Computation Electromagnetics Some areas of...

Computational Electromagnetics Definitions, applications and research

Luis E. Tobón Pontificia Universidad Javeriana

Seminario de investigación Departamento de Electrónica y Ciencias de la Computación

November 30, 2012

Outline

• Definition and some areas of application • Maxwell´s equations and numerical methods • Finite Elements Method

– Discretized EM – The De Rham diagram – Basis functions in tetrahedral element – Galerkin’s method: FETD based on E and H fields – Cases: Heat sink and MW filters – Domain Decomposition: Discontinuous Galerkin’s method – Implicit time integration: Block-Thomas Crank-Nicholson (BT-CN) method – Cases: MW filters and On-Chip interconnection

• Current work: – Improved BT-CN method – FETD based on E and B fields

• Conclusions and future work

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Computational Electromagnetics Definition

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Computational Electromagnetics is the process of modeling the interaction of

electromagnetic fields with physical objects and the environment. Wiki

Computational Electromagnetics deals with the art and science of solving Maxwell´s equations numerically using computers. Jian-Ming Jin

It is used to analyze: • Antenna performance • EM compatibility • EM Wave Propagation • EM devices (RF, MW, photonics)

Areas of Application Devices

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Metamaterials Active and passives Microstrip Antennas

Areas of Application Signal Integrity

Real Scheme

Model

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Areas of Application EM compatibility

Emission spectrum spreading for new generation components

Continuous decrease of power supply voltages.

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Areas of Application EM imaging and sensors

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Oil exploration

acoustics.org

Landmine detection

Computational electromagnetics Maxwell’s equations

Topological Laws Constitutive Laws

James Clerck Maxwell (1831-1879)

Oliver Heaviside (1850 – 1925)

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From Continuous to Discrete!

1. Which fields must be selected? E? H? D? B?

2. What is the correct discrete representation of these fields?

3. What is the definition of discrete constitutive laws?

4. What is the numerical dispersion of these schemes?

Computational electromagnetics Some methods

• Time Domain – FDTD Directly PDE from Maxwell’s Equation (Yee’s cell)

– FETD Weak form of Maxwell’s Equation

• Frequency Domain – FDFD FD in Frequency domain

– FEM Weak form of Helmholtz’ Equation

– MoM Volume and Surface Integral Equations (Electric and Magnetic)

• Hybrid Techniques – FDTD-FETD

– MoM-FEM

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Finite Elements Method

• The De Rham Diagram relates function from the Hilbert-Sobolev spaces by means of differential operators

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Discretized Maxwell’s Equations

Continuous Laws

Co

nst

itu

ve

Top

olo

gica

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Discrete Representation

1-forms

Curl-Conforming

2-forms

Div-Conforming

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Discretized Maxwell’s Equations

Basis Functions Curl-Conforming Tetrahedral element

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6 Edge BF


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12 Edge BF


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8 Face BF

Basis Functions Div-Conforming

Tetrahedral element

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4 Face BF


Tetrahedral element

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12 Face BF


Tetrahedral element

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3 Volume BF

Basis Functions The DeRham diagram

Tetrahedral element

B

D

E

H

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Computational electromagnetics FETD, Galerkin’s method

Weak form of Maxwell’s equations

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Computational electromagnetics Discretization using tetrahedrons

sop.inria.fr

cst.com

Edge basis functions (Ct/Ln)

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Computational electromagnetics Case 1. Heat Sink, model

150 mm

150 mm

60 mm

4 mm

48 mm 48 mm

5 mm

Source: (0,0,-4) mm Ez BHW Fo = 4 GHz

Observer Ez: (63,63,45) mm

PEC cavity

Fmax 10 GHz

30 mm

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Computational electromagnetics Case 1. Heat Sink, discretization and results

Matlab Model using brick elements

Good agreement between commercial software and our results

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Computational electromagnetics Case 3. Strip line

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58.4 mm

110 mm

r=4.8 W=5.7 mm H=3.18 mm T=0.32 mm Zo=50 L=80 mm

16.5 mm

fMax=10 GHz

Source

Voltage in Port 2

Good agreement S21

Computational electromagnetics Case 4. -filter, results

fMax=10 GHz

58.4 mm

110 mm

16.5 mm

r=4.8 W=5.7 mm H=3.18 mm T=0.32 mm Zo=50 L=80 mm

5.7 mm

2.2 mm

7.5 mm

7.5 mm

17.5 mm

7.5 mm

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Computational electromagnetics Case 4. -filter, model

Low Pass Band Pass

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Computational electromagnetics Multi-scale problems

Challenges:

• Spatial discretization – FDTD: too many unknowns – FETD: inversion or factorization of large system matrices

