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Numerical simulation of electrostaticspray painting

Matthias Maischak

I AMf Institute for Applied Mathematics

Hannover University, Germany

BMBF-Projekt 03STM1HV

3D Adaptive Multilevel FEM-BEM Kopplung fur ein ElektrostatischesTeilproblem der Nasslackierung mit Aussenraumaufladung

Head: Prof. Stephan (Hannover)

Partners: Prof. Peukert, Dr. Schmid, Hr. Horn (Erlangen)

Hr. Sonnenschein, Hr. Poppner (DaimlerChrysler)

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Overview

Physical/Engineering problem Model problem

Variational Formulation Method of Characteristics Least-Squares formulation Numerical examples

I AMf Overview 2

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Electrostatic spray painting

I AMf Electrostatic spray painting 3

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Target

Electrode

Outer boundary

T

E

A

u

uA

0

1

I AMf Model configuration (3D) 4

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Electric corona discharge configuration

Applied Voltage between Target and Electrode(s) Electrical field (Poisson equation)

Ion transport (Convection), Space charge density Fluid (Moving Air, CFD) vAir

Particle tracking, Particle charge 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.

I AMf Electric corona discharge configuration 5

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Distribution of scalar potential u

0u = + PElectric field

E = uDensity of ionic current

jIon = bE+ vAir

Charge conservation law

t+ div(jIon) = S

Source term S due to to charge transfer between and P.

0: 8.8542 1012

Electron mobility: 2 104 m2

V s

I AMf Electric corona discharge configuration 6

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Electrostatic Model problem

Poisson equation

u = 0

in (1)

Electrical fieldE = u (2)

Boundary conditions (applied voltage)

u = u0 on T (Target) (3)

u = u1 on E (Electrode) (4)

I AMf Electrostatic Model problem 7

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Artifical boundary conditions on

u

n| =

0 (Outflow b.c.)

Su (infinite domain)(5)

oru| = 0 (Faraday) (6)

Using the Poincare-Steklov operator

S := 12 (W + (I K

)V1

(I K)) we can describe a charge freeexterior domain as a boundary condition for the interior domain.

The symmetric fem-bem coupling formulation approximates the

Poincare-Steklov operator by the Schur-complement of Galerkin

matrices.

I AMf Artifical boundary conditions on 8

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Linear transport equation Neglecting the air flow and the

particle charge we can describe the unipolar ion transport by the

linear transport equation

div(E) =0 in (7)

=A on 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 + 2

0=0 in (9)

=A on E (Inflow-b.c.) (10)

I AMf Transport equation 9

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Kaptzovs assumption

Let Eonset be the field strength where corona emission starts. With

Kaptzovs assumption we obtain on E the boundary condition:

A 0, E n Eonset (11)A(E n + Eonset) = 0 (12)

Peek-field strength Eonset (argument r in cm, result in V/m)

Eonset =3.1 106(1 + 0.308r

) (Cylinder) (13)

Eonset =3.1 106(1 + 0.308r/2

) (Spherical) (14)

I AMf Kaptzovs assumption 10

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Solve initial Laplace equation

?Adjust A on E

?Solve coupled differential equations

?

6

Solve non-linear transport equation

?

-

-

Method of Characteristics or

Non-linear Least-Squares

?Solve Newton-steps

6

Solve Poisson equation

6

I AMf Flowchart of electro static subproblem 11

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Poisson equation

Find u H1D := {v H1() : u|T = u0, u|E = u1}, such that:

(u, v)0 = 10

(, v)0 v H1D,0 (15)

Solver: diagonal preconditioned Conjugate Gradient

Overlaping domain decomposition as preconditioner for CG

Mixed formulation

Find (p, u) H(div; ) L2() such that(q, p)0 + (u, div q)0 = u0, q nT + u1, q nE q H(div; )

(v, divp)0 = 1

0 (, v)0 v L2() (16)

