3.3 Boundary Conditions - uni- · PDF file3.3 Boundary Conditions Open space ... this method has the disadvantage that the geometric ... first need to study the properties of the system

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Ursula van Rienen, Universität Rostock, FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

3.3 Boundary ConditionsOpen space has to be limited to a computational domain

Boundary conditions are needed on it‘s boundary

{ }tan

tan

- electric boundary condition: 0, ,- magnetic boundary condition: 0, Im ,- open boundary condition - periodic boundary condition

EH µ

σ= → ∞= → ∞

CST Microwave Studio ®

The open space holding the problem region under study and its surrounding has to be limited to a computational domain. This is an arbitrary limitation but necessary in all methods which work with a spatial discretization. As soon as a boundary (of the computational domain) has been fixed we need to specify boundary conditions which represent the relation between the fields outside and inside of the computational domain as accurate as possible. The most usual boundary conditions are:• electric boundary condition: Etan = 0, σ ∞, (i.e. the tangential electric field vanishes)• magnetic boundary condition: Htan = 0, Im(µ ) ∞, (i.e. the tangential magnetic field vanishes)• open boundary condition (the radiation to the external space or an infinitely long waveguide, etc.)• periodic boundary condition for periodically continuing structures

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u v

w

G

G( )w k( )w k

( )vb n nP

nP ( )ve n

( )v j ( )v j

3.3.1 Electric Boundary Condition

t n: 0E Bσ → ∞ = =

The electric boundary condition occurs at the transition to perfectly conducting material. The electric conductivity σ tends to infinity. In that case, the tangential component of the electric field E vanishes (as well as the normal component of the magnetic flux density B).The electric boundary condition can easily be implemented in the discretizedequation, i.e. the Maxwell-Grid-Equations:It is only necessary to set the corresponding rows and columns in the matrices to zero.

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Electric Boundary Condition

on

0on

0

t

n A

E G

dB Gdt

∂ ⋅ ⋅ ↓ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ∂ → = − ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

D b

C SD e

t n: 0E Bσ → ∞ = =

This procedure of setting rows and columns to zero is demonstrated here for the example of the discretized induction law: We have to set the columns related to tangential electric field components and the rows related to normal magnetic flux density components in matrix C equal to zero as depicted above.In order not to destroy the band structure of the matrix, these rows and columns remain.

Often the electric boundary condition is also used as symmetry condition: If, by symmetry reasons, we know that the electric field in some grid plane only has normal field components we can replace this plane by an electric boundary and solve the smaller structure which, of course, saves cpu time and storage place.

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3.3.2 Magnetic Boundary Condition

u v

w

G

G( )w k( )w k

( )vb n nP

nP ( )ve n

( )v j ( )v j

t nIm( ) : 0H Dµ → ∞ = =

With the magnetic boundary condition the tangential component of the magnetic field and the normal component of the electric displacement (no surface charges assumed) both vanish. Physically, this corresponds to very high magnetic losses (ideally infinitely high).

Yet, the most frequent application of this boundary condition is with symmetry planes.

In principal, we have two possibilities to implement the magnetic boundary condition:

• The boundary of the computational domain G can be cut short to the dual grid. By that, normal field components Dn result which are perpendicular to the boundary such that similarly to the electric boundary condition the corresponding matrix entries can be set to zero.

Yet, this method has the disadvantage that the geometric modelling now depends on the boundary condition which involves the necessity of re-meshing for different boundary conditions.

• Therefore, usually it is favourable to keep the grid G in it’s original definition. But then, in order to compute the normal D-field components we have to solve a surface integral where we need to integrate along the boundary of half a usual elementary area since the other half lies outside of the grid.

