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team:gdeliege:espaint [2015/08/11 16:59] – created geoffreyteam:gdeliege:espaint [2016/03/30 15:23] (current) – external edit 127.0.0.1
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-==== Electrostatic painting ====+===== Electrostatic painting =====
  
 +=== Problem description ===
 +
 +Electrostatic painting is one of the applications I studied during my PhD.
 +I started from a mathematical model by François Henrotte [1]
 +and used this nice coupled problem to test different potential and mixed formulations
 +of electrostatic equations.
 +
 +The original device is made of a set of thin wires parallel to a grounded iron plate.
 +The wires are brought to a negative potential of high amplitude.
 +The strong electric field around the wires causes the acceleration of free electrons which move away from
 +the cathode and combine with atoms.
 +The negative ions drift toward the anode, i.e. the grounded plate, due to Coulomb forces.
 +In the absence of coating particles, the system is described by classical electrostatic equations coupled with a transient convection equation,
 +$$
 +\begin{eqnarray*}
 +\nabla\cdot\vec{d} &=& \rho_i \\
 +\nabla\times\vec{e} &=& 0 \\
 +\vec{d} &=& \varepsilon_0\vec{e} \\
 +\partial_t \rho_i +\nabla\cdot( \mu_i\vec{e}\rho_i) &=& 0
 +\end{eqnarray*}
 +$$
 +where $\mu_i$ is the ion mobility.
 +
 +The convection equation is integrated in time with an implicit scheme and the electrostatic equations are solved at each time step. I implemented several electrostatic formulations to analyse their influence on the charge conservation: electric scalar potential ($\vec{e}=-\nabla V$), electric vector potential formulation with source field ($\vec{d}=\vec{d}_s+\nabla\times\vec{w}$) and mixed formulation ($\vec{d}$-$V$) [2].
 +It must be noted that electrostatic and magnetostatic mixed formulations have the same stability problems as Stokes equations when the shape functions do not satisfy the Babuska-Brezzi inf-sup condition.
 +Fortunately, a stabilization technique developed in fluid mechanics, the so-called Pressure-Stabilized Petrov-Galerkin formulation, works fine with Maxwell's equations as well [3].
 +
 +=== Finite element simulations ===
 +
 +The geometrical model is a box extending from the middle of a wire to half the distance between two consecutive wires (Fig. 1).
 +Fig. 2 shows the different fields of the vector potential formulation at the end of the simulation. The source field $\vec{d}_s$ is an arbitrary field such that $\nabla\cdot\vec{d}_s=\rho_i$.
 +Fig. 3 (left) shows the currents flowing through the wire and the plate, which reach a steady state after 1.5ms approximately. In a real electrostatic painting problem, one should also model the flux of coating particles and their interactions with ions.
 +Fig. 3 (right) shows that discretizing the electric displacement results in a better charge conservation than the classical scalar potential $V$ formulation, even if $V$ is discretized with second order shape functions.
 +
 +{{ :team:gdeliege:espaint02.png?direct |}}
 +//Figure 1. Simplified geometry of the electrostatic painting device.//
 +
 +{{ :team:gdeliege:espaint01.png?direct&700 |}}
 +//Figure 2. Fields of the vector potential formulation at the end of the simulation (t=2ms): (1) vector potential $\vec{w}$, (2) source field $\vec{d}_s$, (3) electric displacement $\vec{d}=\vec{d}_s+\nabla\times\vec{w}$, (4) ion density $\rho_i$.//
 +
 +{{ :team:gdeliege:espaint05.png |}}
 +//Figure 3. Finite element simulation results : (left) current flowing through the wire and plate surfaces, (right) error on the charge conservation at each time step, calculated as the relative difference between the total charge variation during a time step and the integral of the currents on the wire and plate surfaces.//
 +
 +=== References ===
 +
 +[1] F. Henrotte. //Calcul des efforts électromagnétiques et de leurs effets dans des structures quelconques//. PhD Thesis, Université de Liège, 2000 \\
 +[2] G. Deliége, F. Henrotte, W. Deprez, K. Hameyer. //Finite element modelling of ion convection by electrostatic forces.// IET Science, Measurement & Technology, vol. 151, pp. 398-402, 2004 \\
 +[3] G. Deliége, E. Rosseel, S. Vandewalle. //Iterative solvers and stabilisation for mixed electrostatic and magnetostatic formulations.// Journal of Computational & Applied Mathematics, vol. 215, pp. 348-356, 2008 \\
 +\\
 +[[team:gdeliege|Back to main page]]
  
team/gdeliege/espaint.1439305195.txt.gz · Last modified: 2016/03/30 15:22 (external edit)

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