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19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 1
RICAM Workshop Analysis and Numerics of Acoustic
and Electromagnetic Problems
IGA BEM for Maxwell Eigenvalue Problems
Stefan Kurz, Sebastian Schöps, Felix Wolf
Computational Electromagnetics Laboratory and
Graduate School Computational Engineering
Technische Universität Darmstadt, Germany
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 2
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 3
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 4
Motivation
Particle Accelerators
LHC: 27 km
Source:
CERN
Aerial image of Geneva region
with LHC ring indicated in red
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Motivation
Superconducting Radiofrequency Cavity
TESLA
9-cell cavity
Source:
Fermilab
Nice photograph of TESLA 9-cell cavity
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Motivation
Fields and Design of a TESLA 9-Cell Cavity
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Motivation
Maxwell Eigenvalue Problem
Curvilinear
Lipschitz
polyhedron
(at least)
Relative accuracy 10−9 for the resonance
frequency of the accelerating mode required!
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 8
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 9
IGA BEM
Why Boundary Element Method (BEM)?
+ Only boundary geometry needed
+ Ideally suited to the problem:
simple material, fundamental solution
− Dense matrices
− Eigenvalue problem becomes nonlinear
− (Nasty analysis)
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IGA BEM
Electric Field Integral Equation
Find and
such that
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 11
IGA BEM
Why Isogeometric Analysis (IGA)?
1) Non-Uniform Rational B-Splines
FEM NURBS
F
+ NURBS1) mapping F→ exact geometry
+ CAD systems use NURBS
+ B-Splines efficient in terms of DOFs
− Volumetric spline geometries
need to be constructed manually
J. Corno et al., Isogeometric simulation of Lorentz detuning in superconducting
accelerator cavities, 2015
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 12
IGA BEM
Why Isogeometric Analysis (IGA)?
1) Non-Uniform Rational B-Splines
+ NURBS1) mapping F→ exact geometry
+ CAD systems use NURBS
+ B-Splines efficient in terms of DOFs
− Volumetric spline geometries
need to be constructed manually
J. Corno et al., Isogeometric simulation of Lorentz detuning in superconducting
accelerator cavities, 2015
Error IGA FEM FEM Degree
1e-07 18 304 158 050 1
1e-08 47 520 381 036 1
1e-08 4 480 15 618 2
1e-10 30 628 135 246 2
DOFs required for given accuracy
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 13
IGA BEM
Definition of B-Splines (1)
deg p = 1, dim k = 4 deg p = 2, dim k = 7
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 14
IGA BEM
Definition of B-Splines (2)
• p-open knot vector
• Basis functions defined recursively;
• NURBS basis: weighted by and normalized,
• Derivatives of B-Splines are B-Splines (not for NURBS)
• Tensor product constructions
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 16
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 17
Spaces
The Hilbert-de Rham Complex
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Spaces
Conforming Discretization
Spline spaces on the unit square
A. Buffa & R. Vázquez, Isogeometric analysis for electromagnetic
scattering problems, 2014
removing first
and last element
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 19
Spaces
Mapping to the Physical Domain
• Pullbacks for single-patch domain:
• Extension to multi-patch domain:
Piola
NURBS
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Spaces
The Buffa Spline Complex (1)
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Spaces
The Buffa Spline Complex (2)
The diagram
commutes.
L. Beirao da Veiga et al., Mathematical analysis of variational
isogeometric methods, 2014
quasiinterpolant
single-patch domain
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 22
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 23
Convergence
Approximation Property (1)
Consider
• single patch domain Γ, quasi-uniform knot vector
• spline space of minimal degree p
•
Then
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 24
Convergence
Approximation Property (2)
Consider
• of sufficient regularity
• as its -orthogonal approximation
Then
With these results a discrete inf-sup condition can be established, as in
A. Buffa & R. Hiptmair, The electric field integral equation on Lipschitz
screens: definitions and numerical approximation, 2002
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Convergence
Numerical Test: Plane Wave on a Sphere
DOFs
102 103 104 105
10- 4
10- 3
10- 2
10- 1
100
L2
Err
or
DOFs
102 103 104 105
10- 6
10- 4
10- 2
100
L2
Err
or
deg p = 2 deg p = 3
B-Splines Raviart-Thomas
x-3 x-4
Save ~
61.000 DOFs
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 29
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 30
Contour Integral Method
Problem Definition
Galerkin discretization nonlinear eigenvalue problem:
Find and such that
holomorphic, eigenvalues in
We are going to reduce this to an
equivalent linear eigenvalue problem!
