Numerical simulation of electrostatic spray painting

Transcription

1 Numerical simulation of electrostatic spray painting Matthias Maischak Institute for Applied Mathematics Hannover University, Germany BMBF-Projekt 03STM1HV 3D Adaptive Multilevel FEM-BEM Kopplung für ein Elektrostatisches Teilproblem der Nasslackierung mit Aussenraumaufladung Head: Partners: Prof. Stephan (Hannover) Prof. Peukert, Dr. Schmid, Hr. Horn (Erlangen) Hr. Sonnenschein, Hr. Poppner (DaimlerChrysler)

2 Overview Physical/Engineering problem Model problem Variational Formulation Method of Characteristics Least-Squares formulation Numerical examples Overview 2

3 Electrostatic spray painting Electrostatic spray painting 3

4 Target Γ T Electrode Γ A Γ E ρ A u 1 u 0 Outer boundary Γ Model configuration (3D) 4

5 Electric corona discharge configuration Applied Voltage between Target and Electrode(s) Electrical field (Poisson equation) Ion transport (Convection), Space charge density ϱ Fluid (Moving Air, CFD) v Air Particle tracking, Particle charge ϱ P Corona discharge Corona discharge: Locally high electric field strength leads to ionization of air, ions are transported along field lines. Cloud of space charge near electrode has influence on local field strength. Electric corona discharge configuration 5

6 Distribution of scalar potential u ε 0 u = ϱ + ϱ P Electric field E = u Density of ionic current j Ion = ϱbe + ϱv Air Charge conservation law ϱ t + div(j Ion) = S ϱ Source term S ϱ due to to charge transfer between ϱ and ϱ P. ε 0 : Electron mobility: m V s Electric corona discharge configuration 6

7 Electrostatic Model problem Poisson equation Electrical field u = ϱ ε 0 in Ω (1) E = u (2) Boundary conditions (applied voltage) u = u 0 on Γ T (Target) (3) u = u 1 on Γ E (Electrode) (4) Electrostatic Model problem 7

8 Artifical boundary conditions on Γ u n 0 (Outflow b.c.) Γ = Su (infinite domain) (5) or u Γ = 0 (Faraday) (6) Using the Poincaré-Steklov operator S := 1 2 (W + (I K )V 1 (I K)) we can describe a charge free exterior domain as a boundary condition for the interior domain. The symmetric fem-bem coupling formulation approximates the Poincaré-Steklov operator by the Schur-complement of Galerkin matrices. Artifical boundary conditions on Γ 8

9 Linear transport equation Neglecting the air flow and the particle charge we can describe the unipolar ion transport by the linear transport equation div(eϱ) =0 in Ω (7) ϱ =ϱ A on Γ E (Inflow-b.c.) (8) Non-linear transport equation Inserting the Poisson equation in the linear transport equation we eliminate div E and we obtain E ϱ + ϱ2 ε 0 =0 in Ω (9) ϱ =ϱ A on Γ E (Inflow-b.c.) (10) Transport equation 9

10 Kaptzov s assumption Let E onset be the field strength where corona emission starts. With Kaptzov s assumption we obtain on Γ E the boundary condition: ϱ A 0, E n E onset (11) ϱ A (E n + E onset ) = 0 (12) Peek-field strength E onset (argument r in cm, result in V/m) E onset = ( r ) (Cylinder) (13) E onset = ( r/2 ) (Spherical) (14) Kaptzov s assumption 10

11 Solve initial Laplace equation Adjust ϱ A on Γ E Solve coupled differential equations Solve non-linear transport equation Method of Characteristics or Non-linear Least-Squares Solve Newton-steps Solve Poisson equation Flowchart of electro static subproblem 11

12 Poisson equation Find u H 1 D := {v H1 (Ω) : u ΓT = u 0, u ΓE = u 1 }, such that: ( u, v) 0 = 1 ε 0 (ϱ, v) 0 v H 1 D,0 (15) Solver: diagonal preconditioned Conjugate Gradient Overlaping domain decomposition as preconditioner for CG Mixed formulation Find (p, u) H(div; Ω) L 2 (Ω) such that (q, p) 0 + (u, div q) 0 = u 0, q n ΓT + u 1, q n ΓE q H(div; Ω) (v, div p) 0 = 1 ε 0 (ϱ, v) 0 v L 2 (Ω) (16) Solver: unpreconditioned GMRES Overlapping domain decomposition as preconditioner for GMRES Poisson equation 12

13 Symmetric FEM-BEM-coupling Find (u, σ) H 1 D (Ω) H 1/2 (Γ), such that: Variational formulation: u = ϱ ε 0 in Ω σ = u on Γ n 2σ = W u + (I K )σ on Γ 0 = (I K)u + V σ on Γ Find (u, σ) HD 1 (Ω) H 1/2 (Γ), such that: 2( u, v) 0 + W u, v + (K I)σ, v = 2 Ω ϱ ε 0 v dx (K I)u, ψ V σ, ψ = 0 ψ H 1/2 (Γ) v H 1 D,0(Ω) Symmetric FEM-BEM-coupling 13

