Numerical Integration and High Order Finite Element Method ...durufle/expose/SlidesSandia.pdf ·...
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Numerical Integration and High Order Finite ElementMethod Applied to Maxwell’s Equations
M. Durufle, G Cohen
INRIA, project POEMS
25th april 2007
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 1 / 50
Bibliography and motivation
Y. Maday, E. Ronquist, Spectral Methods
N. Tordjman, mass lumping for wave equation(triangles/quadrilaterals)
Cohen, Monk, mass lumping for Maxwell’s equations (hexahedra)
S. Fauqueux, mixed spectral elements for wave and elasticequations (hexahedra)
S. Pernet, Discontinuous Galerkin methods for Maxwell’sequations (hexahedra)
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 2 / 50
Introduction
Apply techniques of “mass lumping” and “mixed formulation”,which are efficient in temporal domain
Application of these techniques to Helmholtz and time-harmonicMaxwell equationsGain in storage and time, by using these techniques in frequentialdomain
Choose an efficient preconditioning technique to solve linearsystems issued from these equations
Apply the developped algorithms to evaluate accurately radarcross sections of electromagnetic targets
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 3 / 50
Introduction
Apply techniques of “mass lumping” and “mixed formulation”,which are efficient in temporal domain
Application of these techniques to Helmholtz and time-harmonicMaxwell equationsGain in storage and time, by using these techniques in frequentialdomain
Choose an efficient preconditioning technique to solve linearsystems issued from these equations
Apply the developped algorithms to evaluate accurately radarcross sections of electromagnetic targets
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 3 / 50
Introduction
Apply techniques of “mass lumping” and “mixed formulation”,which are efficient in temporal domain
Application of these techniques to Helmholtz and time-harmonicMaxwell equationsGain in storage and time, by using these techniques in frequentialdomain
Choose an efficient preconditioning technique to solve linearsystems issued from these equations
Apply the developped algorithms to evaluate accurately radarcross sections of electromagnetic targets
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 3 / 50
Outline
1 Resolution of Helmholtz equationInterest to use high order methodsEfficient matrix-vector product on hexahedral meshesEfficient iterative solver and preconditioning
2 Time-harmonic Maxwell equationsSpurious modes for Nedelec’s second familySpurious modes for Discontinuous Galerkin methodEfficient matrix-vector product for Nedelec’s first familyEfficient iterative resolution
3 Time-domain Maxwell equationsDescription of DG methodNumerical Results
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 4 / 50
A test case : an optical filter
At right, transmission coefficient according to the frequency
Frequency F = 1.0 is a resonant frequency of the device
Enlightment of the device by a gaussian beam.
PML around the computational domain.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 5 / 50
A test case : an optical filter
At right, transmission coefficient according to the frequency
Frequency F = 1.0 is a resonant frequency of the device
Enlightment of the device by a gaussian beam.
PML around the computational domain.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 5 / 50
A test case : an optical filter
At right, transmission coefficient according to the frequency
Frequency F = 1.0 is a resonant frequency of the device
Enlightment of the device by a gaussian beam.
PML around the computational domain.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 5 / 50
A test case : an optical filter
At right, transmission coefficient according to the frequency
Frequency F = 1.0 is a resonant frequency of the device
Enlightment of the device by a gaussian beam.
PML around the computational domain.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 5 / 50
Advantage to use high order method
Numerical solution for Q5 with 10 points by wavelength
Which order isoptimal to reach an error less than 10% ?
Order 2 3 4 5 6 7Nb dofs 453 000 69 800 52 000 33 200 47 700 42 200
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 6 / 50
Advantage to use high order method
At right, numerical solution for Q2 with 10 points by wavelength
Whichorder is optimal to reach an error less than 10% ?
Order 2 3 4 5 6 7Nb dofs 453 000 69 800 52 000 33 200 47 700 42 200
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 6 / 50
Advantage to use high order method
0.995 1 1.0050
1
2
3
4
5
6
7
F (relative frequency)
|| u ||
ReferenceQ2
Norm of the solution at the ouput, according to the frequency
Which order is optimal to reach an error less than 10% ?
Order 2 3 4 5 6 7Nb dofs 453 000 69 800 52 000 33 200 47 700 42 200
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 6 / 50
Advantage to use high order method
0.995 1 1.0050
1
2
3
4
5
6
7
F (relative frequency)
|| u ||
ReferenceQ2
Norm of the solution at the ouput, according to the frequencyWhich order is optimal to reach an error less than 10% ?
Order 2 3 4 5 6 7Nb dofs 453 000 69 800 52 000 33 200 47 700 42 200
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 6 / 50
Helmholtz equation
−ρω2 u − div(µ∇u) = f ∈ Ω
Use of finite element method leads to the following linear system :
(−ω2Dh + Kh) Uh = Fh
Mass matrix Dh =
∫ΩρϕGL
i ϕGLj dx
Stiffness matrix Kh =
∫Ωµ∇ϕGL
i · ∇ϕGLj dx
Our aim is to develop an efficient iterative solver for an high order ofapproximation r . We need then a fast matrix-vector product(−ω2Dh + Kh) Uh
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 7 / 50
Helmholtz equation
−ρω2 u − div(µ∇u) = f ∈ Ω
Use of finite element method leads to the following linear system :
(−ω2Dh + Kh) Uh = Fh
Mass matrix Dh =
∫ΩρϕGL
i ϕGLj dx
Stiffness matrix Kh =
∫Ωµ∇ϕGL
i · ∇ϕGLj dx
Our aim is to develop an efficient iterative solver for an high order ofapproximation r . We need then a fast matrix-vector product(−ω2Dh + Kh) Uh
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 7 / 50
Helmholtz equation
−ρω2 u − div(µ∇u) = f ∈ Ω
Use of finite element method leads to the following linear system :
(−ω2Dh + Kh) Uh = Fh
Mass matrix Dh =
∫ΩρϕGL
i ϕGLj dx
Stiffness matrix Kh =
∫Ωµ∇ϕGL
i · ∇ϕGLj dx
Our aim is to develop an efficient iterative solver for an high order ofapproximation r . We need then a fast matrix-vector product(−ω2Dh + Kh) Uh
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 7 / 50
Helmholtz equation
−ρω2 u − div(µ∇u) = f ∈ Ω
Use of finite element method leads to the following linear system :
(−ω2Dh + Kh) Uh = Fh
Mass matrix Dh =
∫ΩρϕGL
i ϕGLj dx
Stiffness matrix Kh =
∫Ωµ∇ϕGL
i · ∇ϕGLj dx
Our aim is to develop an efficient iterative solver for an high order ofapproximation r . We need then a fast matrix-vector product(−ω2Dh + Kh) Uh
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 7 / 50
Use of Gauss-Lobatto points
Gauss-Lobatto points for Q5
on the unit square K
Use of these points both for interpolation and numerical quadratureleads to a diagonal mass matrix Dh and a fast matrix-vector product forKh UhSee the thesis of S. Fauqueux, 2003These points permit a fast matrix-vector product
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 8 / 50
Use of Gauss-Lobatto points
Gauss-Lobatto points for Q5
on the unit square K
Use of these points both for interpolation and numerical quadratureleads to a diagonal mass matrix Dh and a fast matrix-vector product forKh UhSee the thesis of S. Fauqueux, 2003These points permit a fast matrix-vector product
