introduction to computational electromagnetics the...

Introduction to Computational Electromagnetics

The Finite-Element Method

Jens Niegemann, Lab for Electromagnetic Fields and Microwave Electronics (IFH), ETH Zürich

Outline

1) Brief Reminder of Maxwell‘s Equations

2) Introduction to the Finite-Element Method in 1D

3) Practical Computation of the Matrices

4) Implementation of Boundary Conditions/Example

5) Higher-order Elements in 1D

6) Extension to Triangular Meshes (2D)

7) Spurious Modes and Vector Elements

8) Summary

2

BRIEF REMINDER OF MAXWELL‘S EQUATIONS

3

A Reminder of Maxwell’s Equations

In differential form and in time-domain (dim.-less units):

Constitutive Relations (in frequency-domain):

𝝏

𝝏𝒕𝑫 𝐫 , 𝐭 = 𝜵 × 𝑯 𝐫 , 𝐭 − 𝐉 𝐫 , 𝐭 𝜵 ⋅ 𝑫 = 𝟎

𝝏

𝝏𝒕𝑩(𝐫 , 𝐭) = − 𝜵 × 𝐄(𝐫 , 𝐭) 𝜵 ⋅ 𝑩 = 𝟎

4

𝑫 𝒓,𝝎 = 𝝐 𝒓,𝝎 𝑬 𝒓,𝝎 Isotropic, dispersive media

𝑩 𝒓 = 𝑯(𝐫 ) Here: No magnetic response!

A Reminder of Maxwell’s Equations

Transformation to frequency domain: 𝐸 𝑟 , 𝑡 = 𝐸 𝑟 𝑒−𝑖𝜔𝑡

Continuity at material interfaces:

Tangential components continuous

Normal components discontinuous

5

−𝒊𝝎 𝝐 𝐫 𝐄(𝐫 ) = 𝜵 ×𝑯(𝐫 ) − 𝐉 𝐫 , 𝐭 𝜵 ⋅ 𝝐 𝑬 = 𝟎

−𝒊𝝎 𝑯(𝐫 ) = − 𝜵 × 𝐄(𝐫 ) 𝜵 ⋅ 𝑯 = 𝟎

𝝐𝟏 = 𝟏 𝝐𝟐 = 𝟐

𝐸1 𝐸2

The Vector Wave-Equation

Combining the two curl-equations, we find the vector wave-equation

A similar expression can also be derived for 𝐻(𝑟 )

To have a unique solution, we need to provide boundary conditions:

1. Dirichlet Conditions: 𝐄 = 𝐜𝐨𝐧𝐬𝐭 on boundary 𝜕Ω

2. Neumann Conditions: 𝜕

𝜕𝒏 𝐄 = 𝐜𝐨𝐧𝐬𝐭 on boundary 𝜕Ω

3. Mixed/Cauchy Conditions: 𝜕

𝜕𝒏 𝐄 + 𝜸 𝐄 = 𝐜𝐨𝐧𝐬𝐭 on boundary 𝜕Ω

6

𝜵 × 𝜵 × 𝑬 𝐫 − 𝝎𝟐 𝝐 𝐫 𝐄 𝐫 = 𝒊𝝎𝐉 𝐫 , 𝐭

INTRODUCTION TO THE FINITE-ELEMENT

METHOD IN 1D

7

Reduction to the 1D-Wave-Equation

System translationally invariant along (y,z)-axes

Assume bounded domain

Here, simple Dirichlet (PEC) boundary conditions:

𝑬 𝒙𝟎 = 𝑬 𝒙𝑵 = 𝟎

8

𝝏𝒙𝟐 𝑬 𝒙 + 𝝎𝟐𝝐 𝒙 𝑬 𝒙 = −𝒊𝝎𝑱(𝒙)

𝐱 𝐱𝟎 𝐱𝐍

𝛀

Two Approaches to Finite Elements

Two different approaches:

1. Method of weighted residuals/Galerkin‘s Method

2. Variational/Ritz-Approach

In typical cases, both lead to the same discretization!

