spectral-galerkin surface integral methods for 3d computational...

Oberwolfach Workshop on Computational Electromagnetics

Spectral-Galerkin surface integral methods

for 3D computational electromagnetics

M. Ganesh (Colorado School of Mines)

1

3D Electromagnetic Scattering configuration

• J random or deterministic particles Dj with surface ∂Dj, j = 1, . . . , J

• Each ∂Dj is a boundaryless simply connected 2-manifold in R3

Random ice crystals : Modeling images from cloud particle imager (CPI)

2

3D Computational Electromagnetics

• Simulate scattering of an incident wave from the ensemble D = ∪Jj=1Dj

Total field generated by random ice crystals

3


• Simulate bistatic (monostatic) radar cross section (RCS) of theensemble D = ∪Jj=1Dj using one (thousands of) incident wave(s)

Bistatic RCS of radom ice crystals

4


• Perform simulation for cases in which experimental RCS is known

RCS of two spheres - IEEE Antennas Propag. 19, 378-390 (1971)

5


• Perform simulation for cases in which experimental RCS is known

Mono. RCS of ogive - IEEE Antennas Propag. Mag. 19, 84-89 (1993)

6

3D Computational Electromagnetics - Inversion

• Given simulated or experimental RCS, find the shape of the obstacleusing RCS data from a few incident waves.(Open Problem: Characterize the required few incident directions.)

Simulated RCS of a bean-shaped perfect conductor with one incident direction

7

Spectral Surface Integral methods - Literature


• Books:

? Potential Theory:

K. E. Atkinson,The Numerical Solution of Integral Equations of the Second Kind,Cambridge University Press, 1997.

? Scattering Theory:

D. Colton & R. Kress,Inverse Acoustic and Electromagnetic Scattering Theory,Springer, 1998.


• Books:

? Potential Theory:

K. E. Atkinson,The Numerical Solution of Integral Equations of the Second Kind,Cambridge University Press, 1997.

? Scattering Theory:

D. Colton & R. Kress,Inverse Acoustic and Electromagnetic Scattering Theory,Springer, 1998.

10


• K. Atkinson in early 1980’s proposed

? a semi-discrete spectral-Galerkinsurface integral equation method for the Laplace equation in 3D:SIAM J. Numer. Anal., 19 (1982), 332-347

? Semi-discrete :Discretization of weakly-singular surface integrals not considered

• R. Kress in late 1980’s, for the Helmholtz equation in 3D,initiated a fully discrete hybrid Nystom-spectral method:

? L. Wienert,Die numerische approximation von randintegraloperatorenfur die Helmholtzgleichung im R3,Ph.D. Thesis (Supervisor: R. Kress),Universitat Gottingen, Germany, 1990.

? Hybrid:Use spectral approximations forweakly-singular surface integrals arising in Nystrom method


• K. Atkinson in early 1980’s proposed

? a semi-discrete spectral-Galerkinsurface integral equation method for the Laplace equation in 3D:SIAM J. Numer. Anal., 19 (1982), 332-347

? Semi-discrete :Discretization of weakly-singular surface integrals not considered

• R. Kress in late 1980’s, for the Helmholtz equation in 3D,initiated a fully discrete hybrid Nystom-spectral method:

? L. Wienert,Die numerische approximation von randintegraloperatorenfur die Helmholtzgleichung im R3,Ph.D. Thesis (Supervisor: R. Kress),Universitat Gottingen, Germany, 1990.

? Hybrid:Use spectral approximations forweakly-singular surface integrals arising in Nystrom method

11


• Nonlinear elasticity model in 3D(mixed BVP with Dirichlet and Neumann boundary data)

? a fully-discrete spectral-Galerkin surface integral equation method:

M. G, I.G. Graham, and J. SivaloganathanSIAM J. Numer. Anal., 31 (1994), 1378-1414

? Mathematical analysis, assuming inner and outer boundary are spheres






• For the Laplace equation in 3D:

? a spectral-collocation surface integral method:M. G, I.G. Graham, and J. SivaloganathanSIAM J. Numer. Anal., 35 (1998), 778-805

? Mathematical analysis because of a new class ofspectral basis functions - sphere assumption avoided






• For the Laplace equation in 3D:

? a spectral-collocation surface integral method:M. G, I.G. Graham, and J. SivaloganathanSIAM J. Numer. Anal., 35 (1998), 778-805

