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team:gdeliege:espaint [2015/08/12 11:10] 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 ==+=== Problem description ===
  
 Electrostatic painting is one of the applications I studied during my PhD. 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 and used this nice coupled problem to test different potential and mixed formulations
 of electrostatic equations. of electrostatic equations.
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 where $\mu_i$ is the ion mobility. where $\mu_i$ is the ion mobility.
  
-The convection equation is integrated in time with an unconditionally stable 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 ($\vec{d}$-$V$). +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]
-In particular, 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. +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 (Pressure-Stabilized Petrov-Galerkinworks fine with Maxwell's equations as well.+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). The geometrical model is a box extending from the middle of a wire to half the distance between two consecutive wires (Fig. 1).
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 {{ :team:gdeliege:espaint01.png?direct&700 |}} {{ :team:gdeliege:espaint01.png?direct&700 |}}
-//Figure 2. //+//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 |}} {{ :team:gdeliege:espaint05.png |}}
-//Figure 3. // +//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]]
  
team/gdeliege/espaint.1439370630.txt.gz · Last modified: 2016/03/30 15:22 (external edit)

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