Adaptive Cross Approximation in the Context of hp ... · UC3M 3D Formulation(cont.) Variational Fomulation Find W 2H(curl)0 such that ZZZ (r F) ( 1 f r r V)d k2 0 ZZZ Fg rVd f r ZZ

UC3MAdaptive Cross Approximation in the Context ofhp-Discretizations

Luis E. Garcıa-Castillo (UC3M)

Ph.D. work of Rosa M. Barrio-Garrido

BCAM June 2014 ACA in the context of hp-discretizations v1.1 1 / 58

UC3MAbstract

In this presentation, Adaptive Cross Approximation (ACA) is used toaccelerate the computations associated to a boundary integral basedmesh truncation technique for finite element analysis of electromagneticscattering and radiation problems. ACA is used in the context of hp finiteelement discretizations. Illustrative results of the performance of ACA inthis context will be shown.

UC3MUniversidad Carlos III de Madrid

3 campuses (actually, 4)3 Schools

I Social & Legal SciencesI Humanities,

Communications andLibrary Sciences

I Polytechnic School


UC3MUniversidad Carlos III de Madrid

Public University in Madrid area; created in 1989

Humanities and engineeringI A total of 17000 studentsI Around 7000 in Engineering (Telecommunications, Industrial, Computer

Science)F 1200 in all telecommunication (electrical engineering) “family”F 103 new telecommunication engineers each year (50 of first cycle)

27 departments in 3 campus: Getafe, Leganes, Colmenarejo.

24 research institutes

116 research groups


UC3MRadio Frequency Group

PersonalI 7 professorsI 10 Ph.D. students (7 granted)I A number of M.S. studentsI 1 laboratory technician + 1 laboratory assistant

FacilitiesI Microwave laboratory and anechoic chamber for antenna measurements

(up to 50 GHz).I LPKF protolaser and microdrilling machine.I Computer clusters.

Research linesI Antennas (special focus on planar antennas)I New materials: EBG (Electronic Band Gap), FSS (Frequency Selective

Surfaces), MetamaterialsI Microwave circuits and subsystemsI Numerical methods for computational electromagneticsI Filters and multiplexers


UC3M


UC3Mq Applications:

Waveguiding structuresI Mode characterization (eigenvalue problem)I S-parameters of discontinuities (deterministic

problem)

Filter analysis and synthesis. Multiplexers.

ScatteringI Computation of Radar Cross Section (RCS) of

objects

RadiationI AntennasI Antennas mounted on platformsI On-board antennas (ships and aircrafts)I Indoor antennasOutline

1 hp-FEM

2 FE-IIEE for Scattering and Radiation EM Problems

3 ACA + hp-FE-IIEE

4 Conclusions


UC3MFeatures of hp-Adaptivity

hp adaptivity: simultaneous variation of the size, h, and polynomialorder of aproximation, p, of the elements of the mesh

given initial mesh

h-refined mesh p-refined mesh hp-refined mesh


UC3MFeatures of hp-Adaptivity (cont.)

hp-Adaptivity Features:I Exponential convergence of the error (even in the

presence of singularities)I Delivery of optimal meshes (even in pre-asymptotic

regime)I Accuracy & Efficiency

hp-adaptivity is a technology developed almost exclusively within theapplied mathematics and computational mechanics community

I complexity of mathematic analysisI complexity of implementationI only one commercial implementation (mechanics and fluid dynamics)


UC3MFeatures of hp-Adaptivity (cont.)

hp-adaptivity for electromagnetics (Maxwell equations) requires specificmathematical analysis and implementations

I no commercial implementationI only a few implementations in academia

+ Self-adaptive hp strategy developed by the authors at ICES (Institute forComputational Engineering and Sciences) of the University of Texas at Austin

F hp-Self-adaptive (fully automatic) strategy for electromagnetic problems in 1D,2D and 3D

F Isoparametric elementsF Segments, Quadrilaterals, Triangular and Hexaedral elements (tetrahedrons

recently added)F Anisotropic refinementsF Support for 1-irregular meshes (i.e., with hanging nodes)