? Mathematical analysis because of a new class ofspectral basis functions - sphere assumption avoided

? A related new matrix-free approximation theory tool was developed:M. G and H. N. MhaskarSIAM J. Numer. Anal., 44 (2006), 1314-1331

12


• The 1990’s open problem of mathematically analyzingKress-Wienert’s method was solved by:

? I.G. Graham, and I. H. SloanNumer. Math., 92 (2002), 289-323




• Acoustic scattering in 3D (sound-soft, sound-hard, absorbing obstacles):

? A fully discrete spectral-Galerkin algorithmwith mathematical analysis and practical realization in:M. G, and I.G. GrahamJ. Comp. Phys., 198 (2004), 211-242




• Acoustic scattering in 3D (sound-soft, sound-hard, absorbing obstacles):

? A fully discrete spectral-Galerkin algorithmwith mathematical analysis and practical realization in:M. G, and I.G. GrahamJ. Comp. Phys., 198 (2004), 211-242

• Electromagnetic scattering in 3D (perfect conductor):

? M. Pieper,Spektralrandintegralmethoden zur Maxwell-Gleichung,Ph.D. Thesis (Supervisor: R. Kress),Universitat Gottingen, Germany, 2007

13


• Naive generalization of spectral-Galerkin algorithmfor 3D acoustic scattering to 3D computational electromagneticsleads to stagnated/quadratic convergence:

? Sample error result for a plane-wave acoustic scatteringfrom a sphere of diameter less than 1λ (J. Comp. Phys, 2004):

# Unknows 36 144 256Error 2.9e-03 2.2e-10 6.2e-14

? Sample error result for a plane-wave electromagnetic scatteringfrom a sphere of diameter less than 1λ (Pieper thesis, 2007):



• Naive generalization of spectral-Galerkin algorithmfor 3D acoustic scattering to 3D computational electromagneticsleads to stagnated/quadratic convergence:

? Sample error result for a plane-wave acoustic scatteringfrom a sphere of diameter less than 1λ (J. Comp. Phys, 2004):


? Sample error result for a plane-wave electromagnetic scatteringfrom a sphere of diameter less than 1λ (Pieper thesis, 2007):


• The stagnated convergence was avoided and spectral convergence wasproved and demonstrated for 3D electromagnetics in recent work:

•M. G and S. Hawkins:

? J. Comp. Physics, 227 (2008), 4543–4562 (Single configuration)

? Numer. Algorithms, 50 (2009), 469–510 (Multiple scattering)

14

Two important spectral-Galerkin tools for 3D electromagnetics

• Find local spherical coordinate parametrizationfor each boundaryless simply connected 2-manifold ∂Dj, j = 1, . . . , J

• For each j = 1, . . . , J, design high-order spectrally accurate local tangentialbasis to approximate tangential vector fields on ∂Dj, j = 1, . . . , J

15

Spectral-Galerkin tool #1: Spherical coordinate representation

• For each particle Dj, j = 1, · · · , J, introduce local coordinates



• Locally, each point (x, y, z)T ∈ ∂Dj, j = 1, · · · , J can be represented as

(x, y, z)T =(qj1(θ, φ), qj2(θ, φ), qj3(θ, φ)

)T, θ, φ ∈ R,

for some nonlinear functionals qji : R2 → R, i = 1, 2, 3



• Locally, each point (x, y, z)T ∈ ∂Dj, j = 1, · · · , J can be represented as

(x, y, z)T =(qj1(θ, φ), qj2(θ, φ), qj3(θ, φ)

)T, θ, φ ∈ R,

for some nonlinear functionals qji : R2 → R, i = 1, 2, 3

• For stochastic obstacles approximate qji map are given via discrete Fourierseries representation with random coefficients

• Thanks to S2 version of the Poincare Conjucture, for each surface ∂Dj,there exist bijective parametrization maps qj : S2 → ∂Dj, j = 1, . . . , J

16


Erythrocytes (Courtesy: NIH) Model: Red blood cell

17


18


• Analytical or Fourier series based representations (required for spectralsurface integral algorithms) are known for substantial class of particles:

? IEEE Antennas Propag. Mag. 35 (1993), 84–89

? J. Quant. Spectrosc. Radiat. Transfer, 100 (2005), 393–405, 2005

? J. Quant. Spectrosc. Radiat. Transfer, 100 (2006), 444–456, 2006

? J. Electromagnetic Waves Appl. 20 (2006), 1827–1836 (2006)

? Inverse Problems, 22, 1509–1532 (2006)

? ... ... ... ... ...