9

The Method of Weighted Residuals

Assuming we have an approximate solution 𝐸 𝑥 :

𝝏𝒙𝟐 𝑬 𝒙 + 𝝎𝟐𝝐 𝒙 𝑬 𝒙 + 𝒊𝝎𝑱 𝒙 = 𝐑𝐞𝐬 𝒙 ≠ 𝟎

Project the residual onto a set of test functions Ψ𝑖 and

demand that the projection vanishes

𝐑𝐞𝐬 𝒙 ⋅ 𝚿𝐢 𝐱 𝒅𝒙𝛀

= 𝟎

The residual is orthogonal to the function space spanned

by Ψ𝑖

10

Weighting function

The Method of Weighted Residuals

Inserting the expression for 𝐑𝐞𝐬 𝒙 :

𝝏𝒙𝟐 𝑬 𝒙 + 𝝎𝟐𝝐 𝒙 𝑬 𝒙 + 𝒊𝝎𝑱 𝒙 𝚿𝐢 𝐱 𝒅𝒙

𝛀

= 𝟎

Split into separate parts:

𝝏𝒙𝟐 𝑬 𝒙 𝚿𝐢 𝐱 𝒅𝒙

𝛀

+𝝎𝟐 𝝐 𝒙 𝑬 𝒙 𝚿𝐢 𝐱 𝒅𝒙𝛀

= −𝒊𝝎 𝑱 𝒙 𝚿𝐢 𝐱 𝒅𝒙𝛀

Integration by parts: 𝑢′𝑣 𝑑𝑥 =Ω

− 𝑢′𝑣 𝑑𝑥 +Ω

𝑢 𝑣 𝑥0

𝑥𝑁

− 𝝏𝒙 𝑬 𝒙 𝝏𝒙𝚿𝐢 𝒙 𝒅𝒙𝛀

+𝝎𝟐 𝝐 𝒙 𝑬 𝒙 𝚿𝐢 𝐱 𝒅𝒙𝛀

+ 𝝏𝒙 𝑬 𝒙 𝚿𝐢 𝐱𝒙𝟎

𝒙𝑵= 𝑭𝒊

11

=:𝑭𝒊

Choice of the Approximate Solution

Assume an expansion into basis functions 𝚽𝐣 𝐱

𝑬 𝒙 = 𝒆𝒋 𝚽𝐣 𝐱

𝑵

𝒋=𝟎

Insertion into the projections:

𝝎𝟐 𝝐 𝚿𝐢 𝚽𝐣𝒅𝒙𝛀

− 𝝏𝒙𝚿𝒊 𝝏𝒙𝚽𝐣 𝒅𝒙𝛀

+ 𝚿𝐢 𝝏𝒙𝚽𝐣 𝒙𝟎

𝒙𝑵𝒆𝒋

𝑵

𝒋=𝟎

= 𝑭𝒊

Special choice: 𝚽𝐣 𝐱 = 𝚿𝐣 𝐱 (Galerkin‘s Method!)

12

Matrix Form

Leads to symmetric expressions:

𝝎𝟐 𝝐 𝚽𝒊 𝚽𝐣𝒅𝒙𝛀

− 𝝏𝒙𝚽𝒊 𝝏𝒙𝚽𝐣 𝒅𝒙𝛀

+ 𝚽𝐢 𝝏𝒙𝚽𝐣 𝒙𝟎

𝒙𝑵𝒆𝒋

𝑵

𝒋=𝟎

= 𝑭𝒊

Can be written as a matrix-equation:

𝝎𝟐𝑴− 𝑺 + 𝑮 𝒆 = 𝑭

Without source, we have a generalized eigenvalue problem

𝑺 − 𝑮 𝒆 = 𝝎𝟐𝑴 𝒆

13

=:𝑴𝒊𝒋 =: 𝑺𝒊𝒋 =:𝑮𝒊𝒋

Mass-Matrix Stiffness-Matrix

PRACTICAL COMPUTATION OF THE MATRICES

14

Choice of the basis functions

With finite elements: Looking for a local basis, i.e. we split

the system into small (finite) elements

Introduce a mapping from each element Ω𝑖 to a reference

element Ωref

15

𝐱 𝐱𝟎 𝐱𝐍

𝛀𝟏

𝐱𝟏 𝐱𝑵−𝟏 …

𝛀𝑵

𝐱𝐤−𝟏

𝛀𝒌

𝐱𝐤

𝟎

𝛀𝐫𝐞𝐟

𝟏

𝒙𝒌 𝝃 = 𝒙𝒌−𝟏 + 𝝃 𝒙𝒌 − 𝒙𝒌−𝟏 = 𝒙𝒌−𝟏 + 𝝃𝚫𝒌

𝝃 𝒙 =𝒙 − 𝒙𝒌−𝟏

𝚫𝐤

Considering an individual element

Introducing an interpolatory basis

→ Blackboard

16

𝟎

𝛀𝐫𝐞𝐟

𝟏

𝑬𝟎𝒌 𝑬𝟏

𝒌

Mapping local to global coefficients

Place nodes at interface between elements

Mapping from local dofs to global dofs 𝑬𝟎(𝟏)

