prof. dr. jasmin smajic modern numerical methods for ... · smajic, numerical methods for...

University of Applied Sciences of Eastern Switzerland

Prof. Dr. Jasmin Smajic

Modern Numerical Methods for Computational ElectromagneticsInstitute of Energy Technology (IET)HSR - University of Applied Sciences of Eastern Switzerland Oberseestrasse 10, Rapperswil, Switzerland [email protected]

Institute of Electromagnetic Fields (IEF)Swiss Federal Institute of Technology (ETH) Gloriastrasse 35, Zurich, [email protected]

mailto:[email protected]

mailto:[email protected]


Smajic, Numerical Methods for Computational Electromagnetics, June 13, 2016

Outline

2

Introduction

Numerical methods for computing 3-D vector fields

Boundary value problem (BVP) – eddy-current analysis

Boundary value problem (BVP) – wave propagation analysis

Overview: FEM, BEM, MMP, FEM-MMP, DG-FEM

Discussion: Possibilities, advantages, drawbacks, problems, etc.

Applications

Electromagnetic transients in high voltage equipment

Electromagnetic fields in transformers and machines

Microwave and optical devices

Outlook



Introduction

3

Partial Differential Equation (PDE):

Boundary Conditions (BC):

( ) 0,

( ) 0,

Df x x

Bf x x

= ∈Ω

= ∈∂Ω

( )Material: , ,Frequency: ( )

Sources: ( , , )s s s

f

E H J

µ σ ε

Mathematical model

Model parameters

Source



Introduction

4

Ω∂∈=

Ω∈=

xxfB

xxfD

,0)(

,0)(

Model discretization

[ ] bxA =Numerical method

PDE & Boundary Condition

Large linearsystem of equations



Introduction

5



Numerical Methods for Computing 3-D Vector Fields

6

BVP for Eddy-current Analysis1

1J. Smajic, “How to Perform Electromagnetic Finite Element Analysis”, the International Association for the Engineering Modelling, Analysis & Simulation Community, NAFEMS Ltd., Hamilton, UK, 2016.




7

BVP for Stationary Current Distribution1





8

BVP for Wave Propagation Analysis1





9

Vector Finite Element Method (FEM)




10





11





12

Vector Finite Element Method (FEM) 2,3

2H. Whitney, “Geometric integration theory”, Princeton University Press, Princeton, NJ, 1957.

3J. C. Nedelec, “Mixed Finite Elements in R3”, Numer. Meth., Vol. 35, pp. 315-341, 1980.




13


2H. Whitney, “Geometric integration theory”, Princeton University Press, Princeton, NJ, 1957.

3J. C. Nedelec, “Mixed Finite Elements in R3”, Numer. Meth., Vol. 35, pp. 315-341, 1980.




14





15





16





17


𝜎𝜎𝑎𝑎𝑎𝑎𝑎𝑎 = 0 𝑆𝑆/𝑚𝑚 ,𝜎𝜎𝑐𝑐𝑐𝑐 = 3.5 107 𝑆𝑆/𝑚𝑚




18


Nedges=73’276

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 =𝑁𝑁𝑛𝑛𝑛𝑛𝑁𝑁𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒2 100% = 0.022%

𝜅𝜅 𝐵𝐵 = 𝐵𝐵 𝐵𝐵−1 = 8.04 1026

𝜅𝜅 𝐶𝐶 = 𝐶𝐶 𝐶𝐶−1 = 1.91 105

𝜅𝜅 𝐾𝐾 = 𝐾𝐾 𝐾𝐾−1 = 𝟑𝟑.𝟑𝟑𝟑𝟑 𝟏𝟏𝟏𝟏𝟐𝟐𝟐𝟐

𝑩𝑩𝒊𝒊𝒊𝒊 𝑪𝑪𝒊𝒊𝒊𝒊





19


Unrealistic improvement:





20





21


Nedges=73’276

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 =𝑁𝑁𝑛𝑛𝑛𝑛𝑁𝑁𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒2 100% = 0.022%

𝜅𝜅 𝐵𝐵 = 𝐵𝐵 𝐵𝐵−1 = 8.04 1026

𝜅𝜅 𝐶𝐶 = 𝐶𝐶 𝐶𝐶−1 = 1.91 105

𝜅𝜅 𝐾𝐾 = 𝐾𝐾 𝐾𝐾−1 = 2.88 107

𝑩𝑩𝒊𝒊𝒊𝒊 𝑪𝑪𝒊𝒊𝒊𝒊





22

Scalar + Vector Finite Element Method (FEM), 𝐻𝐻 −Φ field formulation

4J. P. Webb, B. Forghani, “T-Ω Method Using Hierarchical Edge Elements”, IEE Proceedings – Science, Measurements and Technology, Vol. 142, No. 2, pp. 133-141, 1995.




