Pol Grasland-Mongrain edited Simulation_study.tex  over 8 years ago

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A finite element software (Finite Element Magnetic Method \cite{FEMM}) was used to produce a two dimensional simulation of the magnetic field $\mathbf{B}$. Magnetostatic problem was solved from equations $\nabla \times H = \nabla \times M$, $\nabla B = 0$ and $B=\mu H$, with $H$ the magnetic field intensity, $M$ magnetization of the medium, $B$ the magnetic flux density and $J$ the current density. Medium was considered as linear, and space was meshed with approximately 0.5 cm$^2$ triangles. The magnetic field was supposed to be approximately constant in the sample along the Y axis. The software simulated a N48 NdFeB permanent magnet of 5x5 cm$^2$ placed in a 19x27 cm$^2$ surface of air. Resulting magnetic field is illustrated in Figure \ref{Figure3}-(B), with colors indicating the absolute magnitude and arrows the direction.  Body Lorentz force $\mathbf{f}$ was computed from the cross-product of $\mathbf{j}$ and $\mathbf{B}$. As the electrical current is induced in the XY plane and the magnetic field essentially along X direction, Lorentz force is mainly along Z direction. Z component of the Lorentz force is illustrated in Figure \ref{Figure3}-C, with arrows indicating the Lorentz force vector and color its magnitude. amplitude.  Finally, displacement $\mathbf{u}(\mathbf{r},t)$ was determined analytically along space (pixels of 2x2 mm$^2$) and time (steps of 1 ms) by solving Equation \ref{Equation3} with the Green operator \cite{aki1980quantitative}. It used a medium density $\rho$ of 1000 kg.m$^{-3}$, a bulk modulus $K$ of 2.3 GPa and a shear modulus $\mu$ of 16 kPa, corresponding to a shear wave speed of 4 m.s$^{-1}$.