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Magnetic Boundary Condition

20tH ≡ Imµ ≡∞

2nH1nH

1tH

GGu∆

wE2

v∆

v

u

( )

1 1 2 2

1 1 2 2

2 2

= since 02

v vt u n t u n

vt u n n t

H d s

H H H H

H H H H

⋅

∆ ∆= ∆ + − ∆ −

∆∆ + − ≡

∫

( ) ( ) ( )Symmetric field values: 0 0y y yE x E x E x= − − ⇒ = =

( ) ( ) ( )0Symmetric derivative: 0x x xE x E x E x

x x x∂ ∂ − ∂ =

= ⇒ =∂ ∂ ∂

We have two possibilities to carry out this integration: 1. Integration over half the area, i.e. the integration is carried out as shown

above.2. We can exploit the following symmetry: an electric or magnetic boundary

condition may also be interpreted as symmetry condition of the corresponding field:

• If a symmetric field value shall be prescribed we can also reach this by setting the field value at zero to zero.

• If a symmetric value of the derivative shall be prescribed we can also reach this by setting the derivative value at zero to zero.

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Magnetic Boundary Condition

2tH

Im µ ≡ ∞

2nH1nH

1tHGG

wEv

u

( )1 2

1 2

1

1

2

1 22 sincet u n v n v

n

t u

t v t tnu

H H

H

H d s

H H

HH H H

→ ⋅

= + ∆ − ∆

= + − ∆

∆ − ∆

⋅ ∆ = −

∫

These relationships allow us to extend the dual area outside of grid G in order to get a full area. The resulting scheme is displayed above.The tangential magnetic field on the boundary of G is vanishing. Thus the tangential magnetic field is symmetric with respect to this boundary and we may set Ht1 = - Ht2.We rewrite the contour integral using this relation.Finally we only need to multiply the result by ½ since we had artificially enlarged the contour path (by enlarging the area). This yields, of course, the same result as derived before.

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3.3.3 Open Boundary Condition

In practical applications we often want to study structures which cannot be limited by an ideal electric or ideal magnetic material – their fields extend to infinity. An example for this is the field of a point charge which tends to zero only in infinity.Since our discrete solution space, the grid, always has to be finite we need to deal with those cases where we cannot cover the whole continuous solution domain by a discrete grid.Also, it is often desirable to save cpu time and storage space by solving the discrete problem only for a part of the structure where the main interaction is taking place. The rest domain (the external domain) is taken care of by appropriate boundary conditions. Examples for that are e.g. rf devices which couple to a waveguide. The most simple method to deal with open structures is to choose a computational domain as large that the feedback of the boundaries on the near fields in and around the structure under study are negligible. The big disadvantage of this method is that a major part of the computing resources are used to compute fields which are far off the structure under study!It is more convenient to simulate the influence of the external domain on the fields inside the computational domain approximately by so-called open boundary conditions.These reflections will be discussed in more detail later with help of examples.

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Solution Strategies

Maxwell's equationse.g. curl

Discretization Maxwell-Grid-Equations (MGE)e.g.

Derivation of wave equation,Poisson's equation, etc.

Matrix equationse.g.

Discretizatione.g.

usual approach

procedure with FIT

BE t∂= − ∂ S A

ddt

= −CD e D b

1 TA Sε

− =SD D D S Φ q/ϕ ρ ε∆ = −

Often it is not necessary to solve for the complete set of Maxwell‘s equations. A reduction to the solution of only a subset of the equations or of simplified derived equations usually allows for a reduction of the necessary computing effort (and is thus highly desirable). In the next part about the solution of electromagnetic problems we will study simplifications of Maxwell‘s equations possible for certain problem types and we will learn about the suitable solutions methods , each.

The sketch above illustrates the difference between our method for numerical solution of electromagnetic field problems compared to the usual procedure:In the usual procedure the discretization starts off from derived partial differential equations. This implies that specific discretization for each problem type. With FIT the matrices are set up for a chosen grid and only then transformations and simplifications of the Maxwell-Grid-Equations are done.An additional advantage lies in the fact that all properties guarantying consistency are preserved.

