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IEEE-APS Toronto, July 2010

J.-P. BERENGER*, F. COSTEN**

*Centre d’Analyse de Défense16 bis, Avenue Prieur de la Côte d’Or

94114 Arcueil, France

**School of Electrical and Electronic EngineeringThe University of Manchester

Sackville Street Building, Manchester, M60 1QD, UK

jpberenger@gmail.com; f.costen@cs.man.ac.uk

Application of the Huygens Absorbing Boundary

Condition to Wave-Structure Interaction Problems

IEEE AP-S & URSI

Toronto 12-16 July 2010

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IEEE-APS Toronto, June 2009

Introduction

- The Huygens ABC was introduced in three papers in 2003-2007.

- Can be viewed as just another implementation of analytical ABCs. (Mur, Higdon). However, provides us with more possibilities.

- In this presentation the HABC is combined with a real stretch of the mesh to realize an effective ABC for wave-structure interaction problems. This combination is:

- more effective than any previous analytical ABC. It can challenge the PML ABC in terms of computational cost and accuracy, at least from 2D tests.

- simpler to implement than the PML ABC.

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The Principle of the Huygens ABC

Zero field

Zero

field

Zero

field

Zero field

Any boundary condition

Source field

Outgoing field

Outgoing field

Huygens surface radiating field oppositeto outgoing field

Introduced independently in special cases: Multiple absorbing surfaces (Sudiarta, 2003) and Teleportation (Diaz and Scherbatko, 2004). Later generalized as

Huygens ABC (Berenger, 2007).

The principle is simple: radiating a wave opposite to the outgoing field.

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The Huygens ABC: need of an operator

A problem to implement this simple idea: the outgoing field is not known where the Huygens surface must be enforced (if it were known this would be a perfect ABC!). Huygens Surface

Ey(0) Hz(1/2)

Theory shows that the overall reflection is the same as the one of the operator. => The new ABC is nothing but another implementation of operator ABC’s.

An operator (Higdon, others) is needed to evaluate the field to be radiated by the Huygens surface.

Advantage of the new implementation: Easy combination with other ABC’s (PML or Analytical ABC).

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Wave-Structure Interaction problems

dHig

dHig

Higdon ABC

Object = 300-cell thin plate

The evanescent waves are not absorbed by Higdon operator (nor by Mur ABC)

With Operator ABC the object-ABC separation must be large, typically ~ object size

Ei

ki

Test point

The incident wave is a Unit-Step wave

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Wave-Structure Interaction problems

As expected, the results are like with the Higdon Operator ABC

=> Used “as is”, the HABC is just another implementation of Higdon

ABC, with same drawback (no absorption of evanescent waves).

With a HABC (Higdon) and a PEC behind it (separation = 2 FDTD cells)

dPEC

dPEC

PEC

Object = 300-cell thin plate

HABC

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Combination of Huygens ABC with a stretched mesh

- A Huygens ABC to absorb traveling waves (highest frequencies).

- Outside the HABC only low frequency evanescent waves are present. For a scatterer of size W they decrease in function of distance as:

=> A very coarse mesh region (real stretch of coordinates) can be used to “absorb” evanescent waves (i.e. to permit natural decrease). We can hope the needed coarse mesh be < 10-20 cells in thickness.

Coarse Mesh

w

d

eEdE

0)(

Frequency

Traveling wavesEvanescent waves

Stretch of Mesh HABC (Higdon)

= 2 w

The proposed idea: HABC (Higdon)

No ABC

Coarse Mesh

Coarse Mesh

Coarse Mesh

Scatterer

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Huygens ABC in 2D or 3D

HABC

PEC

Source

HABC

PEC

Source

HABC Extensions

To render the HABC equivalent, rigorously, to an operator ABC, extensions of HABC lines (surfaces in 3D) must be added in corner regions.

Not equivalent to operator ABC Equivalent to operator ABC

-The results presented in the following have been computed with the extensions.