• Such representations are standard in inversion of RCS data

•More representations in near future :R. Hiptmair’s and Optics group at ETH

19

Spectral-Galerkin tool #1: Spherical cap patch representation

• In case of difficulty in obtaining global representation, one may considersurface patches that lead to mapping to spherical caps instead,with partition of unity of the given surface

Spectral-Galerkin tool #1: Spherical cap patch representation

• In case of difficulty in obtaining global representation, one may considersurface patches that lead to mapping to spherical caps instead,with partition of unity of the given surface

• Spectral-FEM:

Another approach for very complicated shaped obstacle is to circumscribethe obstacle as close as possible with a smooth body. Then use

? high-order non-polynomial finite element in the small domain betweenthe given and the smooth surfaces;

? high-order spectral Galerkin surface integral algorithm exterior to thesmooth body

20

Spectral-Galerkin tool #2: High-order tangential basis on ∂Dj

• For each random/deterministic surface ∂Dj in the configuration,

? choose high-order j-th local basis set such that

? each function in the set is tangential to ∂Dj





• Start with high-order finite dimensional space Zjspanned by polynomials of degree less than, say, nj, that aretangential reference surface ∂Bj: Dimension of Zj is Nj = 2n2j, j = 1, . . . , J.





• Start with high-order finite dimensional space Zjspanned by polynomials of degree less than, say, nj, that aretangential reference surface ∂Bj: Dimension of Zj is Nj = 2n2j, j = 1, . . . , J.

• The reference basis set is such that

? for any Cr tangential vector-field g on the reference surface

? there exists an element in Zj

∗ that is a spectrally accurate approximation to g,

∗ with error n−rj

21


• For the j-th particle in the configuration, j = 1, . . . , J,

• construct a smooth tangential transformation Fj such that

? for any Cr tangential vector-field G on ∂Dj

? the Nj-dimensional non-polynomial space FjZj is such that

∗ that the elements in the space are tangential on ∂Dj

∗ and prove that there exists an element in ∂Dj

∗ that is a spectrally accurate approximation to G,

∗ with error n−rj

22


• Standard transformations do not retain the n−rj spectral accuracy


• Standard transformations do not retain the n−rj spectral accuracy

• For full details of construction of Fj with proof of required properties,see M. G and S. Hawkins, J. Comp. Physics, 227 (2008), 4543–4562

• Brief idea: Construct an efficient orthogonal transformation that

? maps each point in the reference surface x

? to the associated point in the given surface x

? by rotation in the plane containing the normal at x and normal at x

23

Spectral-Galerkin tools : Sample advantage in comp EM

Spectral-Galerkin tools : Sample advantage in comp EM

• For full details of single obstacle GH algorithm, seeM. G and S. Hawkins, J. Comp. Physics, 227 (2008), 4543–4562

Performance of industrial standard FISC and recent GH algorithm

Electromagnetic Scattering by a sphere of diameter 48λ

Algorithm Unknowns RMS Err. RMS Err. RMS Err.(MD) (ED) (PW)

GH 48,670 2.9e-11 1.9e-11 9.9e-02

FISC 2,408,448 – – 3.3e-01

GH 51,840 – – 3.7e-03

GH 55,110 – – 5.9e-05

24

Multiple electromagnetic Scattering in three dimensions


• Compute time-harmonic electric and magnetic fields


• Compute time-harmonic electric and magnetic fields

E(x, t) =1√ε0

Re{E(x)e−iωt

}, H(x, t) =

1√µ0

Re{H(x)e−iωt

}? induced by an incident wave, with frequency ω (= c

λ = ck2π),

? impinging on a configuration ∂D with

? J perfectly conducting three dimensional particles ∂D1, . . . , ∂DJ

? situated in a homogeneous medium

? with vanishing conductivity,

? free space permittivity ε0

? and permeability µ0.

25

Exterior Field, Maxwell equations, radiation and boundary conditions

• The scattered field [E(x),H(x)], x ∈ R3 \ ∪Jj=1Dj satisfies:

? the exterior Maxwell equations

curl E(x)− ikH(x) = 0, curl H(x) + ikE(x) = 0, x ∈ R3 \ ∪Jj=1Dj,

? the Silver-Muller radiation condition

lim|x|→∞

[H(x)× x− |x|E(x)] = 0.

