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Institute of Electromagnetic Theory Hamburg University of Technology Efficient numerical solution methods for Maxwell's equations by separations of near field and far field interaction Prof. Dr. Frank Gronwald Chair „Electromagnetic Compatibility“ Institute of Electromagnetic Theory Hamburg University of Technology gronwald@tu-harburg.de

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 2 The Institute of Electromagnetic Theory at the University of Technology Hamburg (TUHH) today aerial view around 1930 Prof. Dr. Christian Schuster Chair “Electromagnetic Theory” Prof. Dr. Frank Gronwald Chair “Electromagnetic Compatibility” Dr. Heinz-D. Brüns Senior Scientist “Numerical Field Computation

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 3 Overview of the talk 1.Introduction and Motivation: Electromagnetic Engineering Applications and Numerical Field Computation 2.Aspects of efficient numerical field computation in electromagnetic theory 3.Electromagnetic near fields and far fields (Coulomb fields and radiation fields) 4.Separations of near field and far field interactions –Method of analytical regularization –Hybrid representations of Green‘s functions –Multilevel Fast Multipole Algorithm 5.Conclusion

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 4 1. Introduction and Motivation - interference analysis of avionic systems some important avionic systems : TCAS = Traffic Collision Avoidance System ATC = Air Traffic Control GPS = Global Positioning System SATCOM = Satellite Communication ADF = Automatic Direction Finder VOR = VHF Omnidirectional Radio Range DME = Distance Measuring Equipment ILS = Instrument Landing System common avionic systems operate in the frequency range 100 kHz to 15 GHz corresponding wavelengths are in the range of 3 km to 2 cm analysis of (unwanted) antenna couplings requires to calculate both near and far fields necessity to numerically solve large scale electromagnetic boundary value problem to obtain interference matrix transmitting antennas receiving antennas interference matrix

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 5 1. Introduction and Motivation – interior problems of Electromagnetic Compatibility complex electric/electronic systems contain various electric/electronic components which might interfer with each other electromagnetic coupling between various electric/electronic components needs to be analyzed and estimated electromagnetic coupling often takes place within a resonating environment (e.g. within a metallic housing, as given by common computer housings) a low cost electronic power meter on a more abstract level: modelling of antenna coupling within a cavity antennas carry electric charges and currents that generate electric and magnetic near fields cavity supports electromagnetic modes that correspond to free electromagnetic (far) fields (i.e., solutions of sourceless Helmholtz equations) necessity to solve electromagnetic boundary value problem with near and far field characteristics

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 6 2. Aspects of efficient numerical field computation in electromagnetic theory Usually, the solution f of an electromagnetic boundary value problem is given by an element of an infinite dimensional function space (such as a Hilbert or Soboloev space): Remark: Often the explicit solution for f is found as the solution of a linear operator equation, see, e.g.: Hanson, G.W. and Yakolev, A.: Operator Theory for Electromagnetics, (Springer, New York, 2002). A numerical solution is a finite approximation f of the form with numerically calculated coefficients An efficient numerical solution requires: number N of approximating basis functions not too large fast numerical calculation of the coefficients

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 7 Important observation: Basis functions that are suitable to approximate a near field (=Coulomb field) often are not suitable to approximate a far field (=radiation field) and vice versa This observation is illustrated by the complementary properties of „rays“ (propagator functions) and „modes“ (eigenfunctions of a compact and self-adjoint operator) that both are often used as approximating basis functions (Felsen, 1984): It follows that in order to numerically solve electromagnetic boundary value problems it is often necessary to separately analyze near field and far field interactions in order to find approximating basis functions which are suitable to approximate both near fields and far fields Note: Efficient numerical computation schemes often are necessary because computer memory and computation time are limited. 2. Aspects of efficient numerical field computation in electromagnetic theory RaysModes Scattering processes yield local information of a system Oscillations yield global infornation of a system Characterize early response in time domainCharacterize late response in time domain Advantageous for high-frequency regime where the mode-density is high and rays of geometrical optics characterize the field Advantageous for low-frequency regime where the mode-density is low and a small number of modes characterizes the field Advantageous to model Coulomb singularitiesAdvantageous to model resonances

