Point Based Animation of Elastic, Plastic and Melting Objects Mark Pauly Andrew Nealen Marc Alexa ETH Zürich TU Darmstadt Stanford Matthias Müller Richard Keiser Markus Gross 李盈璁

Outline Related Work Advantages & Disadvantages Elasticity Model Simulation Loop Time Integration Surface Animation ( 省略 ) Result

Related Work Desbrun & Cani [95,96,99] Physics: Smoothed Particle Hydrodynamics (SPH) Surface: Implicit with suppressed distance blending Tonnesen [98] Physics: Lennard-Jones based forces Surface: Particles with orientation

Advantages & Disadvantages Advantages No volumetric mesh needed Natural adaptation to topological changes Disadvantages Difficulty of getting sharp fracture lines Neighboring Phyxels are not explicitly given Throughout this work we use Spatial Hashing [Teschner et al. 03] for fast neighbor search (when needed)

Elasticity Model Continuum Elasticity Elastic Strain Estimation of Derivatives Discrete Energy Density Elastic Forces

Reference configuration Continuum Elasticity Deformed configuration = Elastic Memory Displacement (vector) field: u(x) = [ u( x,y,z ) v( x,y,z ) w( x,y,z ) ] T u(x)u(x) xx+u(x) x’x’ u(x’)u(x’) x’+u(x’)x’+u(x’)

Elastic Strain → Strain depends on the spatial derivatives of u(x) no strain strain u(x) Next: Compute spatial derivatives of the x component u

Estimation of Derivatives - 1 Computation of the unknown u,x, u,y and u,z at x i by Linear approximation Minimize → WLS/MLS approximation of derivatives uiui xixi xjxj  x = x ij ujuj

Estimation of Derivatives - 2 Set partial derivatives of e with respect to u,x, u,y and u,z to zero to obtain minimizer of e Linear approximation of u j as seen from x i Actual value of u j at point x j Use SVD for the 3x3 Matrix inversion for stability Vector of Unknown Partial Derivatives uiui xixi xjxj  x = x ij ujuj

Discrete Energy Density -1 Strain from  u Stress via material law (Hooke) Energy density (scalar)

Discrete Energy Density -2 Use Smoothed Particle Hydrodynamics (SPH) Method Mass of each Phyxel mi is fix during the simulation Distribute the mass around the Phyxel using a polynomial weighting kernel wij with compact support The density around Phyxel i is From which we compute the volume vi as

Elastic Forces Estimate volume vi represented by phyxel i via SPH Elastic energy of phyxel i Depends on u i and u j of all neighbors j Phyxel i and all neighbors j receive a force

Simulation Loop Verlet Integration (= new displacements) Estimation of Derivatives External Forces (Gravity, Interaction) Computation of Strains, Stresses, Elastic Energy and per Phyxel Body Forces External Forces (Gravity, Interaction) Verlet Integration (= new displacements) Estimation of Derivatives Computation of Strains, Stresses, Elastic Energy and per Phyxel Body Forces

Time Integration Verlet (Explicit) Time Stepping Newtons Second Law of Motion + =

Result 彈性物體 1 、彈性物體 2 彈性物體 1彈性物體 2

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