? the perfect conductor boundary condition

n(x)×E(x) = −n(x)×Einc(x), x ∈ ∪Jj=1∂Dj.

? with known incident electromagnetic wave [Einc,H inc].

26

Far field and RCS : Compute with spectral accuracy

• Given any (unit vector) direction x ∈ S2 (unit sphere),compute fully discrete spectral Galerkin approximations to

? the far fieldE∞(x) = lim

r→∞E(rx)e−ikrr,

? the radar cross section (RCS) of the configuration ∪Jj=1Dj

σ(x) = 4π |E∞(x)|2 /k2,

? the surface current w ∈ Cr(∂D)

? with number of unknowns N = 2n2 (= 2 max{n2j : j = 1, · · · , J}) so that

max{‖w −wn‖∞,∂D, ‖E∞ −En,∞‖∞,S2, ‖σ∞ − σn,∞‖∞,S2

}= O(n−r).

• In addition to full theoretical analysis, demonstrate advantage of spectralconvergence using simulation and comparison with experimental data

27

Surface integral representations of interacting multiple EM fields

• Represent, for example, exterior electric and magnetic fields asintegrals on the J-surfaces in the configuration ∪Jj=1Dj:

E(x) = curl

∫∂D

Φ(x,y)w(y) ds(y), H(x) =1

ikcurl E(x), x ∈ R3 \D.

• Task is to design a Galerkin surface integral algorithm

? to approximate J tangential fields wi = w|∂Di, i = 1, . . . , J

? with spectral accuracy

? by full discrete approximations of J coupled surface integral equations

wi(x) +

J∑j=1

(Mijwj) (x) = −2n(x)×Einc(x), x ∈ ∂Di, i = 1, . . . , J.

• where

(Mijb) (x) = 2

∫∂Dj

n(x)× curlx {Φ(x,y)b(y)} ds(y), b ∈ T (∂Dj), x ∈ ∂Di.

28

Spectrally accurate approximations of surface integral operators Mij

• For high-order fully discrete Galerkin surface integral algorithms

? it is very important to

? to fully discretize all surface integrals Mij, i, j = 1, . . . , J







• It is fundamental to take care singularities in analytically Mij

• This can be achieved by suitable coordinate transformations(with Jacobian or basis functions) cancelling out the singularties

• However, this leads to further complications in the integrands of Mij









• For example, the tangential properties of integrands on ∂Dj are lost

• Another class of functions are required to approximate integrands









• For example, the tangential properties of integrands on ∂Dj are lost

• Another class of functions are required to approximate integrands

• Finally, leading to fully discretize sums M(n)ij with properties

‖Mijb−M(n)ij b‖∞,∂Di = O(n−r), b ∈ T r(∂Dj), i, j = 1, . . . , J

• For full details, seeM. G and S. Hawkins: Numer. Algorithms, 50 (2009), 469–510

29

Boundary Decomposition: disjoint scatterers to each connected scatterer

• Boundary decomposition technique reduces complexity to

? computing scattered field from each connected scatterer∂Dj, j = 1, · · · , J

? as if this is the only scatterer in the configuration with

? new boundary conditions involving

? original incident field and contribution from other J − 1 scatterers







• Represent exterior electric and magnetic fields as

E(x) =

J∑j=1

Ej(x), H(x) =

J∑j=1

Hj(x), Ej(x) = curl

∫∂Dj

Φ(x,y)w(y) ds(y)







• Represent exterior electric and magnetic fields as

E(x) =

J∑j=1

Ej(x), H(x) =

J∑j=1

Hj(x), Ej(x) = curl

∫∂Dj

Φ(x,y)w(y) ds(y)

• The unknown density w on ∂Dj can be computed by the fact that

? for each i = 1, · · · J, [Ej,Hj] represents the unique radiating solution ofthe time harmonic Maxwell equations exterior to Dj,

? subject to the boundary condition

n(x)×Ej(x) = n(x)×Einc(x)−J∑

m = 1m 6= j

n(x)×Em(x) =: f (x) x ∈ ∂Dj.

30

Fully discrete multiple electromagnetic scattering algebric system

• For i = 1, · · · , J, approximate the local surface current as

wi,n(x) =

N∑r=1

wriFi(x)Zr

i (x), x ∈ ∂Di.