UC3MAutomatic hp-Adaptivity in 3D

(a) initial mesh (b) after firststep

(c) after thirdstep

Figure: Illustration of hp-adaptivity showing refinements around a rectangularobstacle in a waveguide (placed vertically in the figure)


UC3MAlgorithm

Fine

OptimalCoarse Mesh

New

Materials & B.C.)(Geometry, Sources,Input Data

PostproccessCompute errorCoarse Solve for Coarse Solve for Fine

Mesh (h/2, p+ 1)εM < T

Mesh φC Mesh φFMesh (h, p)

Yes

No

εM = ||φF − φC ||

Self-Adaptive hp-FEM code

Delivery of optimal meshesApplies to H1-, H(curl)-, and H(div)-conforming discretizationsFlexible: problem independent, nonlinear and eigenvalue problems,extension to goal-oriented approaches.


UC3MH1 2D Variational Formulation

Helmholtz equation

∇t ·[f−1r ∇t φ

]+ k2

0 gr φ = q

Boundary conditions

φ (ρ) = 0 ρ ∈ ΓD;∂φ (ρ)

∂n= 0 ρ ∈ ΓN

∂φ(ρ)

∂n+ j β φ(ρ) = 2 j β φimp

pi(ρ) ρ ∈ Γpi

∂φ(ρ)

∂n+ j k0 φ(ρ) = Ψ(ρ) ρ ∈ ΓS

Pol. φ fr gr q ΓD ΓN

TM Ez µr εr jk0η0Jz ΓPEC ΓPMC

TE Hz εr µr jk0/η0Mz ΓPMC ΓPEC


UC3MH1 2D Variational Formulation (cont.)

q Find φ ∈ H1 such that

b(φ, ω) = f (ω) ∀ω ∈ H10

being H10 := p ∈ H1(Ω), p = 0 on ΓD and

b(φ, ω) =

∫

Ω

∇tw ·[f−1r ∇t φ

]dΩ− k2

0 gr

∫

Ω

w φdΩ

+ j k0 f−1r

∫

ΓS

w φdΓ + j βpi

∫

Γpi

w φdΓ

f (ω) = −∫

Ω

w q dΩ +

∫

ΓS

w Ψ dΓ + 2 j βpi

∫

Γpi

w φimppi

dΓ


UC3MH(curl) 2D Formulation

H-Plane Variational FormulationFind HΩ ∈W, p ∈ V such that

c(FΩ,HΩ) = l(FΩ) ∀FΩ ∈W

W := A ∈ H(curl),Ω), n× A = 0 on ΓD

c(FΩ,HΩ) =

∫

Ω

(∇× FΩ) · ( 1εr∇× HΩ) dΩ− k2

o

∫

Ω

FΩ · µr HΩ dΩ

+ jk2

εrβ10

∫∑

Γip

(n× FΩ) · (n× HΩ) dΓ

l(FΩ) = 2jk2

εrβ10

∫

Γinp

(n× FΩ) · (n× Hin) dΓ


UC3M3D Formulation

Wave equation

∇×(

¯f−1r ∇× V

)− k2

0¯gr V = 0; en Ω

Boundary conditions

n × V = 0; on ΓD (PEC or PMC)

n ×(

¯f−1r ∇× V

)= 0; en ΓN (PEC or PMC)

n ×(

¯f−1r ∇×V

)+ γ n ×n ×V = Uinc; on ΓP (Waveguide Ports)

V fr gr PEC PMC γ Uinc

Form. E E µr εr ΓD ΓN jβz 2γ

frn × n × Einc

Form. H H εr µr ΓN ΓD jk2

0

βz2γ

frn × n × Hinc


UC3M3D Formulation (cont.)