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 8 3. Electromagnetic near fields and far fields - fields generated by a point charge near field (Coulomb) far field (radiation) Advantage: Near field and far field contributions can be written as separate terms Disadvantage: Motion of point charge must be known Disadvantage: Model of point charge not useful for most engineering applications where charge and current distributions are needed

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 9 3. Electromagnetic near fields and far fields - fields generated by continuous sources near field (Coulomb) far field (radiation) near field (Coulomb) Advantage: Near field and far field contributions can be written as separate terms Disadvantage: Current and charge distributions must be known Remark: To obtain current and charge distributions it is often required to solve an integral equation where near and far field contributions still are coupled observer charge and current distribution

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 10 3. Electromagnetic near fields and far fields - contrasted to longitudinal and transverse fields Remark: Quantization of the electromagnetic field requires to quantize true dynamical degrees of freedom only – these are not part of the near (Coulomb) field but part of the far (radiation) field It is then a standard approach to split electromagnetic fields in their longitudinal and tranverse part – for a general vector field, its longitudinal part and transverse part are defined by corresponding split of Maxwell‘s equations: Separation of longitudinal and transverse parts does not correspond to a separation of near (Coulomb) and far (radiation) fields - entangled near and far fields still have to be taken care of in the quantization process, but are there any alternatives? longitudinal part transverse part

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 11 4. Separations of near field and far field interactions There appears to be no canonical method to separate, for the general solution of electromagnetic boundary value problems, near (Coulomb) fields from far (radiation) field To nevertheless employ efficient numerical solution schemes it is nevertheless required to separate near field and far field interactions in some way In the following, three methods for separating near field and far field interactions are introduced: method of analytical regularization (conversion of an integral equation of the first kind to an integral equation of the second kind) hybrid representation of Green‘s function (transforming a canonical Green‘s function into a ray-mode representation) multilevel fast multipole algorithm (effective grouping and translating of near field interactions)

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 12 4. Separations of near field and far field interactions - Method of analytical regularization Electromagnetic boundary value problems often are formulated as electric field integral equations of the form or, if written as linear operator equation, Idea: Split L in two parts L 0 and L 1 where L 0 contains the Coulomb singularity Then: First, construct the „near field solution“ L 0 -1 Second, solve the remaining integral equation of the second kind with the Coulomb singularity removed Solution often possible by iteration:

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 13 4. Separations of near field and far field interactions - Method of analytical regularization Example: Electrically small antenna inside a cavity (Tkachenko & Gronwald, 2003) Consider first a linear wire antenna (length L, radius a, directed along z-axis) in free space which is excited by an incoming wave Approximate solution for the induced current: For a small antenna this solution can be written in the factorized form

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 14 4. Separations of near field and far field interactions - Method of analytical regularization It follows with that the solution for the current along the small antenna within the cavity is given by This result can be used, for example, to calculate the coupling between two antennas within a rectangular cavity For a specific configuration the current transfer ratio is characterized by sharp resonance peaks

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 15 4. Separations of near field and far field interactions - Hybrid representations of Green’s functions General idea: Construct Green‘s functions with both ray and mode properties Illustration by example: Consider the Green‘s function of the Helmholtz equation for the vector potential A inside a three-dimensional rectangular cavity Application of the mirror principle yields: „ray representation“

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 16 4. Separations of near field and far field interactions - Hybrid representations of Green’s functions Now consider the three-dimensional Poisson transformation Application of the Poisson transformation to the ray representation yields the mode representation (Wu & Chang 1987) This is not exactly what we want: We have turned rays into modes but we want to have both ray and mode contributions!

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 17 4. Separations of near field and far field interactions - Hybrid representations of Green’s functions By the application of an Ewald-transformation (Ewald, 1921) it can be shown that (Gronwald, 2005) where and This hybrid representation has very good convergence both in the source region and at resonance!

