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--- team:gdeliege:espaint [2015/08/11 17:01] – geoffrey
+++ team:gdeliege:espaint [2016/03/30 15:23] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
-==== 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]]