On the Accurate Large-Scale Simulation of rrofluids Fe Libo - - PowerPoint PPT Presentation

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On the Accurate Large-Scale Simulation of rrofluids Fe Libo - - PowerPoint PPT Presentation

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On the Accurate Large-Scale Simulation of rrofluids Fe Libo Huang Torsten Hdrich 26 Iron Dominik L. Michels KAUST 1 Real footage 2 Simulation Meshed View Particle View 3 Outline Why it has spikes? Related work Our

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On the Accurate Large-Scale Simulation of rrofluids

Libo Huang Torsten HΓ€drich Dominik L. Michels KAUST 26

Fe

Iron

1

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Real footage

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Particle View Meshed View

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Simulation

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Outline

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  • Why it has spikes?
  • Related work
  • Our method (physically based)
  • Results & Discussion
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random direction No external magnetic field

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nanoparticles Fe3Ο4 rrofluid

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Fe

Iron

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dominant direction

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With external magnetic field

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With external magnetic field

constant magnetic field

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constant magnetic field

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field

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small bump stronger field

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surface tension

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Surface Tension Field Direction

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Simulation

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𝐺 = 𝐺surface + 𝐺

fluid

+ 𝐺magnet

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Finite Element Method Particle

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Particle

  • Approximating

continuous ferrofluid

  • Accurate and

stable magnetic forces

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Challenges

FEM

  • Remeshing the

fluid and air

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Only particles, no re-meshing

  • 1. Smooth magnets, continuous fluid
  • 2. Forces of smooth magnets, accurate, stable
  • 3. Fast multipole method, 𝑃 𝑂2 β†’ 𝑃(𝑂)

Our solution

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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

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From math & physics

  • [Nochetto et al. 2016]
  • [Yoshikawa et al. 2010]
  • [Lavrova et al. 2006, Gollwitzer 2006]

2D dynamic One spike Static

Related Work

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Explicit Scheme Smooth Particle Hydrodynamics [Adami et al. 2012] 𝐺

fluid

SPH Surface Tension [Yang et al. 2017] 𝐺

surface

Magnetic Solver (ours) 𝐺

magnet

𝐺(𝑒, 𝑦) 𝑦(t)

The simulator

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Smooth Magnet

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Point Smooth

Infinite small point Finite size cloud

𝐢(𝑠) ∝ 1 𝑠3 𝐢 𝑠 ∝ Density(𝑠)

Near center Near center

𝐢(0) undefined 𝐢(0) well-defined

Density r Density r

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Point Smooth

Discontinuous Continuous

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magnetic field Input Output: directions

Solve Magnetization

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Solve Magnetization

  • Note: each smooth magnet affects others
  • An optimization problem:
  • Best dominant directions satisfy physics laws.
  • Least square conjugate gradient
  • Fast multipole, 𝑃 𝑂2 β†’ 𝑃(𝑂)

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  • 1. βˆ€ nanoparticle β†’ magnetic field
  • 2. βˆ€ nanoparticle ← magnetic forces

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Force Principles

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Center force (point magnet in smooth field) Fitted force (smooth magnet in smooth field)

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Simulation

Real footage

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Simulation

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Real footage

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Simulation

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Simulation

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Simulation

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3D dynamic ferrofluid simulator using smooth magnet.

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Conclusion

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Questions?

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On the Accurate Large-scale Simulation of Ferrofluids

Libo Huang, Torsten HΓ€drich, Dominik L. Michels

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Unintuitive complex geometry

Why simulating ferrofluids?

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Simulation

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Source Target 𝐺𝑑→𝑒

𝑗

= Ξ›π‘—π‘˜π‘™ 𝑠 𝑛𝑑

π‘˜π‘›π‘’ 𝑙

A third-order tensor (to be measured) gives forces In local coordinates

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field Susceptibility ∝ Nanoparticle Density

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Particle 𝑗 MagnitudeΓ—Direction = 𝑛𝑗 ∈ ℝ3 𝑛𝑗

How to describe ferrofluid?

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𝑐𝑗

fluid = ෍ π‘˜=1 𝑂

𝑐

π‘˜β†’π‘— fluid = ෍ π‘˜=1 𝑂

π»π‘—π‘˜π‘›π‘˜ π»π‘—π‘˜ ∈ ℝ3Γ—3

  • 1. Particles Generate Magnetic Fields
  • 2. Magnetic Fields Influence Particles

𝑛𝑗 = 𝑑(𝑐𝑗

fluid + 𝑐𝑗 external)

𝑑 ∈ ℝ,constant

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𝑐𝑗

fluid = ෍ π‘˜=1 𝑂

π»π‘—π‘˜π‘›π‘˜

𝑛𝑗 = 𝑑(𝑐𝑗

fluid + 𝑐𝑗 external)

A correct particle state 𝑛 generates a field 𝑐fluid, which combined with external field 𝑐external lead to the same state 𝑛.

min

𝑛

𝑛 βˆ’ 𝑑 𝐻𝑛 + 𝑐external

2

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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).

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Fast Multipole Method

Naive 30s FMM 1.5s

𝑐 = 𝐻𝑛

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