23


4J. P. Webb, B. Forghani, “T-Ω Method Using Hierarchical Edge Elements”, IEE Proceedings – Science, Measurements and Technology, Vol. 142, No. 2, pp. 133-141, 1995.




24





25


Neq=16’913

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 =𝑁𝑁𝑛𝑛𝑛𝑛𝑁𝑁𝑒𝑒𝑒𝑒2

100% = 0.081%

𝜅𝜅 𝐾𝐾 = 𝐾𝐾 𝐾𝐾−1 = 1.48 1015




26





27





28

Boundary Element Method (BEM), 𝐻𝐻 −Φ field formulation

( )[ ]

( )[ ] [ ])()(),()(41

),()(41)(

21

0)(

)(

ξξξηησπ

ξηηπ

ξ

δηξξ

ηξξ

ttm HHdSGn

dSKJnJ

+−=∇×−

−∇××+−

∫∫

∫∫

Ω∂

Ω∂

( )[ ]

( )[ ] [ ])()(),()(4

),()(41)(

21

0)(0

)(

ξξξηηπµµ

ξηησπ

ξσ

δηξξ

ηξξ

nn

mm

HHdSKJn

dSGn

+−=∇×⋅−

−∇⋅−−

∫∫

∫∫

Ω∂

Ω∂

Ω∂∈∀ ξη,

currentvirtual−J

chargemagneticvirtual−mσ

ξη

ξη

ξη,

rkj)-(1

re),K(

,⋅⋅+

=

2k 0 σµωµ r=

ξη

ξη,r

1),G( =

J. Smajic, et al., “BEM-based Simulations in Engineering Design,” in “Boundary Element Analysis: Mathematical Aspects and Applications,” (Edited by M. Schanz and O. Steinbach) Lecture Notes in Applied andComputational Mechanics, Vol. 29, pp. 281-352, Springer Verlag, Berlin, Heidelberg, New York, 2007.




29






30



Kernel expansion: Fast Multipole Technique (FMT)

∑∈

⋅⋅=≈

⇒>>−

mIm yyYxxXyxKyxyxKyxK

farfieldyx

),(0000),(00 ),(),(),(),;,(),(

)(0

νµνµνµ

0000 yxyyxx −⋅≤−+− η Far-field condition

Taylor-, Multipole-, Cebysev- expansion

GMRES with clustering

( )( )∑∈

⋅+⋅=⋅=F

T uYFXuNuAv),(

~τσ

τστσ Matrix-vector multiplication




31



GMRES with clustering

( )( )∑∈

⋅+⋅=⋅=F

T uYFXuNuAv),(

~τσ

τστσ Matrix-vector multiplication




32


J. Smajic, Z. Andjelic, M. Bebendorf, “Fast BEM for Eddy-Current Problems Using H-matrices and Adaptive Cross Approximation”, IEEE Transactions on Magnetics, Vol. 43, Issue 4, , pp. 1269 -1272, April 2007.




33


J. Smajic, Z. Andjelic, M. Bebendorf, “Fast BEM for Eddy-Current Problems Using H-matrices and Adaptive Cross Approximation”, IEEE Transactions on Magnetics, Vol. 43, Issue 4, , pp. 1269 -1272, April 2007.




34

Coupled FEM - MMP

J. Smajic, Ch. Hafner, J. Leuthold, “Coupled FEM-MMP for Computational Electromagnetics”, IEEE Transactions on Magnetics, Vol. 52, No. 3, pp. 7207704, March 2016.




35

Coupled FEM - MMP





36

Coupled FEM - MMP





37

Coupled FEM - MMP





38

Coupled FEM - MMP





39

Coupled FEM - MMP





40

Discontinuous Galerkin Time-domain FEM

Ω𝑒𝑒 Ω𝑝𝑝𝐷𝐷𝑗𝑗

Continuous Galerkin

Ω𝑒𝑒Ω𝑝𝑝

𝐷𝐷𝑗𝑗𝑒𝑒

Discontinuous Galerkin

𝐷𝐷𝑗𝑗𝑝𝑝




41





42




Summary

43

FEM: sparse matrices, ill-conditioned matrices, efficiency depends on field formulation.

BEM: dense matrices, matrix compression and preconditioning, singular integrals.

MMP: boundary methods, sources away from boundary, no singular integrals, control of accuracy.

FEM-MMP: possibility to solve nonlinear problems without air-box.

DG-TD-FEM: no large linear system to solve, leapfrog time-stepping scheme, conditionally stable.

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