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3.4 Static Fields

( )1

0

0 0

µ

E d s

H d s E J d A

D d A dV

B d A

σ

ε

σ

ρ

−

= ⋅ =

= + ⋅ = + ⋅

= ⋅ =

= ⋅ =

∫

∫ ∫∫

∫∫ ∫∫∫∫∫

Ce 0

CM b M e j

SM e q

Sb

Electrostatic

ε

=

=

Ce 0

SM e q1

Magnetostatic

0

µ σ− = +

=

CM b M e j

Sb

In the treatise of time-independent fields Maxwell’s-Grid-Equations simplify to the equations given above.

Obviously we obtain two independent linear systems, • one for the electrostatic fields• one for the magneto-static fields.

The electric grid voltage arises in both systems but in the magneto-static case it is just the excitation and there is no feedback as in the time-varying case.

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3.4.1 Electrostatics

ε

=

=

Ce 0

SM e q

grad TE ϕ= − ↔ =e S Φ

( )( )

0

0T T

Tε

≡

Φ = =

Φ =

C S CS Φ

SM S q ( )div gradε ϕ ρ− =

Grid-POISSON-Equation

/T ε=F

SS Φ q

The electrostatic problem is described by the two matrix equations given above.For it‘s analytical solution, the potential approach is used which can be transferred to the grid as we discussed before.Inserting this approach into the two electrostatic equations yields the set of equations displayed next. The first of those is automatically fulfilled by use of the potential approach. The second equation is called Grid-Potential-Equation. It corresponds to the general (analytical) potential equation.

For constant ε (and assuming for simplicity constant step size ∆ ≡ 1) we get the „Grid-POISSON-Equation“.

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Electrostatics


/

is:- real and symmetric- rank

T

PN I J K

ε=

= ⋅ ⋅

F

SS Φ q

F

We see that the Grid-POISSON-Equation equals an inhomogeneous linear systems which has to be solved by appropriate numerical methods.

As we will treat in more detail in the lecture „Numerical Linear Algebra“, we first need to study the properties of the system matrix F in order to choose a suitable solution method. • The matrix F is real and symmetric.• It‘s rank equals the number of grid points NP = I·J·K.

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Electrostatics

( )u

T T T Tu v w v

w

T T Tu u v v w w

− = = − − − − −

= + +

PF SS P P P P

P

P P P P P P

( )( )( )( )( )( )

11 1 11 1- - - -

- - - -

- - - -

Tu u

Tv v I I I I

Tw w IJ IJ IJ

I I

IJ IJIJ

− + − +

−

≈

+ − +

− + − +

− +

− +

− +

+

+

= =

= =

= = +

I

I D D

I

P P I D I D D D

P P I D I D D D

P P I D

D

I D D D

D

I D D


/T ε=F

SS Φ q

First we want to study the schematic structure of F in order not to avoid the computation of all NP

2 matrix elements and (mainly!) to avoid storing this huge number of elements.

In the following we restrict ourselves to the inner computational domain. There we only need to carry out the computation displayed above.

The complete matrix results summing up the three terms shown above. We see that we obtain a band matrix with the entry “6” on the main diagonal and a “-1” on the six side bands, each, in the distance ± Mu, ± Mv, ± Mw (neglecting the boundaries of the grid).

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Electrostatics

-1

-1

-1

-1

-1

6

-1

uM

vM

wM

T= =F SS

Here, we see the band matrix F with the entry “6” on the main diagonal and a “-1” on the six side bands, each, in the distance ± Mu, ± Mv, ±Mw (neglecting the boundaries of the grid).

