Project 1. Has been posted! Previous year projects:

Transcription

1 Project 1 Has been posted! Previous year projects: Bren Meeder Heegun Lee

2 Hair Simulation (and Rendering) Image from Final Fantasy (Kai s hair) Adrien Treuille

3 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

4 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

5 Tinkertoys Now we know how to simulate a bead on a wire. Next: a constrained particle system. E.g. constrain particle/particle distance to make rigid links. Same idea, but

6 Compact Particle System Notation q = WQ q: 3n-long state vector. q = x 1,x 2,,x n Q = f 1,f 2,,f n m1 Q: 3n-long force vector. matrix. T +J! mn mn mn More Notation C = C 1,C2,,Cm! =! 1,! 2,,! m "C J= "q 2C " How do you implement all this? J= "q"t We have a global matrix equation. like bead-on-wire. m1 W: M-inverse (element- wise -1 W = M reciprocal) -Jq - JW Q celeration C m1 ystem Constraint Equations M: 3n x 3n diagonal mass M = for! M M S

7 constraints state Constrained Dynamics: General Case d x = x dt d 1 f + f = W f + f x = x = M dt C(x) = 0 dc C C = = x = J x = 0 dt x C = J x + J x = 0 = J x + JW f + f At any point the set of legal velocities are those which are perpendicular to the rows of J. Conversely, the illegal velocities are spanned by JT i.e. {JT!!! Rc}. JW f = J x JW f 1 T x M x 2 T = x T M x T = x f + f virtual work T = T due to f Since the constraint force is perpendicular to all legal velocities, it must be in the span of JT. = x f = 0 therefore f = J T λ JW J T λ = J x JW f λ = JW J T 1 J x JW f

8 Drift and Feedback In principle, clamping C at zero is enough Two problems: Constraints might not be met initially Numerical errors can accumulate A feedback term handles both problems: C = - $C - %C, instead of C=0 $ and % are magic constants.

9 How do you implement all this? We have a global matrix equation. We want to build models on the fly, just like masses and springs. Approach: Each constraint adds its own piece to the equation.

10 Matrix Block Structure Each constraint contributes one or more blocks to the matrix. C Sparsity: many empty blocks. "C "x i xi "C "x j xj J Modularity: let each constraint compute its own blocks. Constraint and particle indices determine block locations.

11 J! J& C & C fc x v f m x v f m Global Stuff Global and Local Constraint

12 Constraint Structure Each constraint must know how to compute these C x v f m x v f m p1 p2 C "C "C, "x 1 "x "C "C, "x 1"t "x 2"t Distance Constraint C = x1 - x2 - r

13 Constrained Particle Systems particles n time forces nforces consts nconsts x x v v f f m m x v f m F F F FF C C C C C Added Stuff

14 x x v v f f m m Modified Deriv Eval Loop x 1 v f m 2 Clear Force Accumulators x x v v f f m m x v f m Return to solver F F F FF Apply forces Added Step C C C 4 3 C C Compute and apply Constraint Forces

15 Constraint Force Eval After computing ordinary forces: Loop over constraints, assemble global matrices and vectors. Call matrix solver to get!, multiply by JT to get constraint force. Add constraint force to particle force accumulators.

16 Impress your Friends The requirement that constraints not add or remove energy is called the Principle of Virtual Work. The! s are called Lagrange Multipliers. The derivative matrix, J, is called the Jacobian Matrix.

17 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

18 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

19 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

20 Real Hair

21 Real Hair

22 Real Hair Figure 1.1: Left, close view of a hair fiber (root upwards) showing covered by overlapping scales. Right, bending and twisting instabilitie when compressing a small wisp. Typical human head has 150k-200k individual strands. Dynamics not well understood. Subject still open to debate. Deformations of a hair strand involve rotations that are not infinitely so can only be described by nonlinear equations [AP07]. Physical effe from these nonlinearities include instabilities called buckling. For exam a thin hair wisp is held between two hands that are brought closer to (see Figure 1.1, right), it reacts by bending in a direction perpendicu applied compression. If the hands are brought even closer, a second occurs and the wisp suddenly starts to coil (the bending deformation is

23 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

24 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

25 Questions How could we simulate hair? What about......preventing bending?...different kinds of hair?...collisions?...200k hairs?

26 Student Answers - hair models - mass-spring model: string of particles (one attached to scalp) - but we need a length constraint! - prevent bending: attach every two particles - use constraints to prevent over stretching - how about more than one strand per hair - simulate torsion - rather than using just straight lines, could we use bezier curves as basic elements - rather than using individual strands, interpolate a general mesh - how to implement curly hair: - put the particles in a helix and try to revert to that position - attach every nth particle - force a "curl" - equally favors clockwise and counterclockwise curls - use "big particles" that won't collide at a distance - adjusting the number of particles may affect the hair dynamics (beyond just increasing resolution) - for collisions - implement a spatial data structure to save computation - repelling forces, but could be an issue for many particles - have soft spring forces when "cylinders" intersect - to achieve 100k strands - interpolate between hairs - extrapolate from one hair to many

27 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

28 Hair Dynamics Control Mesh Mass-Spring Systems Rigid Links Super Helices

29 Hair Dynamics Control Mesh Mass-Spring Systems Rigid Links Super Helices

30 Control Mesh A Practical Model for Hair Mutual Interactions Johnny T. Chang, Jingyi Jin, Yizhou Yu. ACM SIGGRAPH Symp. on Computer Animation. pp , 2002.

