5.2 Hair Simulation
5.2.1 Result
We use our strand model to simulate some animation of hair. Since there are not many self contact in hair simulation, we turn off the self-collision to get better performance. The Young’s Modulus of the material parameters are set to 1MPa.
The hair under wind is as Fig 5.6. The wind is modeled as a constant wind multi-plied by a sin function. The wind force is set to 10−4kg.
5.2 Hair Simulation 23
(a) (b)
(c)
Figure 5.6: Under Wind: hair deformed by a constant wind force.
The dynamics results from head shaking and moving are as Fig 5.7 and Fig 5.8. The head dynamics is modeled by rigid body dynamics.
5.2 Hair Simulation 24
(a) (b)
(c) (d)
Figure 5.7: Simulated Hair Motion with Head Shaking: We simulate the head as a rotating rigid body. Then we get a dynamics of head shaking.
5.2 Hair Simulation 25
(a) (b)
(c) (d)
(e) (f)
Figure 5.8: Hair Moving: (a)(b)(c) the result that head moves vertically.
(d)(e)(f)the result that head moves horizontally.
5.2 Hair Simulation 26
The difference motion of different Young’s Modulus is shown as Fig 5.9. We can see the stiff hair style has less deformation when head moves.
(a) (b)
Figure 5.9: The difference between different stiffness in moving: (a) stiff (b) soft The difference resulted by D in hair interaction is presented in Fig 5.10.
(a) (b)
Figure 5.10: The difference of hair after head shaking between different interac-tion parameter (a)with D = 2 × 10−5 (b) with D = 2 × 10−6. Other interaction parameters are set to K = 2 × 10−6, Pinner = 3mm. The result with high viscous force is more cluttered.
5.2 Hair Simulation 27
5.2.2 Performance
We does above demo with simulating 100 guide hair of about 1000 segments.
The time step of simulation and collision detection is 3ms. The average simula-tion time is 0.64ms, the average rendering time is 1.4ms, and the average collision detection and response time is 2.3ms. The average FPS is 23.2
5.2 Hair Simulation 28
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)
(m) (n) (o) (p)
Figure 5.2: Dynamics Result: Our method is blue. The CORDE is green. We can observe our result of dynamics is more stiff. This is because we assume the rest segments will move together in the simplification of Special Cosserat Theory.
There is a big difference, but it is still acceptable in the hair simulation.
5.2 Hair Simulation 29
(a) (b)
(c) (d)
Figure 5.3: Curly Strand Dynamics
C H A P T E R 6
Conclusion
We have proposed a dynamic model for simulating high stiffness model more efficiently. Our method is faster, more stable than traditional method, and also can model the curliness phenomena of strand. With our proposed method, we make the simulation of hair with full interaction able to be performed in real time. But there are still some problems needed to be solved.
Limitation
Our method is based on a shape analysis. It means the shape of simulated strand is needed to be continuous in time and space. It needs enough stiffness to keep strand continuous. In the case which we consider about, the shape is con-strained strongly. But in low stiffness case, the shape may be very discontinuous in time. In this case, it will result some obvious artifacts in dynamics.
Future Work
We can observe from the statics experiment that the force transport method has still some force lost in each integration. As the segment number increases, the error increases. Besides, from the performance result, we can observe that the
30
31
computational time for handling interaction is still the bottleneck in hair simula-tion. We would like to solve the hair interaction problem in the future.
C H A P T E R 7
Appendix
Centrifugal Force
Before searching axial force, we consider the internal force generate by angu-lar speed from Equation 3.7. We need to add it into the external force to compute
˙
ω. If we multiply a mass on Equation 3.7’s both side, we get:
F = mass · ¨rl = mass · Z l
0
( ˙ω × d3+ ω × (ω × d3))ds
where ω × (ω × d3) is the centripetal force. What we want to find is the relation between force and angular acceleration, so we move it to the left side:
F − mass · The left side is total force.
Stiffness
The potential energy of quaternion proposed by the CORDE:
Vk = li
2Kkk(Bk(qj + qj+1) · 1
lj(qj+1− qj) − ˆuk)2 (7.2) 32
33
where
K11= K22= Eπr2
4 , K33= Gπr2 2
where E is the Young’s Modulus for bending stiffness, and G is the Shear Modulus for torsional stiffness.
We can do symbolic differentiation to get the force τq on the quaternion:
τq =
But the mass to quaternion is a dense matrix,which is hard to inverse. We use [16]
method to transform a quaternion force to a torque.
0 τ
= 1
2QTτq (7.3)
where τ is the stiffness torque.
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