Variational Fomulation

Find WΩ ∈ H(curl)0 such that

∫∫∫(∇× F) · ( 1

fr∇× V) dΩ− k2

0

∫∫∫F · gr V dΩ

+γ

fr

∫∫∑

Γip

(n×F)·(n×V) dΓ = 2γ

fr

∫∫

Γincp

(n×F)·(n×Uinc) dΓ ∀F ∈W

beingH(curl)0 := A ∈ H(curl),Ω), n× A = 0 on ΓD

I Symbol Γip stands for the i-th waveguide port

I Symbol Γincp denotes the excited port


UC3MScattering and Radiation Problems


UC3MScattering and Radiation Problems (cont.)





Isotropic, onmidirectional and general directive radiation patterns





R. Otin, ”Regularized Maxwell equations and nodal finite elements for electromagneticfield computations in frequency domain,” ISBN:978-84-89925-03-8, Ed.CIMNE, Barcelona (Spain), 2011










UC3MFE-IIEEFinite Element - Iterative Integral Equation Evaluation

ε1 µ1

Medium 1

ε2 µ2

Medium 2

PEC

PMC

PEC

J

n′

n

M

S ′

Meq

Jeq

S

Exterior Domain(ΩEXT)Einc,Hinc

FEM Domain(ΩFEM)

Cauchy b.c. on truncationboundary S.

n× (∇× E) + jk n× n×E = Ψ

Cauchy residual Ψ upgraded ateach iteration cycle usingGreen’s function of exteriordomain

é asymptotically exact b.c. atthe continuous level


UC3MFE-IIEE (cont.)Finite Element - Iterative Integral Equation Evaluation

Assembly ofElement Matrices

Imposition of B.C.Non Related to S

PostprocessSparseSolver

InitialB.C on S

Computation ofElement Matrices

Upgrade of B.C. on

Mesh

FEM code for non−open problems

Ψ(0)(r)

S: Ψ(i+1)(r)

J(i)eq (r ′),M(i)

eq (r ′)⇒

V (r ∈ ΓS)∇×V (r ∈ ΓS)

q Features:Original sparse structure of the FEM matrices is retainedFEM matrix does not change with iterations (only RHS)Modularity: decoupling of FEM (interior) and exterior domains


UC3MScattered FieldsIntegral Expressions

Scattered field:

VIE-FEM =

∫∫©

S′(Leq ×∇G) dS′−jkh

∫∫©

S′

(Oeq

(G +

1k2∇∇G

))dS′

∇×VIE-FEM = jkh∫∫©

S′(Oeq ×∇G) dS′−

∫∫©

S′

(Leq

(k2G + ∇∇G

))dS′

with G the Green’s function of the problem.

For homogeneous media:

G ≡ G (r, r′) =1

4πe−jk(r−r′)

|r− r′| r ∈ ΓS, r′ ∈ ΓS′


UC3MScattered FieldsIntegral Expressions for Scalar Formulation

Scattered field:

φsc(ρ) =

∮

S′

[Lt

eq(ρ′)∂G(ρ,ρ′)

∂n′−jk0I0Oz

eq(ρ′) G(ρ,ρ′)]

dl ′ ρ ∈ ΓS

and its derivative,

φsc(ρ)

dρ=

∮

S′

[−jk0I0Oz

eq(ρ′)∂G(ρ,ρ′)

∂n

+Lteq(ρ′)

∂

∂n

(∂G(ρ,ρ′)

∂n′

)]dl ′ ρ ∈ ΓS

with G the Green’s function of the problem.

For homogeneous media in 2D:

G ≡ G (ρ,ρ′) =j4

H(2)0 (k |ρ− ρ′|) ρ ∈ ΓS, ρ

′ ∈ ΓS′


UC3MConvergence Analysis of Iterative FEMScattering on Sphere

Influence of distance S-S′ for different materials

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Number of Iteration

||bΨ(i)

−b Ψ(i−

1)||/

||bΨ(i)

||

bΨ Convergence of the Relative Error

Perfect Conductor Sphere R=0.91728λ

RS−R

S’=0.050λ

RS−R

S’=0.075λ

RS−R

S’=0.100λ

RS−R

S’=0.125λ

RS−R

S’=0.150λ

RS−R

S’=0.175λ

RS−R

S’=0.200λ

Perfect conducting sphere

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Number of Iteration

||bΨ(i)