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 8 / 50
Use of Gauss-Lobatto points
Gauss-Lobatto points for Q5
on the unit square K
Use of these points both for interpolation and numerical quadratureleads to a diagonal mass matrix Dh and a fast matrix-vector product forKh UhSee the thesis of S. Fauqueux, 2003These points permit a fast matrix-vector product
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 8 / 50
Use of Gauss-Lobatto points
Gauss-Lobatto points for Q5
on the unit square K
Use of these points both for interpolation and numerical quadratureleads to a diagonal mass matrix Dh and a fast matrix-vector product forKh UhSee the thesis of S. Fauqueux, 2003These points permit a fast matrix-vector product
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 8 / 50
Elementary matrices
(0,0) (1,0)
(1,1)(0,1)
KFi Ki
A4
A1
A2
A3
The transformation Fi
(Dh)i,j =
∫KρJi ϕ
GLi ϕGL
j dx
(Kh)i,j =
∫Kµ Ji DF−1
i DF ∗−1i ∇ ϕGL
i · ∇ϕGLj dx
Use of quadrature formulas (ωXk , ξX
k ) on the unit square
Diagonal matrix(Ah)k ,k = ρ Ji(ξ
Xk )ωX
k
Bloc-diagonal matrix
(Bh)k ,k = µ Ji DF−1i DF ∗−1
i (ξXk )ωX
k
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 9 / 50
Elementary matrices
(Dh)i,j =
∫KρJi ϕ
GLi ϕGL
j dx
(Kh)i,j =
∫Kµ Ji DF−1
i DF ∗−1i ∇ ϕGL
i · ∇ϕGLj dx
Use of quadrature formulas (ωXk , ξX
k ) on the unit square
Diagonal matrix(Ah)k ,k = ρ Ji(ξ
Xk )ωX
k
Bloc-diagonal matrix
(Bh)k ,k = µ Ji DF−1i DF ∗−1
i (ξXk )ωX
k
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 9 / 50
Elementary matrices
(Dh)i,j =
∫KρJi ϕ
GLi ϕGL
j dx
(Kh)i,j =
∫Kµ Ji DF−1
i DF ∗−1i ∇ ϕGL
i · ∇ϕGLj dx
Use of quadrature formulas (ωXk , ξX
k ) on the unit squareX can be equal to GL (Gauss-Lobatto quadrature)X can be equal to G (Gauss quadrature)
Use of quadrature formulas (ωXk , ξX
k ) on the unit square
Diagonal matrix(Ah)k ,k = ρ Ji(ξ
Xk )ωX
k
Bloc-diagonal matrix
(Bh)k ,k = µ Ji DF−1i DF ∗−1
i (ξXk )ωX
k
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 9 / 50
Elementary matrices
(Dh)i,j =
∫KρJi ϕ
GLi ϕGL
j dx
(Kh)i,j =
∫Kµ Ji DF−1
i DF ∗−1i ∇ ϕGL
i · ∇ϕGLj dx
Use of quadrature formulas (ωXk , ξX
k ) on the unit square
Diagonal matrix(Ah)k ,k = ρ Ji(ξ
Xk )ωX
k
Bloc-diagonal matrix
(Bh)k ,k = µ Ji DF−1i DF ∗−1
i (ξXk )ωX
k
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 9 / 50
Fast matrix vector product with any points
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕGLi (ξX
j ) R i,j = ∇ϕXi (ξX
j )
Thus, we have : Dh = C AhC∗ Kh = CR BhR∗C∗
For hexahedral elements (tensorization), we have
Complexity of C U : 6 (r + 1)4 operations in 3-DComplexity of R U : 6 (r + 1)4 operations in 3-DComplexity of Ah U and Bh V : 16 (r + 1)3 operations in 3-D
If we use Gauss-Lobatto points to integrate : C = IIn this case : “equivalence theorem” of S. FauqueuxSame storage for Gauss or GL points (Ah and Bh)MV product two times slower with Gauss integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 10 / 50
Fast matrix vector product with any points
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕGLi (ξX
j ) R i,j = ∇ϕXi (ξX
j )
Thus, we have : Dh = C AhC∗ Kh = CR BhR∗C∗
For hexahedral elements (tensorization), we have
Complexity of C U : 6 (r + 1)4 operations in 3-DComplexity of R U : 6 (r + 1)4 operations in 3-DComplexity of Ah U and Bh V : 16 (r + 1)3 operations in 3-D
If we use Gauss-Lobatto points to integrate : C = IIn this case : “equivalence theorem” of S. FauqueuxSame storage for Gauss or GL points (Ah and Bh)MV product two times slower with Gauss integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 10 / 50
Fast matrix vector product with any points
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕGLi (ξX
j ) R i,j = ∇ϕXi (ξX
j )
Thus, we have : Dh = C AhC∗ Kh = CR BhR∗C∗
r is the order of approximationIf C and R are stored as full matrices
Complexity of C U : 2 (r + 1)6 operations in 3-DComplexity of R U : 6 (r + 1)6 operations in 3-D
Complexity of standard matrix vector product : 2 (r + 1)6 operations in3-D
For hexahedral elements (tensorization), we have
Complexity of C U : 6 (r + 1)4 operations in 3-DComplexity of R U : 6 (r + 1)4 operations in 3-DComplexity of Ah U and Bh V : 16 (r + 1)3 operations in 3-D
If we use Gauss-Lobatto points to integrate : C = IIn this case : “equivalence theorem” of S. FauqueuxSame storage for Gauss or GL points (Ah and Bh)MV product two times slower with Gauss integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 10 / 50
Fast matrix vector product with any points
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕGLi (ξX
j ) R i,j = ∇ϕXi (ξX
j )
Thus, we have : Dh = C AhC∗ Kh = CR BhR∗C∗
For hexahedral elements (tensorization), we have
Complexity of C U : 6 (r + 1)4 operations in 3-DComplexity of R U : 6 (r + 1)4 operations in 3-DComplexity of Ah U and Bh V : 16 (r + 1)3 operations in 3-D
If we use Gauss-Lobatto points to integrate : C = IIn this case : “equivalence theorem” of S. FauqueuxSame storage for Gauss or GL points (Ah and Bh)MV product two times slower with Gauss integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 10 / 50
Matrix vector-product faster than standard methods ?
1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
300
Order r
Tim
e
Standard formulationMixed formulation
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
Order
Stor
age
Standard formulationMixed formulation
3-D comparison between the classical matrix-vector algorithm and thefast algorithm (mixed formulation), in 3-D.At left, time according to the order of approximation, at right storage.
1 2 3 4 5 6 7 8 9 100
100
200
300
400
500
600
700
800
900
Order of approximation
Tim
e
Tetrahedral elementsMixed hexahedral
1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
300
350
Order of approximation
Mem
ory
Tetrahedral elementsMixed hexahedral
Comparison between hexahedral and tetrahedral elements, for timecomputation (at left) and storage (at right)
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 11 / 50
Matrix vector-product faster than standard methods ?
1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
300
Order r
Tim
e
Standard formulationMixed formulation
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
Order
Stor
age
Standard formulationMixed formulation
3-D comparison between the classical matrix-vector algorithm and thefast algorithm (mixed formulation), in 3-D.At left, time according to the order of approximation, at right storage.Gain in time for r ≥ 4, gain in storage for r ≥ 2.
1 2 3 4 5 6 7 8 9 100
100
200
300
400
500
600
700
800
900
Order of approximation
Tim
e
Tetrahedral elementsMixed hexahedral
1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
300
350
Order of approximation
Mem
ory
Tetrahedral elementsMixed hexahedral
Comparison between hexahedral and tetrahedral elements, for timecomputation (at left) and storage (at right)
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 11 / 50
Matrix vector-product faster than standard methods ?
1 2 3 4 5 6 7 8 9 100
100
200
300
400
500
600
700
800
900
Order of approximation
Tim
e
Tetrahedral elementsMixed hexahedral
1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
300
350
Order of approximation
Mem
ory
Tetrahedral elementsMixed hexahedral
Comparison between hexahedral and tetrahedral elements, for timecomputation (at left) and storage (at right)
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 11 / 50
Iterative methods used
0 2000 4000 6000 8000 10000
10−6
10−4
10−2
100
102
104
COCGQMRBICGCRBICGSTABGMRES
Evolution of the residual norm for the scattering of a perfectly conductor disc(Dirichlet condition).