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 18 4. Separations of near field and far field interactions - Hybrid representations of Green’s functions Example: Calculation of the Green‘s function G A zz within a canonical cavity number of termsaccuracy Ray sum10 6 10 -2 Mode sum10 8 10 -5 Ewald sum10 2 10 -8 number of termsaccuracy Ray sum10 8 no convergence Mode sum10 6 10 -5 Ewald sum10 2 10 -8 cuboidal cavity with source pointvalues of the Green‘s function G A zz for varying observation point r and fixed wavenumber k=9.42/L convergence properties in source region (x=y=z=0.26L) convergence properties at resonance (k=9.42/L)

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 19 4. Separations of near field and far field interactions - Hybrid representations of Green’s functions Example: Calculation of the mutual coupling Z 12 between two antennas inside a rectangular cavity Mutual coupling is expressed by the mutual impedance which is calculated by a formula of the form and obtained by the numerical solution an integral equation, involving the cavity‘s Green‘s function rectangular cavity with dimensions l x =6m l y =7m l z =3m absolute value of the mutual impedance

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 20 4. Separations of near field and far field interactions - Hybrid representations of Green’s functions The results of the previous slide have been obtained by the Method of Moments The Method of Moments converts a linear operator equation into an algebraic system of equations: First, the original equation is approximated by it follows and the result is a linear algebraic equation for the unknown coefficients,

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 21 The standard Method of Moments: Usually applied to the conversion of an integral equation to a linear system of equations which contains the unknown electric current elements as primary variables Linear system of equation characterized by the interaction between all electric current elements Multilevel Fast Multipole Algorithm: (Rokhlin, 1990; Lu & Chew, 1993) Interaction between regions that are not within each other near field regions can be approximated by a smaller number of interactions (i.e., less interactions need to be computed and stored!) 4. Separations of near field and far field interactions - Multilevel fast multipole algorithm Multilevel Fast Multipole Algorithm: Far field interactions are effectively grouped and translated Standard Method of Moments: Interaction between all current elements is taken into account

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 22 4. Separations of near field and far field interactions - Multilevel fast multipole algorithm The multilevel fast multipole algorithm treats near field and far field interactions differently Near field interactions: Only interactions between neighboring current elements are taken into account, the corresponding interaction matrix becomes sparse Far field interactions: Less interactions due to Aggregation, Translation, and Disaggregation Translation is mathematically performed by the use of „addition theorems“ that arise from the addition of angular momentum in quantum mechanics

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 23 4. Separations of near field and far field interactions - Multilevel fast multipole algorithm Both the standard Method of Moments and the Multilevel Fast Multipole Algorithm are implemented in the program CONCEPT-II which has been developed at the Institute of Electromagnetic Theory, Hamburg University of Technology, since the mid 1980ies CONCEPT-II used as platform to incorporate research results electromagnetic tool to work on academic and industrial projects Screenshots of CONCEPT-II graphical user interface (compare: http:www.tet.tu-harburg.de/concept/)

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 24 4. Separations of near field and far field interactions - Multilevel fast multipole algorithm Example: Calculation of the surface currents on a ship which are generated by a monopole antenna, operating at f = 150 MHz (λ = 2m) Ship dimensions: 120m length 21m height 15m width Discretisation yields 424158 unknowns (=edge currents) Standard Method of Moments would require 2.6 TeraByte of memory MLFMA requires 5.6 GigaByte of memory, on a single workstation the problem can be solved in 2.8 hours

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Institute of Electromagnetic Theory Frank Gronwald Hamburg University of Technology Transparency 25 5. Conclusion Efficient numerical solution methods for electromagnetic boundary value problems often require to separate near field interactions (Coulomb fields) from far field interactions (radiation fields) There appears to be no general and complete method to achieve this separation But it is often possible to isolate the Coulomb singularity in a way such that numerical computations both in the source region and at resonance become possible. Thank you for your attention!

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Chapter 1 Electromagnetic Fields

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