−b Ψ(i−

1)||/

||bΨ(i)

||


Dielectric Sphere R=0.91728λ (εr=2 µ

r=1)

RS−R

S’=0.050λ

RS−R

S’=0.075λ

RS−R

S’=0.100λ

RS−R

S’=0.125λ

RS−R

S’=0.150λ

RS−R

S’=0.175λ

RS−R

S’=0.200λ

Dielectric sphere (lossless case)


UC3MConvergence Analysis of Iterative FEM (cont.)Scattering on Sphere

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Number of Iteration

||bΨ(i)

−b Ψ(i−

1)||/

||bΨ(i)

||


Dielectric Sphere R=0.91728λ (εr=2−0.02j µ

r=1)

RS−R

S’=0.050λ

RS−R

S’=0.075λ

RS−R

S’=0.100λ

RS−R

S’=0.125λ

RS−R

S’=0.150λ

RS−R

S’=0.175λ

RS−R

S’=0.200λ

Dielectric sphere (low losses)

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Number of Iteration||b

Ψ(i)−

b Ψ(i−1)

||/||b

Ψ(i)||


Dielectric Sphere R=0.91728λ (εr=2−2j µ

r=1)

RS−R

S’=0.050λ

RS−R

S’=0.075λ

RS−R

S’=0.100λ

RS−R

S’=0.125λ

RS−R

S’=0.150λ

RS−R

S’=0.175λ

RS−R

S’=0.200λ

Dielectric sphere (high losses)


UC3Mhp+FE-IIEE + ACA

Fine

OptimalCoarse Mesh

New

Materials & B.C.)

(Geometry, Sources,

Input DataPostproccess

Compute errorCoarse Solve for Coarse Solve for Fine


Sparse Solver ||φ(i)-φ(i−1)||<U

Upgrade of B.C. on S: Ψ(i+1)

φ(r′),∂φ(r′)

∂n′ =⇒ φsc(r),∂φsc(r)

∂n

⇑


Yes

No

εM = ||φF − φC ||


YesNo

Initial B.C. on S.

Ψ(0) = Ψinc +Ψ(0)sc

ITERATIVE FEM FOR OPEN PROBLEMS

FEM code for non-open problems

ACA

U is set adaptively as U ≈ T (j)

10at j-th hp-iteration


UC3MIE MatrixNumerically Rank Defficient Matrices

zφ ij =

∫

ΓS

ωi (ρ)

∮

S′

(ω′ j (ρ

′)∂G(ρ,ρ′)

∂n′− ∂ω′ j (ρ

′)

∂n′G(ρ,ρ′)

)dl ′ dΓ


UC3MACA Method

Rm×n = Um×r Vr×n =r∑

i=1

um×1i v1×n

i

Compression:(m × n)→ (m + n)× r


UC3MACA Algorithm

Initialization:1 Set Z(0) = 0.2 Selection of the 1st row of submatrix Z to be computed. R1 = 13 Computation of the 1st row of Z. (R1 = 1)⇒ Z(1, :)4 Selection of the column of Z to be computed next. Index C1 is finded by

selecting the row in Z(1, :) that makes |Z(R1,C1)| = max(|Z(R1, :)|)5 Computation of the corresponding column of Z . C1 ⇒ Z(:,C1)

6 Computation of the 1st row vector of matrix V as:v1 = Z(R1, :)/Z(R1,C1).

7 Computation of the 1st column vector of matrix U as: u1 = Z(:,C1).8 Computation of the estimated Frobenius norm of the approximative

impedance matriz at 1st iteration: ‖Z(1)‖2 = ‖Z(0)‖2 + ‖u1‖2‖v1‖2


UC3MACA Algorithm

k -th Iteration:1 Selection of the row index, Rk , that makes|Z(Rk ,Ck−1)| = max(|Z(:,Ck−1)|), such that Rk 6= [R1,R2, · · ·Rk−1]