GMRES, BICGSTAB and QMR for complex unsymmetric matrices
COCG, BICGCR for complex symmetric matrices
0 2 4 6 8 10 12x 104
10−6
10−4
10−2
100
102COCGGMRESQMRBICGCRBICGSTAB
We choose to use BICGCR for all future experiments
Need of preconditioning techniques to have less iterations
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 12 / 50
Iterative methods used
0 2 4 6 8 10 12x 104
10−6
10−4
10−2
100
102COCGGMRESQMRBICGCRBICGSTAB
Evolution of the residual norm for the scattering of a dielectric disc(ρ = 4).
We choose to use BICGCR for all future experiments
Need of preconditioning techniques to have less iterations
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 12 / 50
Iterative methods used
0 2 4 6 8 10 12x 104
10−6
10−4
10−2
100
102COCGGMRESQMRBICGCRBICGSTAB
We choose to use BICGCR for all future experiments
Need of preconditioning techniques to have less iterations
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 12 / 50
Preconditioning used
Incomplete factorization with threshold on the damped Helmholtzequation :
−k2(α + iβ)u − ∆u = 0
see Y. Saad, Iterative methods for sparse linear systems
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 13 / 50
Preconditioning used
Incomplete factorization with threshold on the damped Helmholtzequation :
−k2(α + iβ)u − ∆u = 0
see Y. Saad, Iterative methods for sparse linear systemsWe use a Q1 subdivided mesh to compute matrix
At left, initial mesh Q3, at right, subdivided mesh Q1
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 13 / 50
Preconditioning used
Incomplete factorization with threshold on the damped Helmholtzequation :
−k2(α + iβ)u − ∆u = 0
see Y. Saad, Iterative methods for sparse linear systems
Multigrid method on the damped Helmholtz equationsee Y. A. Erlangga and al, Report of Delft University Technology,2004
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 13 / 50
Preconditioning used
Incomplete factorization with threshold on the damped Helmholtzequation :
−k2(α + iβ)u − ∆u = 0
see Y. Saad, Iterative methods for sparse linear systems
Multigrid method on the damped Helmholtz equationsee Y. A. Erlangga and al, Report of Delft University Technology,2004
Without damping, both preconditioners does not lead toconvergence.
A good choice of parameter is α = 1, β = 0.5
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 13 / 50
Scattering by a dielectric sphere
Dielectric sphere of radius 2 and with ρ = 4 ω = 2π
First order absorbing boundary condition on a sphere of radius 3
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 14 / 50
Scattering by a dielectric sphere
Number of dofs to reach less than 5 % L2 error
Finite element structured Q2 struct Q4 struct Q6 n.s. Q4 n.s. P4Number of dofs 220 000 85 000 78 000 243 000 180 000
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 14 / 50
Scattering by a dielectric sphere
Finite element structured Q4 non-structured Q4 non-structured P4No preconditioning 708 s 5 795 s 1 597 sILUT(0.01) 91 s 534 s 363 sMultigrid 185 s 729 s 695 s
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 14 / 50
Scattering by a dielectric sphere
Finite element structured Q4 non-structured Q4 non-structured P4No preconditioning 34 Mo 99 Mo 136 MoILUT(0.01) 137 Mo 420 Mo 507 MoMultigrid 50 Mo 143 Mo 327 Mo
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 14 / 50
Scattering by a cobra cavity
Cobra cavity of length 20, and depth 4
First order absorbing boundary condition on the yellow face
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 9 860 s NC NCILUT(0.01) 1 021 s 13 766 s 8 036 sTwo-grid 1 082 s 6 821 s 14 016 s
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 32 Mo 162 Mo 251 MoILUT(0.01) 150 Mo 1 250 Mo 1 400 MoTwo-grid 60 Mo 283 Mo 710 Mo
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 15 / 50
Scattering by a cobra cavity
Number of dofs to reach less than 5 % L2 errorOrder struct Q4 struct Q6 struct Q8 n.s. Q4 n.s. Q6 n.s. P4Nb dofs 330 000 185 000 95 600 567,000 466 000 360 000
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 9 860 s NC NCILUT(0.01) 1 021 s 13 766 s 8 036 sTwo-grid 1 082 s 6 821 s 14 016 s
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 32 Mo 162 Mo 251 MoILUT(0.01) 150 Mo 1 250 Mo 1 400 MoTwo-grid 60 Mo 283 Mo 710 Mo
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 15 / 50
Scattering by a cobra cavity
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 9 860 s NC NCILUT(0.01) 1 021 s 13 766 s 8 036 sTwo-grid 1 082 s 6 821 s 14 016 s
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 32 Mo 162 Mo 251 MoILUT(0.01) 150 Mo 1 250 Mo 1 400 MoTwo-grid 60 Mo 283 Mo 710 Mo
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 15 / 50
Scattering by a cobra cavity
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 9 860 s NC NCILUT(0.01) 1 021 s 13 766 s 8 036 sTwo-grid 1 082 s 6 821 s 14 016 s
Finite element structured Q8 non-structured Q6 non-structured P4No preconditioning 32 Mo 162 Mo 251 MoILUT(0.01) 150 Mo 1 250 Mo 1 400 MoTwo-grid 60 Mo 283 Mo 710 Mo
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 15 / 50
Scattering by a plane
Real part of the diffracted for an oblique incident plane wave
Q4, 7.2 million of dofs
650 iterations and 7 hours with multigrid preconditioning
More than 50 000 iterations without preconditioning ...M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 16 / 50
Outline
1 Resolution of Helmholtz equationInterest to use high order methodsEfficient matrix-vector product on hexahedral meshesEfficient iterative solver and preconditioning
2 Time-harmonic Maxwell equationsSpurious modes for Nedelec’s second familySpurious modes for Discontinuous Galerkin methodEfficient matrix-vector product for Nedelec’s first familyEfficient iterative resolution
3 Time-domain Maxwell equationsDescription of DG methodNumerical Results
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 17 / 50
Nedelec’s second family on hexahedrals
Time-harmonic Maxwell’s equations :
−ω2 ε ~E(x) + curl(1
µ(x)curl(~E(x))) = 0
Space of approximation
Vh = ~u ∈ H(curl,Ω) such as DF ∗i ~u Fi ∈ (Qr )3
Mass lumping and factorization of stiffness matrix
Low-storage and fast matrix-vector product
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 18 / 50
Nedelec’s second family on hexahedrals
Time-harmonic Maxwell’s equations :
−ω2 ε ~E(x) + curl(1
µ(x)curl(~E(x))) = 0
Mass lumping and factorization of stiffness matrix
Low-storage and fast matrix-vector product
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 18 / 50
Nedelec’s second family on hexahedrals
Time-harmonic Maxwell’s equations :
−ω2 ε ~E(x) + curl(1
µ(x)curl(~E(x))) = 0
Mass lumping and factorization of stiffness matrix
Low-storage and fast matrix-vector productM. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 18 / 50
The unwanted oscillations
Dipole source on a cubic cavity. Left, mesh used for the simulations .Right, numerical solution with Q3 finite edge elements withmass-lumping.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 19 / 50
Eigenmodes with the second family