• Compute coefficients win by solving the fully discrete Galerkin system

(wi,n,FiZpi )m(n) +

J∑j=1

(M(n)ij wj,n,FiZp

i )m(n) = (2F i,FiZpi )m(n),

i = 1, · · · , J, p = 1, · · · , N

• where (·, ·)m(n) is a spectrally quadrature approximation

of the outer Galerkin surface integrals (·, ·)m(n)

• Let Mij be the N ×N matrix version of the Galerkin form of M(n)i,j

31

Fully discrete direct and boundary decomposition linear systems

• Coefficient vectors wi in the direct approach solves the NJ−dim system w1

...wJ

+

M11 . . . M1J

... ...MJ1 . . . MJJ

w1

...wJ

=

f1...fJ



...wJ

+

M11 . . . M1J

... ...MJ1 . . . MJJ

w1

...wJ

=

f1...fJ

• In the boundary decomposition algorithm wi is written in series form

wi =

∞∑k=1

wi,k,



...wJ

+

M11 . . . M1J

... ...MJ1 . . . MJJ

w1

...wJ

=

f1...fJ

• In the boundary decomposition algorithm wi is written in series form

wi =

∞∑k=1

wi,k,

where for the J particle problem, for each i = 1, · · · , J, wi,k solve

(I + Mii)wi,1 = fi,

(I + Mii)wi,k = −J∑

j = 1

j 6= i

Mijwj,k−1, k > 1,

• Penultimate equation: scattering of the incident wave by each scatterer

• Last equation: Scattering by each scatterer of the component of the fieldincident on the scatterer that has undergone k − 1 reflections.

32

Spectrally accurate algorithms for the two approaches

• The boundary decomposition (BD) approach is efficient,

? due to the need to solve several single obstacle scattering problems

? each with N ×N dimensional dense complex matrix.

• The BD iterates diverges for nearby located particles.

Spectrally accurate algorithms for the two approaches

• The boundary decomposition (BD) approach is efficient,

? due to the need to solve several single obstacle scattering problems

? each with N ×N dimensional dense complex matrix.

• The BD iterates diverges for nearby located particles.

• The direct approach (treating the whole configuration as a single sytem)

? works for all multiply-connected J particle configurations,

? but requires solutions of systems with NJ ×NJ dense complex matrix

? hence suitable mainly for small J particles.

• Our high-order spectrally accurate algorithm for each obstacle requiresonly about only about 5% of the unknowns,compared to standard algorithms.

• Hence we can use both the approachesfor spectral Galerkin surface integral algorithms

33

Scattering by two moving spheres : Comparison with measurements

Numerical using GH (present; n = 30)) Experimental (—–) and (....) Multipole (IEEE Antennas ... 1971)

• Backscattered RCS for two moving unit spheresat various separation values d with k = 11.048

34

Scattering by two moving spheres : CPU time for 500 separations

CPU time to simulate backscattered RCS for 2 × sph(10λ)

for 500 evenly spaced separation

distances d in [10.01λ, 50λ] with n = 45

Algorithm 4 × DcOp1 CPU time BD Error2

Direct approach 90.0 h -

Reuse diagonal in direct 56.0 h -

BD approach 3 33.0 h 1.7e-10

[1]DcOp - Dual-core 2.0GHz Opteron processor.[2]Error is computed using only converged values.[3]Jacobi iterations did not converge within100 iterations for first three separations.

35

Total electric field |E(·, t)| at t = 0.5/ω behind 2× bean(24λ) and detection of

radiation free (shadow) region (simulated with n = 125; 5-digit accuracy).

36

Total electric field |E(·, t)| at t = 0.5/ω behind 2× fount(24λ) and detection of


37

Total electric field |E(·, t)| at t = 0.5/ω behind 2× ice(10λ) and detection of


38

Total electric field |E(·, t)| at t = 0 exterior to 125× sph(λ)

(simulated with n = 15; 6-digit accuracy).

39

Total electric field |E(·, t)| at t = 0 exterior to 27× bean(λ)


40

Total electric field |E(·, t)| at t = 0.5 exterior to 27× fount(λ)


41

Total electric field |E(·, t)| at t = 0.5 exterior to 8× ice(λ)


42

spectral-galerkin surface integral methods for 3d computational...

Documents