14 V ψ(x) := 1 2π Kψ(x) := 1 2π K ψ(x) := 1 2π W ψ(x) := 1 2π Γ Γ 1 x y ψ(y) ds y single layer potential 1 n y x y ψ(y) ds y double layer potential 1 Γ n x x y ψ(y) ds y adjoint double layer potential 1 n x n y x y ψ(y) ds y hypersingular operator Γ Symmetric FEM-BEM-coupling 14

15 Smoothing of electrical field Let E = u h Clement-Interpolation: Let x Ω be a node of the mesh T h. Then we define U(x) = T T T h,x T i.e. U(x) is the union of all elements connected to the node x. Then the value of E h in the node x is given by U(x) E h (x) := E (x) dy U(x) 1 dy L 2 -Projection: The L 2 -Projection E h of E is given by: Find E h Vh E that: := {E h [H 1 (Ω)] 3 : E h T ist linear, T T h }, such (E h, v h ) L2 (Ω) = (E, v h ) L2 (Ω) v h V E h. (17) Smoothing of electrical field 15

16 Method of Characteristics We compute ϱ(x) along a field line, i.e., along a characteristic line. We write ϱ(s) = ϱ(x(s)). x(s) is determined by b E(x(s)) = dx(s) ds and x(0) = x 0. Then there holds 0 = be ϱ + b ϱ2 ε 0 = ϱ (s) + b ϱ2 (s) ε 0 (18) For ϱ 0 = ϱ(x(0)) the solution reads resp. 1 ϱ = 1 + bs (19) ϱ 0 ε 0 ϱ(s) = ϱ 0 ε 0 ε 0 + bϱ 0 s (20) Method of Characteristics 16

17 The remaining problem is the determination of the characteristic field line: Find x(s), such that be(x(s)) = dx(s) ds, i.e. x(s) = x(0) + s 0 be(x(t)) dt On the finite element mesh follow the direction of the electrical field. Efficient computation of boundary crossing points is essential. Computated values of space charge density are averaged on each element. Resulting average density is smoothed using Clement interpolation. Method of Characteristics 17

18 Least-Squares Formulation Find the minimum of the non-linear least squares functional for functions ϱ V ϱ A J(ϱ; f, ϱ D, ϱ N ) = E ϱ + ϱ2 ε 0 S ϱ 2 L 2 (Ω (21) with V ϱ A = {v H 1 (Ω) : v = ϱ A on Γ E } (22) The first and second Frechet-derivative of (21) read: J (ϱ; σ) =2 (E ϱ + ϱ2 S ϱ )(E σ + 2ϱσ ) dx Ω ε 0 ε 0 J (ϱ; σ, ξ) =2 (E ξ + 2ϱξ )(E σ + 2ϱσ ) dx Ω ε 0 ε (E ϱ + ϱ2 S ϱ ) 2ξσ dx ε 0 ε 0 Ω Least-Squares Formulation 18

19 The Newton algorithm applied to (21) reads: Find ϱ k+1 V ϱ A, such that 0 = J (ϱ k ; σ) + J (ϱ k ; σ, ϱ k+1 ϱ k ) σ V 0 (23) Least-Squares Formulation 19

20 Numerical Examples Mesh Elements Nodes Single electrode (tetrahedral,horn) Single electrode (tetrahedral,sonnenschein) Six electrodes (tetrahedral,sonnenschein) Using single electrode mesh created by Erlangen-group (Horn) Applied Voltage: Electrode radius: Distance Electrode-Target: V cm 0.32 m Measured Ion flow density (central target): 1.4mA/m 2 Computed E onset : V m (Cylinder), V m (Spherical) Charge injection law: ϱ ΓA = ϱ 0 max(0, (E n E onset )/(E max E onset )) Numerical Examples 20

21 Numerical Examples 21

22 0.5000E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 11 Space charge density (MOC), first and 7th iteration Numerical Examples 22

23 0.5000E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 11 Space charge density (MOC), full injection law (Eonset = 0) Piecewise linear E-Field (left), piecewise constant u (right) Numerical Examples 23

24 absolute value absolute value E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+02 Electric field strength (MOC), first and 7th iteration Numerical Examples 24

25 absolute value absolute value E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 11 Space charge density near tip (MOC), first and 7th iteration absolute value absolute value E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+06 Electric field strength near tip, first and 7th iteration Numerical Examples 25

26 0.1693E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 n E ϱ L2 (Ω) ϱ 2 /ε 0 L2 (Ω) E ϱ + ϱ 2 /ε 0 L2 (Ω) E max E E E E E E E E E E E E E E E E E E E E+07 Least-Squares error indicator 26

27 Conclusion/ to do Method of Characteristics gives space charge distribution Need better E-field (with mixed formulation?) Initial mesh should be better adjusted to field lines Use Least squares functional of transport problem for adaptive scheme Conclusion/ to do 27