario that can possibly produce divergence. Stability Analysis of Immediate Buckling Model We cannot say it is unconditionally stable. Artificial cases can be set up to create instabilities But we did not meet any instabilities in practical cases Physical Model of Cloth II Kwang-Jin Choi FXGear Inc. Physical Model of Cloth II Analysis on Damping used in Clothing Simulation How Damping Affects Cloth Animation? No damping With some damping Excessive damping When Implicit Integration is Used… ⎛ ∂f 2 ∂f ⎞ ∆v ⎛ ∂f ⎞ ⎜ M − ∆t − ∆t ⎟ = ⎜ f 0 + ∆t v 0 ⎟ (implcit) ⎝ ∂v ∂x ⎠ ∆t ⎝ ∂x ⎠ ∆v M = f0 , (explicit) ∆t Implicit Filtering Artificial Damping Force It creates artificial damping. Desbrun, M., Schröder, P., and Barr, A. 1999. Interactive animation of structured deformable objects. In Proceedings of the 1999 Conference on Graphics interface '99 Results with Artificial Damping ∆t = 1 sec 90 AD does not hurt the simulation very much Even 1/30 shows no excessive damping Typically used time step size 1/48 ~ 1/300 Because of collision resolution In most cases, extra damping needs to be added Viscous Damping (Drag Force) f + = −k d v ⎛ 2 ∂f ⎞ ∆v ∂f ⎜ M + ∆tk d I − ∆t ⎟ = f 0 + ∆t v 0 ⎝ ∂x ⎠ ∆t ∂x Push all the eigen values toward positive direction Condition number gets closer to 1 Faster convergence Viscous Damping (Drag Force) f + = −k d v Drag No drag A Better Drag Force (Air Drag) ( )− f = −k d A(n vv n v wind ) )n T n A v - v wind A Better Drag Force (Air Drag) f = −k d A(n ( v − v wind ) )n T Air Drag Viscous Drag A Better Drag Force (Air Drag) f = −k d A(n ( v − v wind ) )n T Air Drag No Drag Deformation Rate Damping Damp the deforming motion, not the rigid motion f ij = −k d ( v i − v j ) j4 jn f i = ∑ f ij j3 i j∈N j1 ∂f i j2 = kd I ∂v j ∂f i = − k d nI ∂v i Deformation Rate Damping f ij = − k d ( v i − v j ) ⎛ ∂f 2 ∂f ⎞ ∆v ∂f ⎜ M − ∆t − ∆t ⎟ = f 0 + ∆t v 0 ⎝ ∂v ∂x ⎠ ∆t ∂x ∂f has zero eigenvalue for rigid translation ∂v ⇒ no dragging Comparison to Drag Force f ij = −k d ( v i − v j ) fi = −k d v i Comparison with Different kd’s f ij = −k d ( v i − v j ) f ij = −k d ( v i − v j ) small k d large k d What about Rotation? f ij = − k d ( v i − v j ) vj xi xj vi The above force damps all the relative motions between particle i and j We want to damp only the motion in spring direction (xi-xj) -x Damped Spring T x ij x ij f ij = −k d T (vi − v j ) x ij x ij vj xi xj vi Minimal damping to rotation Damping to bending and stretching can be decoupled Comparison T x ij x ij f ij = −k d ( v i − v j ) f ij = −k d T (vi − v j ) x ij x ij More General Deformation Rate Damping Define deformation D = (| x i − x j | − L) = (| x ij | − L) ∂D x ij Get the deforming direction = ∂x ij | x ij | T ⎛ ∂D ⎞ dx ij x ij T Get the deformation rate D=⎜ & ⎟ = v ij ⎜ ∂x ⎟ dt | x | ⎝ ij ⎠ ij Damp the deformation to the deforming direction T ∂D ⎛ ∂D ⎞ T ∂D & ⎜ ⎟ v ij = −k d x ij x ij f ij = −k d D = −k d v ij ∂x ij ∂x ij ⎜ ∂x ij ⎝ ⎟ ⎠ | x ij | 2 Baraff and Witkin, Large Steps in Cloth Simulation, SIGGRAPH ‘98 Putting All Together ⎛ ∂c i ⎛ ∂c i ⎞ ⎞ T f damp = −kd ∑ ⎜ ⎜ ⎜ ∂x ⎟ v ⎟ − k a nnT ( v − v wind ) ⎟ ⎟ ⎜ ∂x i ⎝ ⎝ ⎠ ⎠ T ∂f damp ∂c i ⎛ ∂c i ⎞ = −kd ∑ ⎜ ⎜ ∂x ⎟ − k a nnT ⎟ ∂v i ∂x ⎝ ⎠ Putting All Together Course Schedule 1st Session 1:45~2:00 Introduction (Ko) 2:00~3:00 Physical Model of Cloth I (Ko) 3:00~3:30 Physical Model of Cloth II (Choi) 2nd Session 3:45~4:30 Collision Handling (Bridson) 4:30~5:00 Cloth Design and Applications (Zhang) 5:00~5:30 Current State-of-the Art / Challenges Ahead (Ko) Current State-of-the-Art and Challenges Ahead Hyeong-Seok Ko Seoul National Univ. Graphics & Media Lab Would it be Possible? Source of Frustrations Poor collision handling Allowed only regular, rectangular grids Not practical for creating complex garments Garments had to be constructed from 2D patterns Not practical for 3D animation purpose Few Controls Collision Handling 2005 Collision Test Collision Test # of vertices Animation Length Computation Time 6029 20 sec 31.5 mins Self-Collision Test Simulation Statistics 0.1m X 50m cloth 28,007 vertices 15 hours Collision Handling in 3 Pieces Garment Construction 2005 Unstructured Triangular Mesh Construction from 2D Patterns Construction from 3D Modeling Construction from 3D Modeling Construction from 3D Modeling Other Controls 2005 Wrinkle Map Rubber Map Rubber Map Effect of Buttons Wind Fields How Far can it Go? Reconstruction of the Clothes Reconstruction of the Clothes The Simulation The Simulation Original Stiffness Intentional Modification Now, We may call it a “Fashion” Challenges Ahead Physical Model and Simulation Modeling nonlinear and hysteretic properties Exploring the continuum approach Increasing the fidelity of simulated cloth Increasing algorithm speed while maintaining reasonable quality Collision Resolution Rapid collision detection Accurate collision detection Robust collision response generation Thank You

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