Xuqiang Shao, Zhong Zhou, Wei Wu

1. Unified Particle-based Simulation of Deformable Solid-Fluid Interaction Xuqiang Shao, Zhong Zhou, Wei Wu State Key Laboratory of Virtual Reality Technology and Systems, Beihang University, Beijing, China

2. Content • Motivation • Related work • Unified particle-based deformable solid-fluid interaction • Results • Conclusion and future work

3. Motivation • Nowadays, realistic simulations of physical phenomena have been widely studied in the field of computer graphics, and wide applications can be found in commercial films, computer games, special effects, etc. • Solid-fluid interaction is common in everyday life, for example, a piece of iron sinks into a pool of water, and a rubber duck floats on the water surface. • Currently, coupled models are widely used in computer graphics, but the variety of the simulated materials and effects is often constrained by the interfaces between the models.

4. Coupled Models Genevaux et al. (2003) simulated the interaction between solids represented by mass-spring networks and an Eulerian grid fluid. Applying spring forces to the mass-less marker particles in the fluid and the nodes of the mass-spring network. Drawbacks: 1) the discretization of the force field is extremely coarse and imprecise. This imprecision can sometimes lead to ”spurious currents” that may appear at the interface between the solids and the fluid 2) Another limitation of the method is its relatively difficult setup. Indeed, multiple interlocked parameters need to be devised, each having dramatic influence on the results. Related Work

5. Related Work • Müller et al. (2004) developed a method in which SPH fluid particles interact with Lagrangian meshes by adding boundary particles to the surface of the mesh. Drawbacks: Since the density of generated particles varies among polygons which do not have a uniform surface area, it does not guarantee the constant particle density near the wall boundary.

6. Related Work • Chentanez et al. [2006] couple the Eulerian fluid and Lagrangian elastic solid Enforced coupling constraints by combining both the pressure projection and implicit integration steps into one set of simultaneous equations.

7. Related Work • Particle-based method • Müller et al. (2004) proposed a fully particle-based technique to model elastic, plastic and melting behavior of objects. In each step, compute the spatial derivatives of the discrete displacement field using a Moving Least Squares (MLS) procedure.

8. Related Work • Keiser et al. (2005) merged the Navier-Stokes equations with the equations for deformable solids to simulate solids, fuids and phase transitions in a uniform particle-based framework.

9. Related Work • Solenthaler et al. [4] use SPH to approximate the Jacobian of the deformation field, which can handle coarsely sampled and coplanar particle congurations. In this method liquids and deformable objects are uniformly represented and processed with SPH.

10. Deformable solid-fluid interaction • SPH SPH is an interpolation method for particle systems. A continuous function A(x) at a position x is interpolated by a weighted sum of contributions from all neighbour particles Field quantity at location xj Scalar quantity at location xi xj(1) xj(2) xi h xj(3) (x-xj(4)) xj(4) Mass of particle j Smoothing kernel with core radius of h Density at location j

11. Fluid behavior is modeled by the Navier-Stokes momentum equation (1) For the elastic solid, the governing equation (2) Using SPH to solve the above two equations (3) (4) (5) where is the viscosity coefficient, the pressure, u the displacement, v the velocity, the stress, I the identity matrix, V the body volume, and is the force particle i exerts on its jth neighbor.

12. Deformable solid-fluid interaction • Particle’s attributes Breaking down deformable solids and fluids into particles with various properties. Each particle individually carries many attributes summarized in Table 1

13. Deformable solid-fluid interaction • Forces between particles We treat deformable solid as a special fluid constrained to solid deformation, thus the two-way interaction can be achieved directly by multiphase SPH solver. Fsolid-fluid, Ffluid-fluid is the summation of Eq. (3) and (4) Fsolid-solidis calculated by Eq. (5)

14. Deformable solid-fluid interaction Using SPH to calculate density of a particle piat location yields the so called summation density (6) The density summation Equation (6) for multi-phase SPH flow will become problematic when a particle has neighbor particles with different rest densities. Based on the work of Solenthaler et al. (2008), We compute the adapted Density of a particle by multiplying the particle density by the mass of the particle (7) (8) (9)

15. Deformable solid-fluid interaction • Surface Tension Model Surface tension is an important factor in small-scale fluid simulations. Following the work of [MCG03,MSKG05], the surface force can be expressed as a body force (10) where is surface tension coefficient, The curvature k of the interface is the divergence of the normal. To calculate surface tension forces between fluid phase and solid phase, we introduce a color function C as (11) We formulate the density-weighted gradient of the color function as (12)

16. Deformable solid-fluid interaction For a continuous function , Taylor series is (13) Neglect the second and higher order terms, and multiply the equation with the gradient of the kernel function and integrate over the entire domain. The summation form of the corrected gradient as (14) Computing the inverse matrix and taking the trace of Equation (14), we find that the approximated divergence used to calculate the curvature can be written as (15)

17. Results

18. Conclusion and future work Conclusion We have proposed a unied particle-based method to simulate deformable solid-fluid interaction. Future Work • In the future, more details will be simulated, for example bubbles. • Simulate more complex phenomena such as interaction between fluid and permeable or erodible solid.

19. Thanks!