- The need of extensions was not discovered in 2003-2007 papers. It is the subject of a submitted paper (F. Costen, JP Berenger, Journ. Comp. Phys.)

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Can the HABC be placed close to the object?

dPEC = 900 cells

dPEC

PEC

Object = 300-cell thin plate

HABC

dHABC

The FDTD cell is uniform in the FDTD domain

- The HABC does absorb very well the traveling waves even if it is quite close to the object (here, 3-cell separation with a 300-cell object).

- The evanescent waves are absorbed (natural decrease) in the large surrounding mesh.

dPEC = 900 cells

dPEC

PEC

Object = 300-cell thin plate

HABC

dHABC

=> YES

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Use of a stretched mesh outside the HABC

max

Fine mesh HABC (Extensions not drawn)

Coarse MeshMesh Transition

0 43 4+ng

Transition Coarse

Distance from Object in cells

Spatial step

ObjectHSG

ng cells growing geometrically

- HSG 3 cells from object- 4 cells of constant size - ng cells that grow geometrically from to max

Characteristic length of decrease of surrounding evanescent waves = size w of object

=> Maximum step max probably of the order of size w.

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An experiment with a stretched mesh

=> 26 stretched cells can replace 900 non-stretched cells

In this experiment the maximum step max is constant (= W / 5) and the transition region varies from ng = 0 to ng = 8 cells.

dPEC < 26 cells 900 non-stretched cells

HABC

PEC

dPEC < 26 cells 900 non-stretched cells

HABC

PEC

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An experiment with a stretched mesh

=> 17 stretched cells can replace 900 non-stretched cells

In this experiment the transition region is constant (ng = 8 cells) and the maximum step max varies from w / 10 to W / 1.

dPEC < 40 cells 900 non-stretched cells

HABC

PEC

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Comparison with the CFS-PML

The Physics behind Optimized CFS-PML and HABC+stretched mesh is the same. Both methods rely on separation of evanescent waves with traveling waves: evanescent waves at low frequency, traveling waves at high frequency

0

1

j

sx

xx

Frequency

Traveling wavesEvanescent waves

f0 = 2 w / c = 2 w

Stretch of Mesh HABC (Higdon)

CFS-PML real stretch CFS-PML normal absorbing PML

02

xf

The Physics

CFS-PML ABC

HABC+STRETCH

-> From this we can expect similar performance in terms of computational cost

-> HABC is more general: CFS-PML requires sinh = constant

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Comparison with the PML

-15 cells thick normal PML

- 5 cells thick CFS-PML

- HABC + Stretched grid

- HABC: big FDTD domain, but filled with big FDTD cells.

- HABC is intermediate between normal PML and Optimized CFS-PML in terms of CPU time (one cell of PML is 1.5-2 times more costly than one cell of vacuum).

- HABC is simpler to implement.

- Further works to improve HABC (higher order operator, optimization of the mesh stretch).

With PML

With HABCDomain sizes:

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Conclusion

• Combination of a HABC with a strongly stretched mesh realizes an effective ABC for the solution of wave-structure interaction problems.

• This ABC can challenge the PML ABC in terms of computational cost reduction. Only the optimized CFS-PML remains slightly better, but HABC has not yet been optimized.

• HABC implementation is simpler than PML implementation.

• The same principle could be used with other problems where evanescent waves are only present at low frequency (waveguides).

• Extension to 3D is currently under investigation.

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- References -The Huygens ABC I. Wayan Sudiarta, An absorbing boundary condition for FDTD truncation using multiple absorbing surfaces, IEEE Trans. Antennas Propag. 51 (2003), 3268-3275.

R.E. Diaz, I. Scherbatko, A simple stackable re-radiating boundary condition (rRBC) for FDTD, IEEE Antennas Propag. Magazine 46 (1) (2004) 124-130.Bérenger J.P. “ On the Huygens Absorbing Boundary Conditions for Electromagnetics” Journal of Computational Physics, n° 226, 2007.