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--- team:gdeliege:espaint [2015/08/12 10:08] – 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 ==
+=== Problem description ===
 Electrostatic painting is one of the applications I studied during my PhD.
-I started from a mathematical model by François Henrotte
+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.
@@ Line 13: / Line 13: @@
 the cathode and combine with atoms.
 The negative ions drift toward the anode, i.e. the grounded plate, due to Coulomb forces.
-The system is described by classical electrostatic equations coupled with a transient convection equation,
+In the absence of coating particles, the system is described by classical electrostatic equations coupled with a transient convection equation,
 $$
 \begin{eqnarray*}
 \nabla\cdot\vec{d} &=& \rho_i \\
-\nabla\times\vec{e} &=&0 \\
+\nabla\times\vec{e} &=& 0 \\
 \vec{d} &=& \varepsilon_0\vec{e} \\
 \partial_t \rho_i +\nabla\cdot( \mu_i\vec{e}\rho_i) &=& 0
@@ Line 23: / Line 23: @@
 $$
 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 ==
+=== References ===
-[1] F. Henrotte. //Calcul des efforts électromagnétiques et de leurs effets dans des structures quelconques//. PhD Thesis, Université de Liège, 2000
+[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]]