Mesh used for the simulations (Q3)
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 20 / 50
Eigenmodes with the second family
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 20 / 50
Eigenmodes with the second family
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 20 / 50
Two types of penalization
Mixed formulation of Maxwell equations
−ω∫
Ω
E · ϕ +
∫Ω
H · rot(ϕ) − iα∑
e face
∫Γe
[E · n][ϕ · n] =
∫Ω
f · ϕ
−ω∫
Ω
H · ϕ +
∫Ω
rot(E) · ϕ− iδ∑
e face
∫Γe
[H × n] · [ϕ× n] = 0
Approximation space for H
Wh = ~u ∈ L2(Ω) so that DF ∗i ~u Fi ∈ (Qr )3
Equivalence with second-order formulation (α = δ = 0)
Dissipative terms of penalization
Penalization in α does not need of a mixed formulation
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 21 / 50
Two types of penalization
Mixed formulation of Maxwell equations
−ω∫
Ω
E · ϕ +
∫Ω
H · rot(ϕ) − iα∑
e face
∫Γe
[E · n][ϕ · n] =
∫Ω
f · ϕ
−ω∫
Ω
H · ϕ +
∫Ω
rot(E) · ϕ− iδ∑
e face
∫Γe
[H × n] · [ϕ× n] = 0
Approximation space for H
Wh = ~u ∈ L2(Ω) so that DF ∗i ~u Fi ∈ (Qr )3
Equivalence with second-order formulation (α = δ = 0)
Dissipative terms of penalization
Penalization in α does not need of a mixed formulation
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 21 / 50
Effects of penalization
0 1 2 3 4 5−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Numerical eigenvaluesAnalytical eigenvalues
0 1 2 3 4 5−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Analytical eigenvaluesNumerical eigenvalues
Case of the cubic cavity meshed with slip tetrahedrals
At left α = 0.1, at right α = 0.5
0 1 2 3 4−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Analytical eigenvaluesNumerical eigenvalues
0 1 2 3 4−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Case of the Fichera corner
At left α = 0.5, at right δ = 0.5
Both penalizations efficient for regular domains
Delta-penalization more robust for singular domains
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 22 / 50
Effects of penalization
Four modes of the Fichera corner
0 1 2 3 4−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Analytical eigenvaluesNumerical eigenvalues
0 1 2 3 4−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Case of the Fichera corner
At left α = 0.5, at right δ = 0.5
Both penalizations efficient for regular domains
Delta-penalization more robust for singular domains
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 22 / 50
Effects of penalization
0 1 2 3 4−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Analytical eigenvaluesNumerical eigenvalues
0 1 2 3 4−0.4
−0.3
−0.2
−0.1
0
0.1
real(ω)
imag
(ω)
Case of the Fichera corner
At left α = 0.5, at right δ = 0.5
Both penalizations efficient for regular domains
Delta-penalization more robust for singular domains
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 22 / 50
Discontinuous Galerkin method
−ω∫
Ki
ε~E · ~ϕ −∫
Ki
H∇× ~ϕ −∫∂Ki
H ~ϕ× ~ν = 0
−ω∫
Ki
µH ψ −∫
Ki
∇× ~E ψ − 12
∫∂Ki
[~E]× ~ν ψ = 0
Let us notice thatH = 1
2(Hi + Hj)
[ ~E] = (~Ei − ~Ej)(1)
Unknowns in L2 ⇒ Gauss points instead of GL points
Mass lumping and fast matrix vector product
Thesis of S. Pernet, in time-domain
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 23 / 50
Discontinuous Galerkin method
−ω∫
Ki
ε~E · ~ϕ −∫
Ki
H∇× ~ϕ −∫∂Ki
H ~ϕ× ~ν = 0
−ω∫
Ki
µH ψ −∫
Ki
∇× ~E ψ − 12
∫∂Ki
[~E]× ~ν ψ = 0
Unknowns in L2 ⇒ Gauss points instead of GL points
Mass lumping and fast matrix vector product
Thesis of S. Pernet, in time-domain
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 23 / 50
Eigenmodes in DG method (3-D)
Constant number of spurious for regular meshes
Increasing number of spurious modes, otherwise
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 24 / 50
Eigenmodes in DG method (3-D)
Constant number of spurious for regular meshes
Increasing number of spurious modes, otherwise
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 24 / 50
Penalization terms, eigenvalues
To the first equation in E , we add :
−iω α∫∂Ki
[E× n] ·ϕ× n dx
We take α = 0.5
0 1 2 3 4 5−2
−1.5
−1
−0.5
0
real(ω)
imag(ω
)
Numerical eigenvaluesAnalytical eigenvalues
Penalization terms reject ALL spurious modes in complex plane
Persistance of some spurious mode near 0
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 25 / 50
Penalization terms, eigenvalues
0 100 200 300 400 500 600 7000
5
10
15
20
25
30
Number eigenvalue
ω2
Eigenvalues, if no penalization is used α = 0Blue points are numeric eigenvalues, red lines analyticeigenvalues.
0 1 2 3 4 5−2
−1.5
−1
−0.5
0
real(ω)
imag(ω
)
Numerical eigenvaluesAnalytical eigenvalues
Penalization terms reject ALL spurious modes in complex plane
Persistance of some spurious mode near 0
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 25 / 50
Penalization terms, eigenvalues
0 1 2 3 4 5−2
−1.5
−1
−0.5
0
real(ω)
imag(ω
)
Numerical eigenvaluesAnalytical eigenvalues
Eigenvalues if penalization is used α = 0.5Blue points are numeric eigenvalues, red squares analytic eigenvalues.
Penalization terms reject ALL spurious modes in complex plane
Persistance of some spurious mode near 0
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 25 / 50
Penalization terms, eigenvalues
0 1 2 3 4 5−2
−1.5
−1
−0.5
0
real(ω)
imag(ω
)
Numerical eigenvaluesAnalytical eigenvalues
Penalization terms reject ALL spurious modes in complex plane
Persistance of some spurious mode near 0
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 25 / 50
Effects of penalization
At left, numerical solution with α = 0, at right with α = 0.5
0 1 2 3 4 5−2
−1.5
−1
−0.5
0
0.5
real(ω)
imag
(ω)
Numerical eigenvaluesAnalytical eigenvalues
Eigenvalues for the Fichera corner, on split tetrahedral mesh.4
Good approximation of singular eigenvalues
No need to add penalization terms in 2-D
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 26 / 50
Effects of penalization
At left, numerical solution with α = 0, at right with α = 0.5
Fine solution on split meshes
Negligible overcost in computational time
0 1 2 3 4 5−2
−1.5
−1
−0.5
0
0.5
real(ω)
imag
(ω)
Numerical eigenvaluesAnalytical eigenvalues
Eigenvalues for the Fichera corner, on split tetrahedral mesh.4
Good approximation of singular eigenvalues
No need to add penalization terms in 2-D
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 26 / 50
Effects of penalization
0 1 2 3 4 5−2
−1.5
−1
−0.5
0
0.5
real(ω)
imag
(ω)
Numerical eigenvaluesAnalytical eigenvalues
Eigenvalues for the Fichera corner, on split tetrahedral mesh.4
Good approximation of singular eigenvalues
No need to add penalization terms in 2-D
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 26 / 50
Nedelec’s first family on hexahedra
Space of approximation
Vh = ~u ∈ H(curl,Ω) so that DF ti ~u Fi ∈ Qr−1,r ,r × Qr ,r−1,r × Qr ,r ,r−1
Basis functions
~ϕ1i,j,k (x , y , z) = ψG
i (x) ψGLj (y) ψGL
k (z) ~ex 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
~ϕ2j,i,k (x , y , z) = ψGL
j (x) ψGi (y) ψGL
k (z) ~ey 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
~ϕ3k,j,i (x , y , z) = ψGL
k (x) ψGLj (y) ψG
i (x) ~ez 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