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Electrostatics

( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( ) ( )

grad uu u

u

uu u u u

u u

vv v

v

vv v v v

v v

ww w

w

ww w w w

w w

n n MdA n A n

n

n M nn M A n M

n M

n n Mn A n

n

n M nn M A n M

n M

n n Mn A n

n

n M nn M A n M

n M

V n n Q n

ε ϕ ε

ε

ε

ε

ε

ε

ρ

Φ − Φ +− ≈ +

∆

Φ − − Φ− − −

∆ −

Φ − Φ ++

∆

Φ − − Φ− − −

∆ −

Φ − Φ ++

∆

Φ − − Φ− − −

∆ −

= =

∫∫

We obtain the same result when we locally set up the general potential equation for a grid in the inner part of grid G.By summation over all 6 subsections of the area integral the equation given above directly follows.

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Electrostatics

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

u u u u

v v v v

w w w w

n n n M n M n M n M

n M n M n M n M

n M n M n M n M Q n

α α αα αα α

Φ + − Φ − + + Φ +

+ − Φ − + + Φ +

+ − Φ − + + Φ + =

( ) ( )( ) ( ) ( )

( ) ( )

( )( ) ( ) ( )

( ) ( )

( )( ) ( ) ( )

( ) ( )

( ) ( )( ) ( )

with

"6"

"1"

u u uu u u

u u u

v v vv v v

v v v

w w ww w

w w w

uu u

u

n n Mn A n A n M

n n M

n n MA n A n M

n n M

n n MA n A n IJ

n n M

nn M A n

n

ε εα

ε ε

ε ε

εα

−= + − +

∆ ∆ −

−+ − +

∆ ∆ −

−+ − =

∆ ∆ −

+ = − =∆

Re-arrangement yields the equation shown above with the factors α(n), etc. as given there.From this notation further properties of the matrix F may be derived. The most important one besides the symmetry is the positive definiteness of F, i.e. all it‘seigenvalues are real and positive. These properties allow for the usage of a number of solution methods for the system . ( ) =εSM SΦ q

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Dirichlet Boundary Conditions

1n−Φ nΦ 1n+Φ

n I−ΦG

v

u

( ) 0nΦ = Φ ( ) 0,i n− ΦF


/T ε=F

SS Φ q

-1

-1

-1

-1

-1

6

-1

Next we regard how to implement boundary conditions in the general potential equation. The most simple case of some domain with perfectly conducting hull is treated first:Usually, the DIRICHLET boundary condition denotes such boundary conditions which is characterized by fixed, prescribed potential values in the boundary plane. In electrostatics this is equivalent to prescribe that the tangential components of the electric field vanish on the boundary: Et = 0. Let us assume that the perfectly conducting hull has the potential Φ0.

For the example displayed above, let n be a boundary point. In order to embed the boundary condition it is only necessary to bring the known potential to the right hand side such that the rank of the matrix diminishes by one. Yet, proceeding like that we would loose the band structure of the matrix. Thus we proceed as follows, instead:• Set Φ(n) = Φ0 in the vector of unknown potentials, i.e. the solution vector.

• Set all values in the n-th row to zero. This makes the matrix asymmetric.• Carrying out the matrix multiplication all values in the n-th column would get multiplied with the known potential Φ0. Thus, we may put –F(i,n)· Φ0 a priori to the right hand side and set all entries in the n-th column to zero which also brings back the matrix symmetry.• Set the diagonal entry F(n,n) to one.

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Neumann Boundary Conditions4Φ

GG

v

u

0Φ1Φ 2Φ

3Φ

u∆

2v∆

0

4 3

4

3 0

outside

inside

v v

v v

ϕ

ϕ

Φ − Φ∂ ≈ ∂ ∆ →Φ − ΦΦ

∂ ≈∂ ∆

= Φ

( )0 0 1 1 2 2 3 3 54 5 6 6 0"6" " 1" " 1" " " " 1" " 1""3" " 1/ 2" " 1/ 2" " " " 1/ 2" " 1/ 2"

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αα α α αα αΦ − Φ − Φ − Φ − Φ =− − − −

− −−− −

− + Φ−

The NEUMANN boundary condition denotes such boundary conditions characterized by fixed, prescribed normal derivatives of the potential in the boundary plane. In electrostatics this is equivalent with vanishing normal components of the electric field in the boundary plane: En = 0. In the example shown above the partial derivative of ϕ with regard to v is relevant.