𝑬𝟏(𝟏)

𝑬𝟎(𝟐)

𝑬𝟏(𝟐)

𝑬𝟎(𝟑)

𝑬𝟏(𝟑)

𝑬𝟎(𝟒)

𝑬𝟏(𝟒)

=

1 0 0 0 00 1 0 0 00 1 0 0 00 0 1 0 00 0 1 0 00 0 0 1 00 0 0 1 00 0 0 0 1

𝐸0𝐸1𝐸2𝐸3𝐸4

17

𝑬𝟎(𝟏)

𝑬𝟏(𝟏)

𝐱 𝐱𝟎 𝐱𝟒 𝛀𝟏 𝐱𝟏 𝐱𝟑 𝒙𝟐 𝛀𝟒 𝛀𝟐 𝛀𝟑

𝑬𝟎(𝟐)

𝑬𝟏(𝟐)

𝑬𝟎(𝟑)

𝑬𝟏(𝟑)

𝑬𝟎(𝟒)

𝑬𝟏(𝟒)

𝑬𝟎 𝑬𝟏 𝑬𝟐 𝑬𝟑 𝑬𝟒

Construction of the global basis

Expansion:

𝑬 𝒙 = 𝑬𝒋 𝚽𝐣 𝐱

𝑵

𝒋=𝟎

= 𝚽𝟎𝐤 𝑬𝟎

𝒌 +𝚽𝟏𝐤 𝑬𝟏

𝒌

𝑵

𝒌=𝟏

Map to degrees of freedom (dofs):

𝑬 𝒙 = (𝚽𝟎

(𝟏) 𝑬𝟎 +𝚽𝟏

(𝟏) 𝑬𝟏) + (𝚽𝟎

(𝟐)𝑬𝟏 + 𝚽𝟏

(𝟐) 𝑬𝟐) + ⋯+ (𝚽𝟎

(𝐤)𝑬𝒌−𝟏 + 𝚽𝟏

(𝒌) 𝑬𝒌)

Collect global dofs

𝑬 𝒙 = 𝚽𝟎

(𝟏) 𝑬𝟎 + (𝚽𝟏

𝟏+𝚽𝟎

𝟐)𝑬𝟏 +⋯+ (𝚽𝟏

(𝐤−𝟏)+𝚽𝟎

(𝐤))𝑬𝒌−𝟏 + 𝚽𝟏

(𝒌) 𝑬𝒌

18

Construction of the global basis

Expansion:

𝑬 𝒙 = 𝑬𝒋 𝚽𝐣 𝐱

𝑵

𝒋=𝟎

= 𝚽𝟎𝐤 𝑬𝟎

𝒌 +𝚽𝟏𝐤 𝑬𝟏

𝒌

𝑵

𝒌=𝟏

→ Blackboard

19

Final Assembly of the global matrices

With the local matrices at hand, we can easily construct the

global matrices.

General shape: Tri-diagonal because of overlap

𝑴 =

𝑿 𝑿 𝟎 𝟎 𝟎𝑿 𝑿 𝑿 𝟎 𝟎𝟎 𝑿 𝑿 𝑿 𝟎𝟎 𝟎 𝑿 𝑿 𝑿𝟎 𝟎 𝟎 𝑿 𝑿

20


Iterate over all elements and add local matrices at the right

place:

𝑴 =

𝑴𝟏,𝟏(𝟏) 𝑴𝟏,𝟐

(𝟏)𝟎 𝟎 𝟎

𝑴𝟐,𝟏(𝟏)

𝑴𝟐,𝟐(𝟏)

𝑿 𝟎 𝟎

𝟎 𝑿 𝑿 𝑿 𝟎𝟎 𝟎 𝑿 𝑿 𝑿𝟎 𝟎 𝟎 𝑿 𝑿

21



place:

𝑴 =

𝑴𝟏,𝟏(𝟏) 𝑴𝟏,𝟐

(𝟏)𝟎 𝟎 𝟎

𝑴𝟐,𝟏(𝟏)

𝑴𝟐,𝟐(𝟏)

+𝑴𝟏,𝟏(𝟐)

𝑴𝟏,𝟐(𝟐)

𝟎 𝟎

𝟎 𝑴𝟐,𝟏(𝟐)

𝑴𝟐,𝟐(𝟐)