Solver: unpreconditioned GMRES

Overlapping domain decomposition as preconditioner for GMRES

I AMf Poisson equation 12

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Symmetric FEM-BEM-coupling

Find (u, ) H1D() H1/2(), such that:

u = 0

in

=u

non

2 = W u + (I K) on

0 = (I K)u + V on Variational formulation:

Find (u, ) H1D() H1/2(), such that:

2(u, v)0 + W u ,v + (K I), v = 2

0v dx v H1D,0()

(K I)u, V , = 0 H1/2()

I AMf Symmetric FEM-BEM-coupling 13

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V (x) := 12

1|x y|(y) dsy single layer potential

K(x) :=1

2

ny

1

|x

y

|

(y) dsy double layer potential

K(x) :=1

2

nx

1

|x y|(y) dsy adjoint double layer potential

W (x) :=

1

2

nx

ny

1

|x y|(y) dsy hypersingular operator

I AMf Symmetric FEM-BEM-coupling 14

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Smoothing of electrical field

Let E = uhClement-Interpolation: Let x be a node of the mesh Th.Then we define

U(x) =

TTh,xT

T

i.e. U(x) is the union of all elements connected to the node x.

Then the value of Eh in the node x is given by

Eh(x) :=

U(x)

E(x) dyU(x)

1 dy

L2

-Projection: The L2

-Projection Eh of E is given by:Find Eh VEh := {Eh [H1()]3 : Eh|T ist linear , T Th}, suchthat:

(Eh, vh)L2() = (E, vh)L2()

vh

VE

h

. (17)

I AMf Smoothing of electrical field 15

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Method of Characteristics

We compute (x) along a field line, i.e., along a characteristic line.

We write (s) = (x(s)). x(s) is determined by b E(x(s)) = dx(s)dsand x(0) = x0. Then there holds

0 = bE + b 2

0= (s) + b

2(s)

0(18)

For 0 = (x(0)) the solution reads

1

=

1

0+

bs

0(19)

resp.

(s) = 0 00 + b0s(20)

I AMf Method of Characteristics 16

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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 interpola-tion.

I AMf Method of Characteristics 17

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Least-Squares Formulation

Find the minimum of the non-linear least squares functional

J(; f, D, N) =E + 2

0 S2L2( (21)

for functions VA withVA = {v H1() : v = A on E} (22)

The first and second Frechet-derivative of (21) read:

J(; ) =2

(E + 2

0 S)(E + 2

0) dx

J(; , ) =2

(E + 20

)(E + 20

) dx

+ 2

(E + 2

0 S) 2

0dx

I AMf Least-Squares Formulation 18

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The Newton algorithm applied to (21) reads:

Find k+1 VA , such that

0 = J(k; ) + J(k; , k+1 k) V0 (23)

I AMf Least-Squares Formulation 19

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Numerical Examples

Mesh Elements Nodes

Single electrode (tetrahedral,Horn) 347193 68647

Single electrode (tetrahedral,Sonnenschein) 1650233 281829

Six electrodes (tetrahedral,Sonnenschein) 3030803 545429

Using single electrode mesh created by Erlangen-group (Horn)

Applied Voltage: 26300 V

Electrode radius: 0.0075 cm

Distance Electrode-Target: 0.32 m

Measured Ion flow density (central target): 1.4mA/m2

Computed Eonset: 1.4 107 Vm (Cylinder),1.86 107 Vm (Spherical)

Charge injection law: |A = 0 max(0, (E n Eonset)/(Emax Eonset))I AMf Numerical Examples 20

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I AMf Numerical Examples 21

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0.1000E11

0.4663E11

0.2175E10

0.1014E09

0.4729E09

0.2205E08

0.1028E07

0.4795E07

0.2236E06

0.1043E05

0.4862E05

0.2267E04

0.1057E03

0.4931E03

0.2299E02

0.1072E01

0.5000E01

0.1000E11

0.4663E11

0.2175E10

0.1014E09

0.4729E09

0.2205E08

0.1028E07

0.4795E07

0.2236E06

0.1043E05

0.4862E05

0.2267E04

0.1057E03

0.4931E03

0.2299E02

0.1072E01

0.5000E01

Space charge density (MOC), full injection law (Eonset = 0)

