On the Accurate Large-Scale Simulation of rrofluids Fe Libo Huang Torsten Hädrich 26 Iron Dominik L. Michels KAUST 1
Real footage 2
Simulation Meshed View Particle View 3
Outline • Why it has spikes? • Related work • Our method (physically based) • Results & Discussion 4
No external magnetic field rrofluid Fe Iron 26 nanoparticles Fe 3 Ο 4 random direction 5
With external magnetic field dominant direction 6
With external magnetic field constant magnetic field 7
constant magnetic field 8
field 9
small bump stronger field 10
surface tension 11
Simulation Field Direction Surface Tension 12
𝐺 = 𝐺 surface + 𝐺 fluid + 𝐺 magnet 13
Particle Finite Element Method 14
Challenges Particle FEM • Approximating • Remeshing the continuous fluid and air ferrofluid • Accurate and stable magnetic forces 15
Our solution Only particles, no re-meshing 1. Smooth magnets, continuous fluid 2. Forces of smooth magnets, accurate, stable 3. Fast multipole method, 𝑃 𝑂 2 → 𝑃(𝑂) 16
Related Work From visual computing Ferrofluid: [Ishikawa et al. 2012, 2013] Rigid magnet: [Thomaszewski et al. 2008] Rigid magnet: [Kim et al. 2018] Post processing Rigid magnet Rigid magnet 17
Related Work From math & physics • [Nochetto et al. 2016] • [Yoshikawa et al. 2010] • [Lavrova et al. 2006, Gollwitzer 2006] 2D dynamic One spike Static 18
The simulator Explicit Scheme Smooth Particle 𝐺 Hydrodynamics fluid [Adami et al. 2012] SPH Surface Tension 𝐺(𝑢, 𝑦) 𝐺 𝑦(t) surface [Yang et al. 2017] Magnetic Solver 𝐺 magnet (ours) 19
Smooth Magnet 20
Smooth Point r r 0 0 Density Density Infinite small point Finite size cloud 𝐶(𝑠) ∝ 1 Near center 𝐶 𝑠 ∝ Density(𝑠) Near center 𝑠 3 𝐶(0) undefined 𝐶(0) well-defined 21
Point Smooth Discontinuous Continuous 22
Solve Magnetization Output: directions Input magnetic field 23
Solve Magnetization • Note: each smooth magnet affects others • An optimization problem: • Best dominant directions satisfy physics laws. • Least square conjugate gradient • Fast multipole, 𝑃 𝑂 2 → 𝑃(𝑂) 24
Force Principles 1. ∀ nanoparticle → magnetic field 2. ∀ nanoparticle ← magnetic forces 25
Center force Fitted force (point magnet (smooth magnet in smooth field) in smooth field) 26
Simulation Real footage 27
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Simulation 29
Real footage 30
Simulation 31
Simulation 32
Simulation 33
Conclusion 3D dynamic ferrofluid simulator using smooth magnet. 34
On the Accurate Large-scale Simulation of Ferrofluids Libo Huang, Torsten Hädrich, Dominik L. Michels Questions? 35
Unintuitive Why simulating complex ferrofluids? geometry 39
Simulation 40
Target Source = Λ 𝑗𝑘𝑙 𝑠 𝑛 𝑡 𝑘 𝑛 𝑢 𝑗 𝑙 𝐺 𝑡→𝑢 In local coordinates A third-order tensor (to be measured) gives forces 49
Susceptibility ∝ Nanoparticle Density field 50
How to describe ferrofluid? Particle 𝑗 𝑛 𝑗 Magnitude × Direction = 𝑛 𝑗 ∈ ℝ 3 51
1. Particles Generate Magnetic Fields 𝑂 𝑂 fluid = fluid = 𝑐 𝑗 𝑐 𝐻 𝑗𝑘 𝑛 𝑘 𝑘→𝑗 𝑘=1 𝑘=1 𝐻 𝑗𝑘 ∈ ℝ 3×3 2. Magnetic Fields Influence Particles fluid + 𝑐 𝑗 external ) 𝑛 𝑗 = 𝑑(𝑐 𝑗 𝑑 ∈ ℝ ,constant 52
A correct particle state 𝑛 generates a field 𝑐 fluid , which combined with external field 𝑐 external lead to the same state 𝑛 . 𝑂 fluid = 𝑐 𝑗 𝐻 𝑗𝑘 𝑛 𝑘 𝑘=1 fluid + 𝑐 𝑗 external ) 𝑛 𝑗 = 𝑑(𝑐 𝑗 𝑛 − 𝑑 𝐻𝑛 + 𝑐 external min 2 𝑛 53
Center force: All nanoparticles moved to particle center to calculate force (bounded but inaccurate). Fitted force: All nanoparticles contribute to the force. Pre-calculated, stored as fitted polynomial (accurate surface force). 54
Fast Multipole Method Naive 30s 𝑐 = 𝐻𝑛 FMM 1.5s 55
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