Elastic objects for computer graphic field using MPS method (sap-0179) Masahiro Kondo, ∗Masayuki Tanaka, †Takahiro Harada, ‡Seiichi Koshizuka§ The University of Tokyo

1 Introduction

such that the angular momentum conserves. The moment of inertia of a particle is denoted by I in Equ. (3).

Physics simulations play an important role on computer graphics in recent years. We newly adopt MPS(Moving Particle Semiimplicit), which can calculates elastic objects based on continuum mechanics. Interaction is easily evaluated by particle collisions. M¨ uller et al. simulated the elastic, plastic and melting object using particles[Muller et al. 2004]. They created an animation from the motion of the particles. Moving least square approximation was used in their method. In elastic analysis, the stress must be calculated. It is known that local particle oscillation occurs if stress is evaluated at particles [Vignjevic et al. 2000]. M¨ uller et al. calculated the stress tensor at the particle position, which may cause local particle oscillation. If stress terms are calculated between particles, the oscillation does not occur. We use MPS elastic method[Kondo et al. 2005], in which stress is calculated between particles. This method can analyze the elastic body motion.

3 Interaction between different objects The interaction between two objects can be expressed by particle collision. If the distance between two particles in the different objects is shorter than the collision distance, the particles are judged to collide. The repulsive force is calculated by |rij | − |ri | ⃗rij (⃗vj − ⃗vi ) · ⃗rij ⃗rij f⃗ic = −k1 + k2 ∆t2 |rij | ∆t|rij | |rij |

(4)

where k1 and k2 are the dimensionless coefficients. Fig. 1 shows the motion of nine elastic pawns. The polygons after deformation are reproduced in the same way as M¨ uller et al.

2 Motion equation of particles The continuum equation of the small deformation elastic body is ρ

∂vt ∂ = (λϵkk δst + 2µϵst ) ∂t ∂xs

(1)

Particle motion is calculated using discretized form of the above equation. Each particle has ⃗xi , ⃗vi , Ri and ωi , which denote the position, the velocity, the rotation matrix and the angular velocity of the particle i, respectively. Rigid body rotation, which should not affect the elastic analysis, is removed by considering the rotation of each particle. The governing equation (1) is discretized to ∂⃗vi m d X (ϵkk )i + (ϵkk )j ⃗rij m =λ Wij 0 ∂t ρ n0 j |rij |rij | | + 2µ

0 0 − Rj ⃗rij m d X 2⃗rij − Ri⃗rij Wij 0 2 0 ρ n j |rij |

4 Conclusion (2)

The stress in the first term of right hand side of Equ. (2) is calculated at the particle position using the volumetric strain evaluated 0 P |rij |−|rij | by (ϵkk )i = nd0 j wij . |r 0 | ij

The stress in the second term is evaluated between particles. This is effective to suppress the local oscillation. The torque given to each particle is calculated as 0 0 − Rj ⃗rij 1X 1 d 2⃗rij − Ri⃗rij ∂⃗ ω 0 =− Ri⃗rij × 2µ Wij 0 2 0 ∂t 2 j ρn |rij |

(3) ∗ e-mail:

[email protected] † e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected]

We use MPS elastic method for the simulation of elastic objects. Local oscillation is suppressed because the stress is evaluated between particles.

References

where m, d, n0 are the mass of one particle, the spatial dimension 0 and the normalizing constant. ⃗rij = ⃗xj − ⃗xi and ⃗rij =⃗ x0j − ⃗x0i re are the current and initial relative positions. Wij = rij − 1 is the weight function.

I

Figure 1: Elastic Pawns

KONDO , M., KOSHIZUKA , S., AND S UZUKI , Y. 2005. Application of symplectic scheme to three-dimensional elastic analysis using mps method. Transactions of the Japan Society of Mechanical Engineers A 72, 65–71. M ULLER , M., K EISER , R., N EALEN , A., PAULY, M., G ROSS , M., AND A LEXA , M. 2004. Point based animation of elastic, plastic and melting objects. ACM SIGGRAPH Symposium on Computer Animation, 141–151. V IGNJEVIC , R., C AMPBELL , J., AND L IBERSKY, L. 2000. A treatment of zero-energy modes in the smoothed particle hydrodynamics method. Computational methods in applied mechanics and engineering 184, 67–85.