2 Computation of the corresponding row of Z. (Rk )⇒ Z (Rk , :)

3 Compute the error commited in the approximation of the Rk row of the Zmatrix through the previous iterations, as:erow = Z(Rk , :)−

∑k−1l=1 u(Rk , l)vl

4 Selection of the column index, Ck , that makes|erow(Rk ,Ck )| = max(|Z(Rk , :)|), such that Ck 6= [C1,C2, · · ·Ck−1]

5 Computation of the corresponding column of Z. Ck ⇒ Z(:,Ck ).6 Computation of the k th row vector of matrix V. vk = erow/erow(Rk ,Ck ).7 Compute the error commited in the approximation of the Ck column of

the Z matrix through the previous iterations, as:ecolumn = Z(:,Ck )−∑k−1

l=1 V(l ,Ck )ul


UC3MACA Algorithm (cont.)

k -th Iteration:8 Computation of the k th column vector of matrix U. uk = ecolumn.9 Computation of the estimated Frobenius norm of the approximative

impedance matriz at k th iteration:‖Z(k)‖2 = ‖Z(k−1)‖2 + 2

∑k−1j=1 |uT

j uk | · |vjvTk |+ ‖uk‖2‖vk‖2.

10 Check convergence. If ‖uk‖‖vk‖ ≤ δ‖Z(k)‖ ⇒ stop algorithm.Otherwise, goes to step 1) of the (k + 1)th iteration.


UC3MACA ValidationAn Example: PEC Circular Cylinder

200 250 300 350 400 450 500−60

−50

−40

−30

−20

−10

0

10

! (ï¿½)

E z (dB)

ove

r bou

ndar

y S

Ez exacto

Ez iter hp 1

Ez iter hp 16



Fine

OptimalCoarse Mesh

New

Materials & B.C.)

(Geometry, Sources,






φ(r′),∂φ(r′)


∂n

⇑


Yes

No

εM = ||φF − φC ||


Yes

No

Initial B.C. on S.




ACA




UC3MError Control in ACADependence with FE-IIEE Error Threshold U

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

10−5

10−4

10−3

10−2

10−1

100

101

Exact error on S versus ε. Number of FE-IIEE iterations (threshold U) as parameter


UC3MError Control in ACADependence with FE-IIEE Error Threshold U (cont.)

ACA Error Control Features:Error control via εRobustness: crucial feature for the double nested loop

Adaptive Selection of ε

With ε ≈ U(i)

10⇒ we assure ACA does not alter (FE-IIEE + hp-adaptivity)

double loop error control.

Typically ε ∈[

U(i)

10− U(i)

]is OK



Fine

OptimalCoarse Mesh

New

Materials & B.C.)

(Geometry, Sources,






φ(r′),∂φ(r′)


∂n

⇑


Yes

No

εM = ||φF − φC ||


Yes

No

Initial B.C. on S.




ACA




UC3MError Control in ACA+AdaptivityDependende with hp Error Threshold T

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

101

Iter. hp 25

Iter. hp 20

Iter. hp 16

Iter. hp 11

Estimated error on S versus ε. hp-iteration number (threshold T ) as parameter


UC3MError Control in ACA+AdaptivityDependende with hp Error Threshold T (cont.)

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

101

Iter. hp 25

Iter. hp 20

Iter. hp 16

Iter. hp 11

Exact error on S versus ε. hp-iteration number (threshold T ) as parameter


UC3MACA Compression RatesDependence with ε and hp Error Threshold T

10−3 10−2 10−1 100 10165

70

75

80

85

90

95

100

Error control parameter in ACA, ε

Com

pres

sion

leve

l %

Iter. hp 25

Iter. hp 20

Iter. hp 16

Iter. hp 11

ACA compression rate versus ε. hp-iteration number (threshold T ) as parameter


UC3MACA Compression RatesSensibility to Type of Refinements

0 200 400 600 800 1000 1200 1400 16000

10

20

30

40

50

60

70

80

90

100

Number of unknowns on S

Compression level (%)

h refinement, ε = 0.001h refinement, ε = 0.01h refinement, ε = 0.1

p refinement, ε = 0.001p refinement, ε = 0.01p refinement, ε = 0.1

ACA compression rate versus number of dof on S.Type of refinement (h or p) and ε as parameter