31 Control Mesh

32 Hair Dynamics Control Mesh Mass-Spring Systems Rigid Links Super Helices

33 Recall...

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35

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37 Disadvantages Torsional Rigidity Non-stretching of the strands

38

39 decided to start with the mass-spring system since we had a working code fro in-house cloth simulator. There we started by adapting the existing partic

40 k = implicit integration? n+1 Implicit integrator adds stability Loss of angular momentum Good Jacobian (filter) very important

41 k Is infinity! Well, how do we preserve length then? use non-linear correction

42 non-linear post correction

43 non-linear post correction

44 non-linear post correction

45 non-linear post correction Post solve correction Successive relaxation until convergence Guaranteed length preservation Cheap simulation of k infinity

46 non-linear post correction How to implement? Cloth simulation literatures Provot 1995 (position only) Bridson 2002 (impulse) Hair-specific relaxation possible

47 Predictor-corrector scheme Implicit Filter (Predictor) Sharpener (Corrector) Implicit Filter (Predictor)

48 1.First pass-implicit integration First implicit solve to get new velocity

49 2.First pass-implicit integration Advance position with the predicted mid-step velocity

50 3.Non-linear Correction Apply non-linear corrector to get position (length) right

51 4.Impulse Change velocity due to length preservation Velocity may be out of sync after impulse

52 5.Second implicit integration Filters out velocity field Velocity field in sync again

53

54 Hair Dynamics Control Mesh Mass-Spring Systems Rigid Links Super Helices

55 Featherstone Algorithm joint i joint i+1 link i link 1 link n joint n joint 1 link 0 (base) inboard outboard outboard joint link i!1 O fi!1 I f i!1 inboard joint τoi!1 m i!1 g τii!1 O

56 Rigid Links Fewer degrees of freedom. Torsional forces. Difficult Implementation. Constraints Difficult.

57 Hair Dynamics Control Mesh Mass-Spring Systems Rigid Links Super Helices

58 Super Helices other interesting approaches to handle strand-strand interactions include wisp level interactions [PCP01b, BKCN03b], layers [LK01b] and strips [CJY02b]. We demonstrate the effectiveness of the proposed Oriented Strand methodology, through impressive results in production of Madagascar and Shrek The Third at PDI/DreamWorks, in Section 5.1. Why just use straight rods? 1.4 Super-Helices: a compact model for thin geometry Figure 1.5: Left, a Super-Helix. Middle and right, dynamic simulation of natural hair of various types: wavy, curly, straight. These hairstyles were animated using N = 5 helical elements per guide strand.

59 ng the internal friction coefficient. Super Helices the terms needed in equation (1.23) have been given in equations ( the The Dynamics of Super-Helices gging latter into the former, one arrives at explicit equations o the generalized coordinate q(t). Although straightforward in princ 3 culation is involved. It can nevertheless be worked out easily using a culation language such as Mathematica [Wol99]: the first step is to im reconstruction of Super-Helices as given in Appendix 1.4.3; the se o work out the right-hand sides of equations (1.24), using symbolic in enever necessary; the final step is to plug everything back into equati is leads to the equation of motion of a Super-Helix: Figure 1.6: Left, geometry of Super-Helix. Right, animating Super-Helices with L different natural curvatures andn twist: a) straight, b) wavy, c) curly, d) strongly i M[s, q] q + K (q q ) = A[t, q, q ] + J [s, q,t] F (s,t) ds. iq curly. In this example, each Super-Helix is composed 0of 10 helical elements. We shall first present the model that we used to animate individual hair strands this equation, the bracket notation is used to emphasize that all func (guide strands). This model has a tunable number of degrees of freedom. It is enbuilt byupon explicit formula termstheories of their arguments. the Cosserat and in Kirchhoff of rods. In mechanical engineering literature, a rod is defined as an elastic material that is effectively one dimensional: equation (1.25), the inertia matrix M is a dense square matrix of its length is much larger than the size of its cross section.

60 Super Helices Figure 1.8: Fitting γ for a vertical oscillatory motion of a disciplined, curly hair clump. Left, comparison between the real (top) and virtual (bottom) experiments. Right, the span A of the hair clump for real data is compared to the simulations for different values of γ. In this case, γ = kg m3 s 1 gives qualitatively

61 Super Helices

62 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

63 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

64 decided to start with the mass-spring system since we had a working code fro in-house cloth simulator. There we started by adapting the existing partic

65 guide strands having closetips ro o guidetwo strands having close roots but distant criterion onbetween the distance ti terion on the distance tips, see between Figure 4, (d) Rendering Interpolation Extrapolation 4: Semi-interpolating scheth gure 4: Figure Semi-interpolating scheme for generating

66 Overview More Constraints Hair Real Hair Questions Hair Dynamics Hair Rendering

67 Conclusion