𝑿 𝟎

𝟎 𝟎 𝑿 𝑿 𝑿𝟎 𝟎 𝟎 𝑿 𝑿

22



place:

𝑴 =

𝑴𝟏,𝟏(𝟏) 𝑴𝟏,𝟐

(𝟏)𝟎 𝟎 𝟎

𝑴𝟐,𝟏(𝟏)

𝑴𝟐,𝟐(𝟏)

+𝑴𝟏,𝟏(𝟐)

𝑴𝟏,𝟐(𝟐)

𝟎 𝟎

𝟎 𝑴𝟐,𝟏(𝟐)

𝑴𝟐,𝟐(𝟐)

+𝑴𝟏,𝟏(𝟑)

𝑴𝟏,𝟐(𝟑)

𝟎

𝟎 𝟎 𝑴𝟐,𝟏(𝟑)

𝑴𝟐,𝟐(𝟑)

𝑿

𝟎 𝟎 𝟎 𝑿 𝑿

23



place:

𝑴 =

𝑴𝟏,𝟏(𝟏) 𝑴𝟏,𝟐

(𝟏)𝟎 𝟎 𝟎

𝑴𝟐,𝟏(𝟏)

𝑴𝟐,𝟐(𝟏)

+𝑴𝟏,𝟏(𝟐)

𝑴𝟏,𝟐(𝟐)

𝟎 𝟎

𝟎 𝑴𝟐,𝟏(𝟐)

𝑴𝟐,𝟐(𝟐)

+𝑴𝟏,𝟏(𝟑)

𝑴𝟏,𝟐(𝟑)

𝟎

𝟎 𝟎 𝑴𝟐,𝟏(𝟑)

𝑴𝟐,𝟐(𝟑)

+𝑴𝟏,𝟏(𝟒)

𝑴𝟏,𝟐(𝟒)

𝟎 𝟎 𝟎 𝑴𝟐,𝟏(𝟒)

𝑴𝟐,𝟐(𝟒)

24


Stiffness matrix is assembled similarly!

What is still missing is the matrix G = 𝚽𝐢 𝝏𝒙𝚽𝐣 𝒙𝟎

𝒙𝑵

𝑮 =

𝑮𝟎,𝟎 𝑮𝟎,𝟏 𝟎 𝟎 𝟎

𝟎 𝟎 𝟎 𝟎 𝟎𝟎 𝟎 𝟎 𝟎 𝟎𝟎 𝟎 𝟎 𝟎 𝟎𝟎 𝟎 𝟎 𝑮𝑵,𝑵−𝟏 𝑮𝑵,𝑵

Plays a certain role for some boundary conditions!

25

IMPLEMENTATION OF BOUNDARY CONDITIONS

26

Dirichlet Boundary Conditions

Assuming PEC boundary conditions: 𝐸 𝑥0 = 𝐸 𝑥𝑁 = 𝑐

𝑿 𝑿 𝟎 𝟎 𝟎𝑿 𝑿 𝑿 𝟎 𝟎𝟎 𝑿 𝑿 𝑿 𝟎𝟎 𝟎 𝑿 𝑿 𝑿𝟎 𝟎 𝟎 𝑿 𝑿

𝑬𝟎𝑬𝟏𝑬𝟐𝑬𝟑𝑬𝟒

=

𝑭𝟎𝑭𝟏𝑭𝟐𝑭𝟑𝑭𝟒

To enforce boundary conditions, we replace the r.h.s. with

the fixed value and modify the equation

27



𝟏 𝟎 𝟎 𝟎 𝟎𝑨 𝑿 𝑿 𝟎 𝟎𝟎 𝑿 𝑿 𝑿 𝟎𝟎 𝟎 𝑿 𝑿 𝑩𝟎 𝟎 𝟎 𝟎 𝟏

𝑬𝟎𝑬𝟏𝑬𝟐𝑬𝟑𝑬𝟒

=

𝒄𝑭𝟏𝑭𝟐𝑭𝟑𝒄

Matrix is no longer symetric, but can be fixed!

28



𝑿 𝑿 𝟎𝑿 𝑿 𝑿𝟎 𝑿 𝑿

𝑬𝟏𝑬𝟐𝑬𝟑

=𝑭𝟏 − 𝒄𝑨

𝑭𝟐𝑭𝟑 − 𝒄𝑩

Matrix is symetric again!

The fixed unknowns can be removed from the system. Can

save significant number of dofs in 2D and 3D

29

HIGHER-ORDER ELEMENTS IN 1D

30

Considering an individual element

Instead of linear interpolation, we can also use quadratic

On the reference element, we now need three nodes!