UC3MACA PerformanceComputational Cost Estimates

GrN(12p3 + 39p2 + 57p + 30)︸︷︷︸Computation of coefficients U i V j

+ Gf (r)N(2p + 4)︸︷︷︸ACA loop

︸︷︷︸tUV

+ 2GrN(p + 1)I︸︷︷︸U×V×u′

j

︸︷︷︸tACA

< O(pN2I)︸︷︷︸tINT

where f (r) = [r + (r − 1) + (r − 2) + · · ·+ 1]

N: number of dof on Sp: polynomial orderr : numerical rank ACA blocks

G: number of ACA groupsI: number of FE-IIEEiterations

ACA UsageDependence with I is mitigated: tUV dominatesQuadratic dependence (N2) becomes (GrN) with ACA.


UC3MACA PerformanceTypes of Results Shown

Under Different Refinement Patternsh-refinements with object of electrical size R/λ constanth-refinements with R/λ increasing (such that h/λ ≈ constantp-refinements with object of electrical size R/λ constantp-refinements with R/λ increasing (such that h/λ ≈ constant

Under Different ACA Grouping CriteriaInfluence of the number of groups G with the electrical size of the object


UC3MACA with R/λ and p constanth-Refinements

103 10410−1

100

101

102

103

104

Number of unknowns on S: N (for tint) and Nfar (for tuv & taca)

CPU

−tim

e (s

ecs)

tINT (p = 2)

O(N2.25)

tINT (p = 4)

O(N2.45)

tINT (p = 8)

O(N2.10)

p = 2(−) tACA, (−−) tUV

O(N1.05)

p = 4(−) tACA, (−−) tUV

O(N1.20)

p = 8(−) tACA, (−−) tUV

O(N1.50)

p = 8

p = 2

p = 4

p = 4

p = 2

p = 8

p = 2I = 1 − 8 iterr = 1 − 3Compression = 80% − 97%Error = 7% − 0.07%

p = 4I = 4 − 18 iterr = 2.00 − 4.91Compression = 80% − 97%Error = 0.4% − 8.1 x 10−5%

p = 8I = 22 − 26 iterr = 5.07 − 15.26Compression = 75% − 95%Error = 5 x 10−6% − 4 x 10−7%

r , p,G, I are constant ⇒ tUV and tACA are of O(N1.x ) and tUV ≈ tACABCAM June 2014 ACA in the context of hp-discretizations v1.1 50 / 58

UC3MACA with R/λ constant and p ↑p-Refinements

10310−1

100

101

102


CPU

−tim

e (s

ecs)

tINT

(−) tACA(−−) tUV

I = 5 − 23 iterr = 2 − 7.4Compression = 82%Error = 0.73% − 3.6 x 10−7%

p = 2

p = 8

p = 2

p = 8

1st tranche: N2.3

2nd tranche: N3.9

2nd tranche: N3.9

1st tranche: N2.85

Better scalability for moderate p. Computation time decreased around one order ofmagnitude


UC3MACA with R/λ ↑ and p constanth-Strategy

102 10310−2

10−1

100

101

102

103


CPU

−tim

e (s

ecs)

tINT (p = 4)

Max: N3.5

p = 2I = 2 − 33 iterr = 2 − 2.7 − 2Compression = 70% − 90%Error ! 0.7%

p = 4tACA max: N1.6

tUV max: N1.45

p = 2tACA max: N1.55−N2.4

tUV max: N1.55

p = 8tACA max: N2

tUV max: N1.75

tINT (p = 8)

Max: N4.6

tINT (p = 2)

Max: N5

p = 4I = 3 − 45 iterr = 2.8Comp. = 65% − 86%Error ! 0.07%

p = 8I = 5 − 68 iterr = 3.3Comp. = 49% − 83%Error ! 0.005%

p = 2R = "

p = 4R = "

p = 8R = "

p = 2R = 25"

p = 4R = 25"

p = 8R = 25"