ψGi , ψ
GLi lagragian functions linked respectively with Gauss points and
Gauss-Lobatto points.See. G. Cohen, P. Monk, Gauss points mass lumping
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 27 / 50
Nedelec’s first family on hexahedra
Space of approximation
Vh = ~u ∈ H(curl,Ω) so that DF ti ~u Fi ∈ Qr−1,r ,r × Qr ,r−1,r × Qr ,r ,r−1
Basis functions
~ϕ1i,j,k (x , y , z) = ψG
i (x) ψGLj (y) ψGL
k (z) ~ex 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
~ϕ2j,i,k (x , y , z) = ψGL
j (x) ψGi (y) ψGL
k (z) ~ey 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
~ϕ3k,j,i (x , y , z) = ψGL
k (x) ψGLj (y) ψG
i (x) ~ez 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
ψGi , ψ
GLi lagragian functions linked respectively with Gauss points and
Gauss-Lobatto points.See. G. Cohen, P. Monk, Gauss points mass lumping
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 27 / 50
Nedelec’s first family on hexahedra
Space of approximation
Vh = ~u ∈ H(curl,Ω) so that DF ti ~u Fi ∈ Qr−1,r ,r × Qr ,r−1,r × Qr ,r ,r−1
Basis functions
~ϕ1i,j,k (x , y , z) = ψG
i (x) ψGLj (y) ψGL
k (z) ~ex 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
~ϕ2j,i,k (x , y , z) = ψGL
j (x) ψGi (y) ψGL
k (z) ~ey 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
~ϕ3k,j,i (x , y , z) = ψGL
k (x) ψGLj (y) ψG
i (x) ~ez 1 ≤ i ≤ r 1 ≤ j , k ≤ r + 1
ψGi , ψ
GLi lagragian functions linked respectively with Gauss points and
Gauss-Lobatto points.See. G. Cohen, P. Monk, Gauss points mass lumping
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 27 / 50
Elementary matricesMass matrix :
(Mh)i,j =
∫K
Ji DF−1i εDF ∗−1
i ϕi · ϕk dx
Stiffness matrix :
(Kh)i,j =
∫K
1Ji
DF ti µ
−1 DFi ∇ × ϕi · ∇ × ϕk dx
Use of Gauss-Lobatto quadrature (ωGLk , ξGL
k )
Block-diagonal matrix
(Ah)k,k =[Ji DF−1
i εDF ∗−1i
](ξGL
k )ωGLk
Block-diagonal matrix
(Bh)k,k =[ 1
JiDF t
i µ−1 DFi
](ξGL
k )ωGLk
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 28 / 50
Elementary matricesMass matrix :
(Mh)i,j =
∫K
Ji DF−1i εDF ∗−1
i ϕi · ϕk dx
Stiffness matrix :
(Kh)i,j =
∫K
1Ji
DF ti µ
−1 DFi ∇ × ϕi · ∇ × ϕk dx
Use of Gauss-Lobatto quadrature (ωGLk , ξGL
k )
Block-diagonal matrix
(Ah)k,k =[Ji DF−1
i εDF ∗−1i
](ξGL
k )ωGLk
Block-diagonal matrix
(Bh)k,k =[ 1
JiDF t
i µ−1 DFi
](ξGL
k )ωGLk
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 28 / 50
Elementary matricesMass matrix :
(Mh)i,j =
∫K
Ji DF−1i εDF ∗−1
i ϕi · ϕk dx
Stiffness matrix :
(Kh)i,j =
∫K
1Ji
DF ti µ
−1 DFi ∇ × ϕi · ∇ × ϕk dx
Use of Gauss-Lobatto quadrature (ωGLk , ξGL
k )
Block-diagonal matrix
(Ah)k,k =[Ji DF−1
i εDF ∗−1i
](ξGL
k )ωGLk
Block-diagonal matrix
(Bh)k,k =[ 1
JiDF t
i µ−1 DFi
](ξGL
k )ωGLk
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 28 / 50
Fast matrix vector product
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕi(ξGLj ) R i,j = ∇ × ϕGL
i (ξGLj )
Then, we have : Mh = C AhC∗ Kh = CR BhR∗C∗
Complexity of C U : 6 (r + 1)4 operations in 3-D
Complexity of R U : 12 (r + 1)4 operations in 3-D
Complexity of Ah U + Bh U : 30 (r + 1)3 operationsComplexity of standard matrix vector product 18r3 (r + 1)3
Matrix-vector product 67% slower by using exact integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 29 / 50
Fast matrix vector product
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕi(ξGLj ) R i,j = ∇ × ϕGL
i (ξGLj )
Then, we have : Mh = C AhC∗ Kh = CR BhR∗C∗
Complexity of C U : 6 (r + 1)4 operations in 3-D
Complexity of R U : 12 (r + 1)4 operations in 3-D
Complexity of Ah U + Bh U : 30 (r + 1)3 operationsComplexity of standard matrix vector product 18r3 (r + 1)3
Matrix-vector product 67% slower by using exact integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 29 / 50
Fast matrix vector product
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕi(ξGLj ) R i,j = ∇ × ϕGL
i (ξGLj )
Then, we have : Mh = C AhC∗ Kh = CR BhR∗C∗
Complexity of C U : 6 (r + 1)4 operations in 3-D
Complexity of R U : 12 (r + 1)4 operations in 3-D
Complexity of Ah U + Bh U : 30 (r + 1)3 operationsComplexity of standard matrix vector product 18r3 (r + 1)3
Matrix-vector product 67% slower by using exact integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 29 / 50
Fast matrix vector product
Let us introduce the two following matrices, independant of thegeometry :
C i,j = ϕi(ξGLj ) R i,j = ∇ × ϕGL
i (ξGLj )
Then, we have : Mh = C AhC∗ Kh = CR BhR∗C∗
Complexity of C U : 6 (r + 1)4 operations in 3-D
Complexity of R U : 12 (r + 1)4 operations in 3-D
Complexity of Ah U + Bh U : 30 (r + 1)3 operationsComplexity of standard matrix vector product 18r3 (r + 1)3
Matrix-vector product 67% slower by using exact integration
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 29 / 50
Spurious free method
Approximate integration leads to a spurious-free method
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 30 / 50
Spurious free method
Approximate integration leads to a spurious-free method
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 30 / 50
Convergence of the method
Scattering by a perfectly conductor sphere E × n = 0
Convergence on tetrahedral meshes split in hexahedra
5.2 5.4 5.6 5.8 6 6.2 6.4 6.6−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
log10(Number dof)
log 10
(erro
r)
Q1Q2Q3
Loss of one order, convergence O(hr−1) in H(curl,Ω) norm
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 31 / 50
Convergence of the method
Convergence of Nedelec’s first family on regular meshes
−1.4 −1.2 −1 −0.8 −0.6−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
log10(h/r)
log10(er
ror)
Q1Q2Q3Q4Q5
Optimal convergence O(hr ) in H(curl,Ω) norm
Convergence on tetrahedral meshes split in hexahedra
5.2 5.4 5.6 5.8 6 6.2 6.4 6.6−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
log10(Number dof)
log 10
(erro
r)
Q1Q2Q3
Loss of one order, convergence O(hr−1) in H(curl,Ω) norm
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 31 / 50
Convergence of the method
Convergence on tetrahedral meshes split in hexahedra
5.2 5.4 5.6 5.8 6 6.2 6.4 6.6−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
log10(Number dof)
log 10
(erro
r)
Q1Q2Q3
Loss of one order, convergence O(hr−1) in H(curl,Ω) norm
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 31 / 50
Is the matrix-vector product fast ?
Comparison between standard formulation and discrete factorization
Order 1 2 3 4 5Time, standard formulation 55s 127s 224s 380s 631Time, discrete factorization 244s 128s 106s 97s 96sStorage, standard formulation 18 Mo 50 Mo 105 Mo 187 Mo 308 MoStorage, discrete factorization 23 Mo 9.9 Mo 6.9 Mo 5.7 Mo 5.0 Mo
Comparison between tetrahedral and hexahedral elements
1 2 3 4 5 6 7 850
100
150
200
250
300
350
400
450
500
Order of approximation
Tim
e (in
s)
Hexahedral elementsTetrahedral elements
1 2 3 4 5 6 7 80
50
100
150
200
250
Order of approximation
Mem
ory
(in M
o)
Hexahedral elementsTetrahedral elements
At left, time computation for a thousand iterations of COCGAt right, storage for mesh and matrices
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 32 / 50
Is the matrix-vector product fast ?
