Fig. 1. Our approach can accurately reproduce the observation that real ferrofluid is literally climbing up a steel helix placed above a strong electromagnet. This figure shows the final results of our simulation of this scenario rendered from different viewpoints. We present an approach to the accurate and efficient large-scale simulation of the complex dynamics of ferrofluids based on physical principles. Ferrofluids are liquids containing magnetic particles that react to an external magnetic field without solidifying. In this contribution, we employ smooth magnets to simulate ferrofluids in contrast to previous methods based on the finite element method or point magnets. We solve the magnetization using the analytical solution of the smooth magnets' field, and derive the bounded magnetic force formulas addressing particle penetration. We integrate the magnetic field and force evaluations into the fast multipole method allowing for efficient large-scale simulations of ferrofluids. The presented simulations are well reproducible since our approach can be easily incorporated into a framework implementing a Fast Multipole Method and a Smoothed Particle Hydrodynamics fluid solver with surface tension. We provide a detailed analysis of our approach and validate our results against real wet lab experiments. This work can potentially open the door for a deeper understanding of ferrofluids and for the identification of new areas of applications of these materials.
Numerical integration procedure updating the state (i.e. vertex positions and velocities) of the magnetic fluid. Input: Current fluid state. Output: Updated fluid state. 1 Advection; Section 4.1. 2 Enforce harmonic velocity by Helmholtz decomposition; Section 4.2. 3 Calculate surface tension and gravity potential. Section 4.3. 4 Calculate magnetic pressure using Algorithm 2. 5 Solve the pressure using a BEM; Section 4.5. 6 Apply the negative gradient of the pressure to the velocity. ALGORITHM 2: Evaluation procedure of the magnetic pressure. Input: Vertex positions. Output: Magnetic pressure at vertices. 1 Update the external magnetic field. 2 Evaluate the external magnetic scalar potential; Section 4.4.1. 3 Calculate magnetic double layer charges at vertices; Section 4.4.2. 4 Evaluate magnetic pressure discontinuities at vertices; Section 4.4.4.
The simulation of large open water surface is challenging using a uniform volumetric discretization of the Navier-Stokes equations. Simulating water splashes near moving objects, which height field methods for water waves cannot capture, necessitates high resolutions. Such simulations can be carried out using the Fluid-Implicit-Particle (FLIP) method. However, the FLIP method is not efficient for the long-lasting water waves that propagate to long distances, which require sufficient depth for a correct dispersion relationship. This paper presents a new method to tackle this dilemma through an efficient hybridization of volumetric and surface-based advection-projection discretizations. We design a hybrid time-stepping algorithm that combines a FLIP domain and an adaptively remeshed Boundary Element Method (BEM) domain for the incompressible Euler equations. The resulting framework captures the detailed water splashes near moving objects with the FLIP method, and produces convincing water waves with correct dispersion relationships at modest additional costs.
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