• Time integration

– explicit scheme: very small time steps – implicit scheme: inversion or factorization of large matrices

www.imec.be

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Introduction

coarse subdomains fine subdomains

Domain decomposition for multiscale structures

Explicit RK

Implicit CN-BT CN-GS

Hybrid IMEX

LocalTS

Low to High

High

Low

Brick

Tetra

Prism

Hexa

/Sm

alle

st

Larg

est

10

105

103

10

100

10-1

Multiscale Factor = Largest /Smallest

Maxwell’s equations Galerkin’s weak form

perform integration by parts

surface integration around subdomain 29

Domain Decomposition Method Discontinuous Galerkin FETD

Galerkin’s weak form with integration by parts

Riemann solver for interface between adjacent subdomains

surface integration

Domain Decomposition Method Riemann solver

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Large system matrices are divided into several middle sized matrices by the hybrid SETD/FETD method

31 5 X 5 X 4 subdomains Interfaces between subdomains

Domain Decomposition Method Discretized system of equations

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Domain Decomposition Method Time integration, Crank-Nicholson for sequential domains

(i-1)-th subdomain (i+1)-th subdomain i-th subdomain

Reflections

Transmissions

Sequential order of subdomains:

Crank-Nicholson implicit method:

Block diagonal!!

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Domain Decomposition Method Time integration, Block-Thomas Crank-Nicholson method

1. Block LU decomposition

2. Solve for L (forward)

3. Solve for U (Backward)

0.2mm

0.8mm

0.1mm

PEC Cavity

Port 1. Active Fmax 30 GHz

Port 2. Passive

4.9 mm

Microstrip Z0=50W=0.065mm T=0.67 m

Dielectric: Duroid

r=2.2

Capacitor 0.065mm x 0.06 mm x 0.08 m

0.013 mm

0.013 mm

Chip Inside:

multiscale factor = 10000 34

Domain Decomposition Method Case 4. Microwave filter High Pass

10 times less unknowns than

FDTD

FDTD

FDTD

4 times faster than HFSS 8 times faster than CST 9 times faster than FDTD 35

Domain Decomposition Method Case 4. Microwave filter High Pass

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Thickness of plates in layers 1, 2 and 3 is 6 m.

Multiscale Factor

Domain Decomposition Method Case 5. High Q Band Pass Microwave filter

In Port 1

In Port 2

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Resonant frequency li fr 0.45mm 1.34GHz 0.65mm 1.22GHz 0.85mm 1.14GHz

li

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This analysis takes less than 1.5 hours, 5.9 hours for one simulation using FDTD

Resonance Tunning


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multiscale factor = 667

Real model

Simplified model

Domain Decomposition Method Case 6. Interconnect Layered structure

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FDTD grid

PPW=40

cells: 511 X 323 X 60

total DoF: > 50 million

SETD / FETD mesh

PPW=40

44 subdomains

total DoF: 152,356


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• Relatively big difference for

S31 and S41

– S31 and S41 are very small

quantities (< -50 dB)

– Interfaces bring artificial

dissipation and dispersion


Domain Decomposition Method Case 7. Packaging-to-Chip interconnect

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GND

Connectors

IC

Port 1

Port 2

Port 3

Port 4

Port 5

Port 6

Active port: Port 1 (50 Ohms)

Passive port: Port 4 (50 Ohms)

Vs: BHW fc=2.6 GHz

11 mm 6 mm

Domain Decomposition Method Case 7. Packaging-to-Chip interconnect

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SPrism DG-FETD

Total DoF: 69548

CPU time: 9 min

Mem. Cost: 192 MB

FDTD

Total DoF: 1.4 M

CPU Time: 36 min

HFSS (30 freq.)

CPU time: 11:26 min

Mem. Cost: 66 M

7 Layer-Domains

Optimization BT-CN

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LU decomposition and Block-Thomas

1 2

S1 S2 M1 M2 T12

T21

L U

3 S3 M3

T32

T23

New LDU Decomposition

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LDU decomposition

1 2

S1 S2 M1 M2 T12

T21

3 S3 M3

T32

T23

Volumes

Interfaces

No Transpose

Volume

Interface

Surface to volume

Volume to surface

Connection between interfaces in same domain. Usually are zeros

New LDU Decomposition

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1 2

S1 S2 M1 M2 T12

T21

3 S3 M3

T32

T23

Volumes

Interfaces BT Volume source to interfaces

Interfaces source to volume

LDU-Block decomposition

1

2

3

Advantages: 1. Highly parallelizable 2. Smaller matrices 3. Memory cost 4. CPU time 5. General formulation ?