Comparison between tetrahedral and hexahedral elements
1 2 3 4 5 6 7 850
100
150
200
250
300
350
400
450
500
Order of approximation
Tim
e (in
s)
Hexahedral elementsTetrahedral elements
1 2 3 4 5 6 7 80
50
100
150
200
250
Order of approximation
Mem
ory
(in M
o)
Hexahedral elementsTetrahedral elements
At left, time computation for a thousand iterations of COCGAt right, storage for mesh and matrices
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 32 / 50
Comparison DG method vs first family
Both methods are spectrally correct
Both methods have a fast MV product
DG needs more dof, because DG Q3 is less accurate thanFamily1 Q4
DG needs more storage for direct solvers (about 4 times than firstfamily)
DG can deal easily non-conforming meshes
DDM methods are faster with DG
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 33 / 50
Comparison DG method vs first family
Both methods are spectrally correct
Both methods have a fast MV product
DG needs more dof, because DG Q3 is less accurate thanFamily1 Q4
DG needs more storage for direct solvers (about 4 times than firstfamily)
DG can deal easily non-conforming meshes
DDM methods are faster with DG
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 33 / 50
Comparison DG method vs first family
Both methods are spectrally correct
Both methods have a fast MV product
DG needs more dof, because DG Q3 is less accurate thanFamily1 Q4
DG needs more storage for direct solvers (about 4 times than firstfamily)
DG can deal easily non-conforming meshes
DDM methods are faster with DG
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 33 / 50
Preconditioning used
Incomplete factorization with threshold on the damped Maxwellequation :
−k2(α + iβ)εE − ∇× (1µ∇× E) = 0
ILUT threshold ≥ 0.05 in order to have a low storage
Incomplete factorization with threshold on the damped Maxwellequation :
−k2(α + iβ)εE − ∇× (1µ∇× E) = 0
Multigrid method on the damped Maxwell equationUse of the Q1 mesh to do the multigrid iteration
Without damping, both preconditioners does not lead toconvergence.
A good choice of parameter is α = 0.7, β = 0.35
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 34 / 50
Preconditioning used
Incomplete factorization with threshold on the damped Maxwellequation :
−k2(α + iβ)εE − ∇× (1µ∇× E) = 0
ILUT threshold ≥ 0.05 in order to have a low storageUse of a Q1 subdivided mesh to compute matrix
=⇒
Incomplete factorization with threshold on the damped Maxwellequation :
−k2(α + iβ)εE − ∇× (1µ∇× E) = 0
Multigrid method on the damped Maxwell equationUse of the Q1 mesh to do the multigrid iteration
Without damping, both preconditioners does not lead toconvergence.
A good choice of parameter is α = 0.7, β = 0.35
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 34 / 50
Preconditioning used
Incomplete factorization with threshold on the damped Maxwellequation :
−k2(α + iβ)εE − ∇× (1µ∇× E) = 0
Multigrid method on the damped Maxwell equationUse of the Q1 mesh to do the multigrid iteration
Without damping, both preconditioners does not lead toconvergence.
A good choice of parameter is α = 0.7, β = 0.35
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 34 / 50
Transparent conditionSilver-Muller condition is a first-order ABC :
E × n + n × H × n = 0
Use of a transparent condition based on integral representation formulas:
Epot(x) =
∫Γ
ik (G(x , y)+1k2 ∇y∇y G(x , y)) (n×H)(y) dy +
∫Γ
(n×E)(y)×∇y G(x , y) dy
new boundary condition E × n + n × H × n = Epot × n + n × Hpot × n
Needs of a virtual boundary Γ
Σ
Incident Plane Wave
Γ
Ω
µ = µ0ε = ε0
GMRES iterations to solve linear system
C. Hazard, M. Lenoir, On the solution of time-harmonic scatteringproblems for Maxwell’s equations
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 35 / 50
Transparent condition
Silver-Muller condition is a first-order ABC :
E × n + n × H × n = 0
Use of a transparent condition based on integral representation formulas:
Needs of a virtual boundary Γ
Σ
Incident Plane Wave
Γ
Ω
µ = µ0ε = ε0
GMRES iterations to solve linear system
C. Hazard, M. Lenoir, On the solution of time-harmonic scatteringproblems for Maxwell’s equations
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 35 / 50
Transparent condition
Silver-Muller condition is a first-order ABC :
E × n + n × H × n = 0
Use of a transparent condition based on integral representation formulas:
Needs of a virtual boundary Γ
Σ
Incident Plane Wave
Γ
Ω
µ = µ0ε = ε0
GMRES iterations to solve linear system
C. Hazard, M. Lenoir, On the solution of time-harmonic scatteringproblems for Maxwell’s equations
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 35 / 50
Radar cross section
Computation of far field of the electromagnetic objects by the formula
σ(u) =k2
4π
∫Σ
eiku·OM[
u× (n× H) + (u ⊗ u − I)(E× n)]
dM
Bistatic RCS : the vector of observation u varies
Monostatic RCS : the wave vector k varies and u = k
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 36 / 50
Radar cross section
Computation of far field of the electromagnetic objects by the formula
σ(u) =k2
4π
∫Σ
eiku·OM[
u× (n× H) + (u ⊗ u − I)(E× n)]
dM
Bistatic RCS : the vector of observation u varies
Monostatic RCS : the wave vector k varies and u = k
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 36 / 50
Scattering by a dielectric sphere
Sphere of radius 2 with ε = 3.5 µ = 1
Outside boundary on a sphere of radius 3.
How many dofs/time to reach an error less than 0.5 dB
−350 −300 −250 −200 −150 −100 −50 0−5
0
5
10
15
20
25
30
35
40
θ (in degrees)
Rcs (
dB m
2 )
Analytical RCSNumerical RCS
Finite Element Q2 Q4 Q6 Q8Nb dofs 940 000 88 000 230 000 88 000No preconditioning 19 486 s 894 s 4 401 s 1 484 sILUT(0.05) - 189 s 1 035 s 307 sTwo-grid 4 4344 s 488 s 1 095 s 952 s
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 37 / 50
Scattering by a dielectric sphere
How many dofs/time to reach an error less than 0.5 dB
−350 −300 −250 −200 −150 −100 −50 0−5
0
5
10
15
20
25
30
35
40
θ (in degrees)
Rcs (
dB m
2 )
Analytical RCSNumerical RCS
Finite Element Q2 Q4 Q6 Q8Nb dofs 940 000 88 000 230 000 88 000No preconditioning 19 486 s 894 s 4 401 s 1 484 sILUT(0.05) - 189 s 1 035 s 307 sTwo-grid 4 4344 s 488 s 1 095 s 952 s
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 37 / 50
Scattering by a cobra cavity
Cobra cavity of length 10, and depth 2
Outside boundary at a distance of 1
How many dofs/time to reach an error less than 0.5 dB
−450 −400 −350 −300 −250 −200 −150 −1000
10
20
30
40
50
60
θ
Rcs (
dB m
2 )
Finite Element Q4 Q6Nb dofs 412 000 187 000No preconditioning 14 039 s 12 096 sILUT(0.05) 2 247 s 846 sTwo-grid 9 294 s 10 500 s
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 38 / 50
Scattering by a cobra cavity
How many dofs/time to reach an error less than 0.5 dB
−450 −400 −350 −300 −250 −200 −150 −1000
10
20
30
40
50
60
θ
Rcs (
dB m
2 )
Finite Element Q4 Q6Nb dofs 412 000 187 000No preconditioning 14 039 s 12 096 sILUT(0.05) 2 247 s 846 sTwo-grid 9 294 s 10 500 s
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 38 / 50
Outline
1 Resolution of Helmholtz equationInterest to use high order methodsEfficient matrix-vector product on hexahedral meshesEfficient iterative solver and preconditioning
2 Time-harmonic Maxwell equationsSpurious modes for Nedelec’s second familySpurious modes for Discontinuous Galerkin methodEfficient matrix-vector product for Nedelec’s first familyEfficient iterative resolution
3 Time-domain Maxwell equationsDescription of DG methodNumerical Results
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 39 / 50
Discontinuous Galerkin Method
Let Ω =
Ne⋃i=1
Ki . Find ~E(., t) ∈ [L2(Ω)]3, ~H(., t) ∈ [L2(Ω)]3 s.t.