(r , p const.) + (G ↑ with R/λ) ⇒ (tUV) ↑ moderately, (I ↑)⇒ (tACA) ↑ (tINT) ↑ ↑ ↑strongly influenced by I ↑


UC3MACA with R/λ ↑ and p ↑p-Strategy

10310−2

10−1

100

101

102

103

104


CPU

−tim

e (s

ecs)

tINT

(−) tACA(−−) tUV

I = 2 − 88 iterr = 2 − 4Compression = 85% − 90%Error = 0.39% − 6.5 x 10−5%

p = 2R = !

p = 8R = 25!

p = 2R = !

p = 8R = 25!

2nd tranche: N4.2

1st tranche: N3.1

1st tranche: N2

2nd tranche: tACA: N2.8

tUV: N2.65

(p ↑) + (r ↑ with accuracy) + (G ↑ with R/λ) ⇒ (tUV) ↑ (I ↑)⇒ (tACA) ↑ (tINT) ↑↑ ↑ strongly influenced by I ↑


UC3MACA with R/λ ↑ and p ↑Influence of ACA Group Criterion

10310−2

10−1

100

101

102

103

104


CPU

−tim

e (s

ecs)

tINT

1st tranche: N3.1

2nd tranche: N4.2

p = 2R = !

Reference 1st tranche: N2

Ref. 2nd tranche: tACA: N2.8

tUV: N2.65

1st tranche: N1.75


tUV: N2.65

1st tranche: N1.85


tUV: N2.57

(−−) tUV(−) tACAG : number of ACA groups

p = 3R = 5!

G " (bigger groups)

G # (smaller groups)

Reference: (G = 16 − 86)I = 2 − 88 iterr = 2.00 − 4.00Compression = 85% − 90%Error = 0.39% − 6.5 x 10−5%

G ", bigger groups (G = 8 − 24)I = 8 − 105 iterr = 5.72 − 7.28Compression = 94%Error = 4.2 x 10−2% (5!, p=3) − 1.6 x 10−5%

G #, smaller groups (G = 28 − 124)I = 2 − 88 iterr = 2.00 − 3.62Compression = 75% − 85%Error = 0.4 − 6.3 x 10−5%

Optimum: G ≈√

10R/λBCAM June 2014 ACA in the context of hp-discretizations v1.1 54 / 58

UC3MConclusions & Future Research

ConclusionsACA provides robust error control which is crucial for adaptivity andspecifically for the double loop of hp-FE-IIEEACA reduces computation time by at least one order of magnitude formoderate-large electrical size objects.ACA reduces order of complexity almost in all situations (h and/or p)ACA competitive performance in comparison with FMM

TODO List3D Implementation in HOFEM (Higher Order Finite Element Mode) code


UC3M,,


UC3MFor Further Reading

L. Demkowicz, Computing with hp Finite Elements. I. One- andTwo-Dimensional Elliptic and Maxwell Problems. Chapman &Hall/CRC Press, Taylor and Francis, 2007.

R. Fernandez-Recio, L. E. Garcıa-Castillo, I. Gomez-Revuelto, andM. Salazar-Palma, “Fully coupled hybrid FEM-UTD method usingNURBS for the analysis of radiation problems,” IEEE Transactions onAntennas and Propagation, vol. 56, no. 3, pp. 774–783, Mar. 2008.

M. Bebendorf and S. Rjasanow, “Adaptive low-rank approximation ofcollocation matrices,” Computing, vol. 70, no. 1, pp. 1–24, Mar. 2003.

R. M. Barrio-Garrido, L. E. Garcıa-Castillo, I. Gomez-Revuelto, andM. Salazar-Palma, “Medidas experimentales de la complejidadcomputacional de un codigo autoadaptativo hp para problemas abiertosacelerado mediante ACA,” in XXVIII Simposium Nacional de la URSI,Santiago de Compostela, Espana, sep 2013.


UC3MThank you for your attention !!

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