→ Blackboard

31

𝑬𝟎(𝟏)

𝑬𝟐(𝟏)

𝐱 𝐱𝟎 𝐱𝟒 𝛀𝟏 𝐱𝟏 𝐱𝟑 𝒙𝟐 𝛀𝟒 𝛀𝟐 𝛀𝟑

𝑬𝟎 𝑬𝟏 𝑬𝟐 𝑬𝟑 𝑬𝟒

𝟎 𝛀𝐫𝐞𝐟 𝟏

𝑬𝟎𝒌 𝑬𝟐

𝒌 𝑬𝟏𝒌

𝑬𝟏(𝟏)

Plot of the basis functions

32

Comparison of Convergence

33

Convergence: 𝐸 − 𝐸 ≤ 𝑐 ℎ𝑝+1

EXTENSION TO TRIANGULAR MESHES (2D)

34

TM-Polarization

In TM-Polarization, we can describe the entire dynamics

via the 𝐸𝑧(𝑥, 𝑦)-field

Galerkin-Procedure analogous to 1D

35

𝝏𝒙𝟐 𝑬 𝒙, 𝒚 + 𝝏𝒚

𝟐 𝑬 𝒙, 𝒚 + 𝝎𝟐𝝐 𝒙, 𝒚 𝑬 𝒙, 𝒚 = −𝒊𝝎𝑱(𝒙, 𝒚)

→ Blackboard

Elements in 2D

Typically, one uses triangular or quadrilateral elements

First step: Mesh generation

Similarly to 1d-case, we need

1. Mapping to reference element

2. Local matrices

3. Mapping global to local nodes

→ Blackboard

36

Elements in 2D

Assembly is almost identical to the 1D-Case, but matrix not

tri-diagonal!

Matrices are much larger now: Use specialized sparse

matrix solvers or iterative methods!

Convergence properies as before: 𝐸 − 𝐸 ≤ 𝑐 ℎ𝑝+1

It is also possible to use curved elements!

37

SPURIOUS MODES AND VECTOR ELEMENTS

38

Vectorial Problems

In 3D, we can no longer reduce the problem to a scalar

field:

Idea: Repeat all steps as before, but use basis-functions

for each component independently:

Φ𝑖 ∈Φ𝑖

00

,0Φ𝑖

0,

00Φ𝑖

Freitag, 16. März 2012 39

𝜵 × 𝜵 × 𝑬 𝐫 − 𝝎𝟐 𝝐 𝐫 𝐄 𝐫 = 𝒊𝝎𝐉 𝐫 , 𝐭

Problems with this approach

Does not work very well: In eigenvalue problems one find

spurious modes.

The physical reason is, that we are not solving the full

Maxwell‘s equations, but only the curl-equations.

By taking the divergence of the wave-equation

we find:


𝜵 × 𝜵 × 𝑬 𝐫 = 𝝎𝟐 𝝐 𝐫 𝐄 𝐫

𝝎𝟐 𝛁 𝝐 𝐫 𝐄 𝐫 = 𝟎

Reasons for the Spurious Modes

For 𝜔 ≠ 0, we have the divergence condition build in!

But, for 𝜔 = 0, we will also find solutions, which are

unphysical!

The null-space is of infinite size, since every function

is a solution for 𝜔 = 0.


𝑬 𝒓 = 𝛁𝚽

Poor Numerical Approximation of the Null-Space

The large null-space in itself is not a problem!

Due to poor numerical approximation, we find spurious

non-zero eigenvalues inbetween the correct values

Mathematically, the underlying problem is related to the

discontinuity of the fields at interfaces!

With our current basis functions, we enforce continuity

since we share field-values on nodes or edges!


A simple example

Physical Modes of a rectangular cavity


A simple example

Modes in the null-space of a rectangular cavity


Solution

The common solution: Use more suitable basis functions

Here, Whitney elements (vector elements):


→ Blackboard

Plot of the First-Order Whitney Elements

Freitag, 16. März 2012 46 Departement/Institut/Gruppe

SUMMARY

47

Summary

Finite-Element Method (FEM) is a very powerful and highly

accurate volume method

It leads to linear systems of equations with very sparse

matrices which are easily assembled

For vectorial problems, one should work approp. elements

There is a large variety of commerical (Comsol, JCMWave,

HFSS, Microwave Studio) and open-source frameworks

(Deal.II, FreeFem, Dune, …) available!

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introduction to computational electromagnetics the...

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