Piecewise linear E-Field (left), piecewise constant u (right)I AMf Numerical Examples 23

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0.7883E+02

0.1893E+03

0.4547E+03

0.1092E+04

0.2622E+04

0.6298E+04

0.1512E+05

0.3632E+05

0.8723E+05

0.2095E+06

0.5031E+06

0.1208E+07

0.2902E+07

0.6969E+07

0.1674E+08

0.4019E+08

0.9653E+08

absolute value

0.8214E+02

0.1938E+03

0.4574E+03

0.1080E+04

0.2548E+04

0.6012E+04

0.1419E+05

0.3348E+05

0.7901E+05

0.1865E+06

0.4400E+06

0.1038E+07

0.2451E+07

0.5783E+07

0.1365E+08

0.3221E+08

0.7601E+08

absolute value

Electric field strength (MOC), first and 7th iteration


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0.1000E11

0.4663E11

0.2175E10

0.1014E09

0.4729E09

0.2205E08

0.1028E07

0.4795E07

0.2236E06

0.1043E05

0.4862E05

0.2267E04

0.1057E03

0.4931E03

0.2299E02

0.1072E01

0.5000E01

absolute value

0.1000E11

0.4663E11

0.2175E10

0.1014E09

0.4729E09

0.2205E08

0.1028E07

0.4795E07

0.2236E06

0.1043E05

0.4862E05

0.2267E04

0.1057E03

0.4931E03

0.2299E02

0.1072E01

0.5000E01

absolute value

Space charge density near tip (MOC), first and 7th iteration

0.1915E+06

0.2825E+06

0.4168E+06

0.6149E+06

0.9072E+06

0.1339E+07

0.1975E+07

0.2914E+07

0.4299E+07

0.6343E+07

0.9358E+07

0.1381E+08

0.2037E+08

0.3006E+08

0.4434E+08

0.6543E+08

0.9653E+08

absolute value

0.1975E+06

0.2865E+06

0.4156E+06

0.6030E+06

0.8747E+06

0.1269E+07

0.1841E+07

0.2671E+07

0.3874E+07

0.5621E+07

0.8154E+07

0.1183E+08

0.1716E+08

0.2490E+08

0.3612E+08

0.5239E+08

0.7601E+08

absolute value

Electric field strength near tip, first and 7th iteration


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0.0000E+00

0.1058E+03

0.2117E+03

0.3175E+03

0.4233E+03

0.5292E+03

0.6350E+03

0.7408E+03

0.8467E+03

0.9525E+03

0.1058E+04

0.1164E+04

0.1270E+04

0.1376E+04

0.1482E+04

0.1588E+04

0.1693E+04

0.0000E+00

0.8859E+02

0.1772E+03

0.2658E+03

0.3544E+03

0.4429E+03

0.5315E+03

0.6201E+03

0.7087E+03

0.7973E+03

0.8859E+03

0.9745E+03

0.1063E+04

0.1152E+04

0.1240E+04

0.1329E+04

0.1417E+04

n E L2() 2/0L2() E + 2/0L2() Emax1 1.2160E+04 1.2068E+03 1.1657E+04 1.0297E+08

2 1.0299E+04 1.1297E+03 9.8192E+03 8.2916E+07

3 1.0067E+04 1.1132E+03 9.5929E+03 8.0682E+07

4 1.0117E+04 1.1163E+03 9.6416E+03 8.1190E+07

5 1.0095E+04 1.1141E+03 9.6202E+03 8.0963E+07

I AMf Least-Squares error indicator 26

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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

I AMf Conclusion/ to do 27

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