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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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 3 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 4 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 5 / 58
UC3M
BCAM June 2014 ACA in the context of hp-discretizations v1.1 6 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 6 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 7 / 58
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)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 8 / 58
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)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 9 / 58
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)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 10 / 58
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.
BCAM June 2014 ACA in the context of hp-discretizations v1.1 11 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 12 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 13 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 14 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 15 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 16 / 58
UC3MScattering and Radiation Problems
BCAM June 2014 ACA in the context of hp-discretizations v1.1 17 / 58
UC3MScattering and Radiation Problems (cont.)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 18 / 58
UC3MScattering and Radiation Problems (cont.)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 19 / 58
UC3MScattering and Radiation Problems (cont.)
Isotropic, onmidirectional and general directive radiation patterns
BCAM June 2014 ACA in the context of hp-discretizations v1.1 20 / 58
UC3MScattering and Radiation Problems (cont.)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 21 / 58
UC3MScattering and Radiation Problems (cont.)
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 22 / 58
UC3MScattering and Radiation Problems (cont.)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 23 / 58
UC3MScattering and Radiation Problems (cont.)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 24 / 58
UC3MScattering and Radiation Problems (cont.)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 25 / 58
UC3MScattering and Radiation Problems (cont.)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 26 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 27 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 28 / 58
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′
BCAM June 2014 ACA in the context of hp-discretizations v1.1 29 / 58
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′
BCAM June 2014 ACA in the context of hp-discretizations v1.1 30 / 58
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)
||
bΨ Convergence of the Relative Error
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)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 31 / 58
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)
||
bΨ Convergence of the Relative Error
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)||
bΨ Convergence of the Relative Error
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)
BCAM June 2014 ACA in the context of hp-discretizations v1.1 32 / 58
UC3Mhp+FE-IIEE + ACA
Fine
OptimalCoarse Mesh
New
Materials & B.C.)
(Geometry, Sources,
Input DataPostproccess
Compute errorCoarse Solve for Coarse Solve for Fine
Mesh (h/2, p+ 1)εM < T
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
⇑
Mesh φC Mesh φFMesh (h, p)
Yes
No
εM = ||φF − φC ||
Self-Adaptive hp-FEM code
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 33 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 34 / 58
UC3MACA Method
Rm×n = Um×r Vr×n =r∑
i=1
um×1i v1×n
i
Compression:(m × n)→ (m + n)× r
BCAM June 2014 ACA in the context of hp-discretizations v1.1 35 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 36 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 37 / 58
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.
BCAM June 2014 ACA in the context of hp-discretizations v1.1 38 / 58
UC3MACA ValidationAn Example: PEC Circular Cylinder
200 250 300 350 400 450 500−60
−50
−40
−30
−20
−10
0
10
! (�)
E z (dB)
ove
r bou
ndar
y S
Ez exacto
Ez iter hp 1
Ez iter hp 16
BCAM June 2014 ACA in the context of hp-discretizations v1.1 39 / 58
UC3Mhp+FE-IIEE + ACA
Fine
OptimalCoarse Mesh
New
Materials & B.C.)
(Geometry, Sources,
Input DataPostproccess
Compute errorCoarse Solve for Coarse Solve for Fine
Mesh (h/2, p+ 1)εM < T
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
⇑
Mesh φC Mesh φFMesh (h, p)
Yes
No
εM = ||φF − φC ||
Self-Adaptive hp-FEM code
Yes
No
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 40 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 41 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 42 / 58
UC3Mhp+FE-IIEE + ACA
Fine
OptimalCoarse Mesh
New
Materials & B.C.)
(Geometry, Sources,
Input DataPostproccess
Compute errorCoarse Solve for Coarse Solve for Fine
Mesh (h/2, p+ 1)εM < T
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
⇑
Mesh φC Mesh φFMesh (h, p)
Yes
No
εM = ||φF − φC ||
Self-Adaptive hp-FEM code
Yes
No
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 43 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 44 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 45 / 58
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
BCAM June 2014 ACA in the context of hp-discretizations v1.1 46 / 58
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
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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.
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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
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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
Number of unknowns on S: N (for tint) and Nfar (for tuv & taca)
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
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UC3MACA with R/λ ↑ and p constanth-Strategy
102 10310−2
10−1
100
101
102
103
Number of unknowns on S: N (for tint) and Nfar (for tuv & taca)
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 ↑
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UC3MACA with R/λ ↑ and p ↑p-Strategy
10310−2
10−1
100
101
102
103
104
Number of unknowns on S: N (for tint) and Nfar (for tuv & taca)
CPU
−tim
e (s
ecs)
tINT
(−) 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 ↑
BCAM June 2014 ACA in the context of hp-discretizations v1.1 53 / 58
UC3MACA with R/λ ↑ and p ↑Influence of ACA Group Criterion
10310−2
10−1
100
101
102
103
104
Number of unknowns on S: N (for tint) and Nfar (for tuv & taca)
CPU
−tim
e (s
ecs)
tINT
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
2nd tranche: tACA: N2.8
tUV: N2.65
1st tranche: N1.85
2nd tranche: tACA: N2.7
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
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UC3M,,
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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.
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UC3MThank you for your attention !!