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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
3D Computational Electromagnetics
• 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
3D Computational Electromagnetics
• Perform simulation for cases in which experimental RCS is known
RCS of two spheres - IEEE Antennas Propag. 19, 378-390 (1971)
5
3D Computational Electromagnetics
• 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
8
9
Spectral Surface Integral methods - Literature
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.
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.
10
Spectral Surface Integral methods - Literature
Spectral Surface Integral methods - Literature
• 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
Spectral Surface Integral methods - Literature
• 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
Spectral Surface Integral methods - Literature
Spectral Surface Integral methods - Literature
• 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
Spectral Surface Integral methods - Literature
• 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
Spectral Surface Integral methods - Literature
• 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
? A related new matrix-free approximation theory tool was developed:M. G and H. N. MhaskarSIAM J. Numer. Anal., 44 (2006), 1314-1331
12
Spectral Surface Integral methods - Literature
Spectral Surface Integral methods - Literature
• 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
Spectral Surface Integral methods - Literature
• 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
Spectral Surface Integral methods - Literature
• 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
• Electromagnetic scattering in 3D (perfect conductor):
? M. Pieper,Spektralrandintegralmethoden zur Maxwell-Gleichung,Ph.D. Thesis (Supervisor: R. Kress),Universitat Gottingen, Germany, 2007
13
Spectral Surface Integral methods - Literature
Spectral Surface Integral methods - Literature
• 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):
# Unknows 72 288 512Error 7.9e-03 5.9e-04 3.0e-04
Spectral Surface Integral methods - Literature
• 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):
# Unknows 72 288 512Error 7.9e-03 5.9e-04 3.0e-04
• 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
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
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
• 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
Spectral-Galerkin tool #1: Spherical coordinate representation
Erythrocytes (Courtesy: NIH) Model: Red blood cell
17
Spectral-Galerkin tool #1: Spherical coordinate representation
18
Spectral-Galerkin tool #1: Spherical coordinate representation
• 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
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.
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.
• 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
Spectral-Galerkin tool #2: High-order tangential basis on ∂Dj
• 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
Spectral-Galerkin tool #2: High-order tangential basis on ∂Dj
• Standard transformations do not retain the n−rj spectral accuracy
Spectral-Galerkin tool #2: High-order tangential basis on ∂Dj
• 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
Multiple electromagnetic Scattering in three dimensions
• Compute time-harmonic electric and magnetic fields
Multiple electromagnetic Scattering in three dimensions
• Compute time-harmonic electric and magnetic fields
E(x, t) =1√ε0
Re{E(x)e−iωt
}, H(x, t) =
1õ0
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
? with spectral accuracy
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
? with spectral accuracy
• 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
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
? with spectral accuracy
• 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
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
? with spectral accuracy
• 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
• 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
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)
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)
• 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
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
• In the boundary decomposition algorithm wi is written in series form
wi =
∞∑k=1
wi,k,
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
• 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
radiation free (shadow) region (simulated with n = 125; 5-digit accuracy).
37
Total electric field |E(·, t)| at t = 0.5/ω behind 2× ice(10λ) and detection of
radiation free (shadow) region (simulated with n = 110; 6-digit accuracy).
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)
(simulated with n = 35; 6-digit accuracy).
40
Total electric field |E(·, t)| at t = 0.5 exterior to 27× fount(λ)
(simulated with n = 35; 6-digit accuracy).
41
Total electric field |E(·, t)| at t = 0.5 exterior to 8× ice(λ)
(simulated with n = 60; 6-digit accuracy).
42