First, we enlarge the grid virtually to the outside of the computational domain with the same step size outside as the last step size inside the grid. Additionally, we assume a potential Φ4 outside of the grid with a value which is determined by the condition that it‘s change across the boundary vanishes – as demanded by definition. From this, we directly get that Φ4 = Φ3 and we can write down the factors of our general potential equation.In a last step we divide the factors by 2 in order to have a factor of 1 for the „inner“ potential Φ3 and thus to maintain the symmetric of the matrix. This division by 2 can be reached by setting the step sizes outside the grid to zero.


Open Boundary Conditions

Open boundary condition of first order Open boundary condition of second order

Off-center point charge

Source of pictures: T. Weiland, CAD-Skript, Darmstadt 2002

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Example: Plug

Electric Field

( , , ) grid32 47 39

58,656 grid points

r zϕ −× × =

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3.4.2 Magnetostatics

1µ− =

=

CM b j

Sb 0

curl :B A= = ⇒ = =b Ca Sb SC a 0

1µ− =CM Ca j

The set of two matrix equations displayed on top are the basic equations formagnetostatics. The right hand side shall comprise all current types, i.e. impressed current and conduction current.The second equation is automatically fulfilled if we use a potential approach B = curl A with a vector potential A. A discrete vector is defined which holds all components of the vector potential needed to be defined as primary edge voltages in order to apply the curl operator.Inserting this vector potential approach in the curl equation leads to the equation given above which involves both curl operators – the one on the primary grid as well as the one on the dual grid.As shown previously, in the electrostatic case we had only Np unknowns in the discrete potential vector φ. The approach with the vector potential, appropriate for magnetostatics, affords 3 Np unknowns in the vector related to the vector potential. For some special cases where a 2D discretization is sufficient it may be reduced to 2 Np unknowns.

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Magnetostatics: Hi-scheme

div 0 div grad div iB µ µ Hϕ= → =

divm iq µ H=

m

1. Solve curl with arbitrary, simple, even non-physical2. Determine div .3. Solve div grad = q . This is fully analogous to electrostatics.

4. Compute grad .

i i

m i

i

H J Hq µ H

µ

H H

ϕϕ

==

= −

gradhH ϕ= − : gradi h iH H H H ϕ= + = −

It is desirable to find an approach which allows for magnetostatics, too, to work with a scalar potential instead of the vector potential in order to reduce the problem size from 3 Np to Np.Of course a scalar potential can not lead to a complete description of the magneto static field because this field is not curl-free (inhomogeneity on the right hand side). Yet, we may use the fact that the solution of an inhomogeneous partial differential equation may be written as the general solution of the homogeneous equation plus some special solution of the inhomogeneous equation.For magnetostatics this means that any arbitrary (even non-physical) field which correctly describes the currents (i.e. the inhomogeneity) is appropriate as special inhomogeneous solution. This can be used in order to directly compute such a special solution field with direct method but low numerical effort.For the homogeneous solution we use the scalar potential approach as before in electrostatics since all gradient fields are solutions of a homogeneous curl equation. This scalar potential is then plugged in to the approach H = Hi + Hhwhich is used to replace B in Gauss’ law for magnetism. As result we obtain an inhomogeneous scalar potential equation the solution of which solves our magneto-static problem. In analogy to electrostatics, the inhomogeneity describes the sources of the Hi-field (and thus unphysical “magnetic charges”).The necessary steps 1. – 4. To solve the magneto-static problem are listed above.

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3.3 Boundary Conditions - uni- · PDF file3.3 Boundary Conditions Open space ... this method has the disadvantage that the geometric ... first need to study the properties of the system