∂
∂t
∫Ki
ε ~EKi · ~ϕKi dx −∫
Ki
∇∧ ~HKi · ~ϕKi dx
+
∫Ki
σ~EKi · ~ϕKi dx +
∫Ki
~J · ~ϕKi dx =∫∂Ki
α[~nKi ∧ (~E ∧ ~nKi )]Ki∂Ki· ~ϕKi dσ +
∫∂Ki
β[~H ∧ ~nKi ]Ki∂Ki· ~ϕKi dσ,
∀~ϕKi ∈ H(curl ,Ki)
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 40 / 50
Discontinuous Galerkin Methods for Time-Domain
∂
∂t
∫Ki
µ ~HKi · ~ψKi dx +
∫Ki
∇∧ ~EKi · ~ψKi dx =∫∂Ki
γ[~E ∧ ~nKi ]Ki∂Ki· ~ψKi dσ +
∫∂Ki
δ[~nKi ∧ (~H ∧ ~nKi )]Ki∂Ki· ~ψKi dσ,
∀~ψKi ∈ H(curl ,Ki)
+ metallic boundary condition on Γb = ∂Ω and initial conditions,
where ~EKi = ~E |Ki, ~HKi = ~H |Ki
, ~ϕKi = ~ϕ|Ki, ~ψKi = ~ϕ|Ki
and α, β, γ, δ realconstant parameters.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 41 / 50
Discontinuous Galerkin Methods for Time-Domain
∂
∂t
∫Ki
µ ~HKi · ~ψKi dx +
∫Ki
∇∧ ~EKi · ~ψKi dx =∫∂Ki
γ[~E ∧ ~nKi ]Ki∂Ki· ~ψKi dσ +
∫∂Ki
δ[~nKi ∧ (~H ∧ ~nKi )]Ki∂Ki· ~ψKi dσ,
∀~ψKi ∈ H(curl ,Ki)
+ metallic boundary condition on Γb = ∂Ω and initial conditions,
where ~EKi = ~E |Ki, ~HKi = ~H |Ki
, ~ϕKi = ~ϕ|Ki, ~ψKi = ~ϕ|Ki
and α, β, γ, δ realconstant parameters.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 41 / 50
Discontinuous Galerkin Methods for Time-Domain
∂
∂t
∫Ki
µ ~HKi · ~ψKi dx +
∫Ki
∇∧ ~EKi · ~ψKi dx =∫∂Ki
γ[~E ∧ ~nKi ]Ki∂Ki· ~ψKi dσ +
∫∂Ki
δ[~nKi ∧ (~H ∧ ~nKi )]Ki∂Ki· ~ψKi dσ,
∀~ψKi ∈ H(curl ,Ki)
+ metallic boundary condition on Γb = ∂Ω and initial conditions,
where ~EKi = ~E |Ki, ~HKi = ~H |Ki
, ~ϕKi = ~ϕ|Ki, ~ψKi = ~ϕ|Ki
and α, β, γ, δ realconstant parameters.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 41 / 50
Discrete Energy
EKi (t) =∑
Ki⊂Ω
∫Ki
(ε~EKi ) · ~EKi dx +
∫Ki
(µ~HKi ) · ~HKi dx
1 −β = γ = 12 , α ≥ 0 and δ ≥ 0 =⇒
∂E∂t
(t) =∑
Γ∈Fi , Γ=Ki∩Kj
−α‖[~nKi ∧ (~E ∧ ~nKi )−δ‖[~nKi ∧ (~H ∧ ~nKi )]‖2Γ∑Γ∈Γb, Γ⊂Ki
−α‖~nKi ∧ (~EKi ∧ ~nKi )‖2Γ − δ‖~nKi ∧ (~HKi ∧ ~nKi )‖2Γ
=⇒ Decreasing energy: Dissipative scheme.
2 −β = γ = 12 , α = 0 et δ = 0 =⇒ ∂
∂tE(t) = 0
=⇒ Energy conservation: Conservative scheme.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 42 / 50
Discrete Energy
EKi (t) =∑
Ki⊂Ω
∫Ki
(ε~EKi ) · ~EKi dx +
∫Ki
(µ~HKi ) · ~HKi dx
1 −β = γ = 12 , α ≥ 0 and δ ≥ 0 =⇒
∂E∂t
(t) =∑
Γ∈Fi , Γ=Ki∩Kj
−α‖[~nKi ∧ (~E ∧ ~nKi )−δ‖[~nKi ∧ (~H ∧ ~nKi )]‖2Γ∑Γ∈Γb, Γ⊂Ki
−α‖~nKi ∧ (~EKi ∧ ~nKi )‖2Γ − δ‖~nKi ∧ (~HKi ∧ ~nKi )‖2Γ
=⇒ Decreasing energy: Dissipative scheme.
2 −β = γ = 12 , α = 0 et δ = 0 =⇒ ∂
∂tE(t) = 0
=⇒ Energy conservation: Conservative scheme.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 42 / 50
Discrete Energy
EKi (t) =∑
Ki⊂Ω
∫Ki
(ε~EKi ) · ~EKi dx +
∫Ki
(µ~HKi ) · ~HKi dx
1 −β = γ = 12 , α ≥ 0 and δ ≥ 0 =⇒
∂E∂t
(t) =∑
Γ∈Fi , Γ=Ki∩Kj
−α‖[~nKi ∧ (~E ∧ ~nKi )−δ‖[~nKi ∧ (~H ∧ ~nKi )]‖2Γ∑Γ∈Γb, Γ⊂Ki
−α‖~nKi ∧ (~EKi ∧ ~nKi )‖2Γ − δ‖~nKi ∧ (~HKi ∧ ~nKi )‖2Γ
=⇒ Decreasing energy: Dissipative scheme.
2 −β = γ = 12 , α = 0 et δ = 0 =⇒ ∂
∂tE(t) = 0
=⇒ Energy conservation: Conservative scheme.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 42 / 50
Discrete Energy
EKi (t) =∑
Ki⊂Ω
∫Ki
(ε~EKi ) · ~EKi dx +
∫Ki
(µ~HKi ) · ~HKi dx
1 −β = γ = 12 , α ≥ 0 and δ ≥ 0 =⇒
∂E∂t
(t) =∑
Γ∈Fi , Γ=Ki∩Kj
−α‖[~nKi ∧ (~E ∧ ~nKi )−δ‖[~nKi ∧ (~H ∧ ~nKi )]‖2Γ∑Γ∈Γb, Γ⊂Ki
−α‖~nKi ∧ (~EKi ∧ ~nKi )‖2Γ − δ‖~nKi ∧ (~HKi ∧ ~nKi )‖2Γ
=⇒ Decreasing energy: Dissipative scheme.