New LDU Decomposition, algorithm

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0. Pre-Processing:

Solve and store:

1. Algorithm Volume to interface:

Parallel

Parallel

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2. Algorithm Interface solution:

No needed

1. Solved as a whole 2. Apply BT


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1 2

S1 S2 M1 M2 T12

T21

3 S3 M3

T32

T23

Sparse Sparse LUPQR decomposition

3. Algorithm Interface to Volume:


Models

Total 42555 SD1: 21214 SD2: 21341

Total 63849 SD1: 21214 SD2: 21421 SD3: 21214

Total 85270 SD1: 21214 SD2: 21421 SD3: 21421 SD4: 21214

Total 106691

Total 128112

Total 385k

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Cases of study

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Field accuracy case 1

Perfect agreement

Fmax = 670 MHz 20 ppw

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Field accuracy case 2

Perfect agreement

Accuracy is not an issue

Fmax = 670 MHz 20 ppw

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Computational cost, Memory DoF per SD fixed, Number of SD changed

Out of memory

It is not the limit 4 times less memory

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Computational cost, time DoF per SD fixed, Number of SD changed

The new method is always faster

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Computational cost, memory Number SD fixed, DoF per SD changed

No a general solution!!

> 4k DoF on interface

Out of memory < 3k Interface linear system is solved as a whole

Block-Thomas algorithm for interface linear system

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Computational cost, memory Block-Thomas for Interface Linear System

> 6k DoF on interface

320 MB 376 s

590 MB 726 s

No limit of memory, yet More study is required

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Maxwell’s Equations

Continuous Laws

Co

nst

itu

ve

Top

olo

gica

l

Discrete Representation

1-forms

Curl-Conforming

2-forms

Div-Conforming

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Discrete Maxwell’s Equations Wave Equation Formulation

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Discrete Maxwell’s Equations EH Formulation

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Sparse square matrices length(h) >> length(e)

Discrete Maxwell’s Equations EB Formulation

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Sparse square matrices size(Mee) ≈ size(Mbb)

62 size(Mee) ≈ size(Mbb)

dense

Discrete Maxwell’s Equations EB-Hodge Formulation

Validation Eigenvalues

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=2.5mm

1.0 cm

0.75 cm

0.5 cm

Mode Analytical

result (GHz) E1H2 (GHz)

Error (%) E1B1 (GHz) Error (%) E2B2 (GHz) Error (%)

TE101 24,9830 24,3452 2,5529 24,7779 0,8210 24,9871 -0,0164

TM110 33,5191 31,9491 4,6839 32,9020 1,8410 33,5275 -0,0251

TE011 35,9334 34,1833 4,8704 35,3638 1,5852 36,0425 -0,3036

TE201 35,9334 34,3456 4,4187 35,5245 1,1379 36,0542 -0,3362

TM111 39,0252 36,5466 6,3513 38,0133 2,5929 39,0603 -0,0899

TE111 39,0252 37,3066 4,4038 38,7656 0,6652 39,0767 -0,1320

TM210 42,3986 39,5894 6,6257 41,5081 2,1003 42,4103 -0,0276

TE102 42,6897 40,6803 4,7070 42,9204 -0,5404 42,6889 0,0019

Large Error E DoF: 309 H DoF: 2770

Same as Wave E DoF: 309 B DoF: 818

Same as Wave E DoF: 1910 B DoF: 3552

Eigenvalues Maxwell´s equations

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First Mode Second Mode

Transient solutions Maxwell´s equations, Explicit

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Transient solutions Maxwell´s equations, Explicit

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Transient solutions Maxwell´s equations, Implicit

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Conclusions

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• FETD for Mawxell´s equations was defined.

• Correct basis functions to approximate E, B, H, and D fields according with the De Rham diagram were presented.

• Efficient and accurate locally implicit DG-FETD schemes have been discussed: – The spatial discretization is based on discontinous Galerkin’s method – The time stepping consists of the Crank-Nicolson method with free-iterative Block-

Thomas algorithm.

• It was showed the DG-FETD’s capacity of solving large systems and layered structures for multiscale simulations.

• Implicit time integration for sequential domains is improved performing a new memory efficient and highly paralellizable LDU decomposition.

• A new implementation of FEM based on E-B fields shows improvements in accuracy and computational costs, for both frequency and time responses.

Nest work

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1. Numerical dispersion analysis of FETD based on EB fields

2. Implementation of EB-scheme in hexahedral and prismatic elements

3. Realistic cases of application

1. On-chip

2. Oil exploration

3. Photonic device (photonic crystal or metamaterial)

4. Writing

Acknowledgments

Prof. Qing Liu’s group Dr. Jiefu Chen Pratt School of Engineering Duke University Pontificia Universidad Javeriana, Cali Universidad del Quindío Fulbright Colciencias Intel Co. Family and friends!!!

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