2 −β = γ = 12 , α = 0 et δ = 0 =⇒ ∂
∂tE(t) = 0
=⇒ Energy conservation: Conservative scheme.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 42 / 50
Discrete Energy
EKi (t) =∑
Ki⊂Ω
∫Ki
(ε~EKi ) · ~EKi dx +
∫Ki
(µ~HKi ) · ~HKi dx
1 −β = γ = 12 , α ≥ 0 and δ ≥ 0 =⇒
∂E∂t
(t) =∑
Γ∈Fi , Γ=Ki∩Kj
−α‖[~nKi ∧ (~E ∧ ~nKi )−δ‖[~nKi ∧ (~H ∧ ~nKi )]‖2Γ∑Γ∈Γb, Γ⊂Ki
−α‖~nKi ∧ (~EKi ∧ ~nKi )‖2Γ − δ‖~nKi ∧ (~HKi ∧ ~nKi )‖2Γ
=⇒ Decreasing energy: Dissipative scheme.
2 −β = γ = 12 , α = 0 et δ = 0 =⇒ ∂
∂tE(t) = 0
=⇒ Energy conservation: Conservative scheme.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 42 / 50
Discrete Energy
EKi (t) =∑
Ki⊂Ω
∫Ki
(ε~EKi ) · ~EKi dx +
∫Ki
(µ~HKi ) · ~HKi dx
1 −β = γ = 12 , α ≥ 0 and δ ≥ 0 =⇒
∂E∂t
(t) =∑
Γ∈Fi , Γ=Ki∩Kj
−α‖[~nKi ∧ (~E ∧ ~nKi )−δ‖[~nKi ∧ (~H ∧ ~nKi )]‖2Γ∑Γ∈Γb, Γ⊂Ki
−α‖~nKi ∧ (~EKi ∧ ~nKi )‖2Γ − δ‖~nKi ∧ (~HKi ∧ ~nKi )‖2Γ
=⇒ Decreasing energy: Dissipative scheme.
2 −β = γ = 12 , α = 0 et δ = 0 =⇒ ∂
∂tE(t) = 0
=⇒ Energy conservation: Conservative scheme.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 42 / 50
Discrete Formulation (Gauss Points)
BεEn+1 − En
∆t+ Rh Hn+1/2 + Bσ
En+1 + En
2
+ αDh En + β Sh Hn+1/2 + Jn = 0,
BµHn+1/2 − Hn−1/2
∆t+ Rh En + γ S∗h En + δD∗h Hn−1/2 = 0,
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 43 / 50
Discrete Formulation (Gauss Points)
BεEn+1 − En
∆t+ Rh Hn+1/2 + Bσ
En+1 + En
2
+ αDh En + β Sh Hn+1/2 + Jn = 0,
BµHn+1/2 − Hn−1/2
∆t+ Rh En + γ S∗h En + δD∗h Hn−1/2 = 0,
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 43 / 50
Discrete Formulation (Gauss Points)
BεEn+1 − En
∆t+ Rh Hn+1/2 + Bσ
En+1 + En
2
+ αDh En + β Sh Hn+1/2 + Jn = 0,
BµHn+1/2 − Hn−1/2
∆t+ Rh En + γ S∗h En + δD∗h Hn−1/2 = 0,
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 43 / 50
Main Features of this Approximation
Bε, Bσ, Bµ: 3× 3 block-diagonal symmetric mass matrices,Rh: very sparse matrix which needs no storage,Sh, S∗h: jump block-diagonal symmetric matrices which need nostorage,Dh, Dh
∗: jump block-diagonal symmetric matrices which must bestored.
−→ The dissipative terms induce a (reasonable) additonal storage.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 44 / 50
Main Features of this Approximation
Bε, Bσ, Bµ: 3× 3 block-diagonal symmetric mass matrices,Rh: very sparse matrix which needs no storage,Sh, S∗h: jump block-diagonal symmetric matrices which need nostorage,Dh, Dh
∗: jump block-diagonal symmetric matrices which must bestored.
−→ The dissipative terms induce a (reasonable) additonal storage.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 44 / 50
Main Features of this Approximation
Bε, Bσ, Bµ: 3× 3 block-diagonal symmetric mass matrices,Rh: very sparse matrix which needs no storage,Sh, S∗h: jump block-diagonal symmetric matrices which need nostorage,Dh, Dh
∗: jump block-diagonal symmetric matrices which must bestored.
−→ The dissipative terms induce a (reasonable) additonal storage.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 44 / 50
Main Features of this Approximation
Bε, Bσ, Bµ: 3× 3 block-diagonal symmetric mass matrices,Rh: very sparse matrix which needs no storage,Sh, S∗h: jump block-diagonal symmetric matrices which need nostorage,Dh, Dh
∗: jump block-diagonal symmetric matrices which must bestored.
−→ The dissipative terms induce a (reasonable) additonal storage.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 44 / 50
Main Features of this Approximation
Bε, Bσ, Bµ: 3× 3 block-diagonal symmetric mass matrices,Rh: very sparse matrix which needs no storage,Sh, S∗h: jump block-diagonal symmetric matrices which need nostorage,Dh, Dh
∗: jump block-diagonal symmetric matrices which must bestored.
−→ The dissipative terms induce a (reasonable) additonal storage.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 44 / 50
Another Feature of Numerical Dissipation: PMLStabilization
0 50 100 150 200−10
−5
0
5
Time (in seconds)
log 10
(|| u
||)
No penalization, α = 0Penalization, α = 0.1
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 45 / 50
Numerical Examples
Dielectric spherical torus
Figure: Configuration of the experiment
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 46 / 50
Numerical Examples
Dielectric spherical torus
Figure: Ey component of the electric field at a point of the domain afterpropagation across 10λ (left) and 120λ (right).
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 47 / 50
Numerical Examples
Dielectric spherical torus
CPU time: FETD (Q3) : 300 s, FDTD (20pts/λ) : 1100 s.
Storage FDTD (20pts/λ)/FETD (Q3) = 10.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 48 / 50
Numerical Examples
Dielectric spherical torus
CPU time: FETD (Q3) : 300 s, FDTD (20pts/λ) : 1100 s.
Storage FDTD (20pts/λ)/FETD (Q3) = 10.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 48 / 50
Numerical Examples
Airplane
Frequency: 0.75 Ghz (30λ).
Mesh: 78 000 elements, 30 000 000 DOF (Q4).
Storage: 1.2 Go.
CPU time (30λ): 30 h on a monoprocessor Linux system, 2GoRam, 3.2 GHz.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 49 / 50
Numerical Examples
Airplane
Frequency: 0.75 Ghz (30λ).
Mesh: 78 000 elements, 30 000 000 DOF (Q4).
Storage: 1.2 Go.
CPU time (30λ): 30 h on a monoprocessor Linux system, 2GoRam, 3.2 GHz.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 49 / 50
Numerical Examples
Airplane
Frequency: 0.75 Ghz (30λ).
Mesh: 78 000 elements, 30 000 000 DOF (Q4).
Storage: 1.2 Go.
CPU time (30λ): 30 h on a monoprocessor Linux system, 2GoRam, 3.2 GHz.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 49 / 50
Numerical Examples
Airplane
Frequency: 0.75 Ghz (30λ).
Mesh: 78 000 elements, 30 000 000 DOF (Q4).
Storage: 1.2 Go.
CPU time (30λ): 30 h on a monoprocessor Linux system, 2GoRam, 3.2 GHz.
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 49 / 50
Numerical Examples
Airplane
Figure: The surfacic mesh (before splitting)
Figure: Snapshots of the currents on the plane with (right) and without (left)dissipation
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 50 / 50
Numerical Examples
Airplane
Figure: Snapshots of the currents on the plane with (right) and without (left)dissipation
M. Durufle, G Cohen (INRIA, project POEMS)Numerical Integration and High Order Finite Element Method Applied to Maxwell’s Equations25th april 2007 50 / 50