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Contour Integral Method
Beyn‘s Version of Keldysh’s Theorem
T as before, holomorphic . Then
with normalized left and right eigenvectors
W.-J. Beyn, An integral method for solving nonlinear eigenvalue
problems, 2012
# eigenvalues in contour
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Contour Integral Method
Beyn‘s Method
Construct a diagonalizable matrix B computable from T
with same eigenvalues as within D
1. Find such that
has maximal rank
2. Compute SVD1) of
3. Compute
4. is given by
1) Singular Value Decomposition
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Contour Integral Method
Beyn’s Method (cont’d)
contour points # eigenvalues in contour
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Contour Integral Method
Adaptive Method: Introduction
- 1 0 1
- 1
0
1
- 1 0 1
- 1
0
1
- 1 0 1
- 1
0
1
• BEM solution is
expensive
• Solve sloppily,
with few contour
points
• Limited accuracy
• Increase number
of contour points
• Expensive
• Accuracy
saturates
• Adaptive method
• Compute distance
of points
• Solve for disjoint
domains, containing
only one EV1) each
1) Eigenvalue
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 38
Contour Integral Method
Adaptive Method: Performance
100 150 200 250 300 35010-15
10-13
10-11
10-9
direct
adaptive
Error
Evaluations of polynomial
• Matrix-valued
polynomial,
order m = 60,
polynomial
degree 5
• Octave‘s
polyeig
as reference
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 39
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 40
„Fast Methods“
Calderón Preconditioning
• Matrix T rapidly ill-conditioned for
• EFIE1) operator preconditions itself,
eigenvalues accumulate around -1/4
• Need a Gram matrix to link domain and range of
• Discrete div- and curl-conforming spaces in stable L2 duality
• Classical BEM: Raviart-Thomas ↔ Buffa-Christiansen
• IGA BEM: Suitable B-Spline spaces under investigation
Li et al., Subdivision based isogeometric analysis technique for electric
field integral equations for simply connected structures, 2016, Fig. 20
1) Electric Field Integral Equation
Fig. 20 from
Li2016:
Relative
residual versus
number of
GMRES
iterations.
Calderón vastly
outperforms
diagonal
preconditioning
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 41
„Fast Methods“
Adaptive Cross Approximation (ACA)
• Represent T by a hierarchical matrix
• Block-partition T in such a way that index offset
corresponds to geometric distance
• Consider bounding boxes Q containing supports of
B-Spline basis functions as geometric objects
• Create a geometrically balanced binary cluster tree
• Approximate admissible blocks adaptively
by low-rank matrices
B. Marussig et al., Fast isogeometric boundary element
method based on independent field approximation, 2015, Fig. 8
Fig. 8 from
Marussig2015:
NURBS curve, two
B-Spline basis
functions, control
polygon of Bézier
segments,
bounding boxes of
basis functions’
supports
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 42
B. Marussig et al.
2015, Figs. 24, 25
„Fast Methods“
ACA: Crankshaft Example
Fig. 24 from Marussig2015:
Image of considered
crankshaft
Fig. 25 from Marussig2015: Compression rate
versus order m of T, for different ACA
approximation qualities. For m= 107, a
compression by about a factor of 10 can be
achieved. The curves show the expected n log n
asymptotic behavior.
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 43
Outline
• Motivation
• IGA BEM
• Spaces
• Convergence
• Contour Integral Method
• „Fast Methods”
• Conclusions and Outlook
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 44
Conclusions and Outlook
• IGA-BEM seems natural to reconcile CAD1) and CAE2)
• Benefit from the smoothness of geometry and fields in
accelerator applications
• Convert nonlinear into linear eigenvalue problem by
Contour Integral Method
Outlook:
• Implement and investigate integration with fast methods
• Complete numerical analysis for multi-patch domains
1) Computer-Aided Design2) Computer-Aided Engineering
19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 45
Further References
1. J. Asakura et al., A numerical method for nonlinear eigenvalue problems using contour
integrals, 2009
2. A. Buffa et al., Isogeometric methods for computational electromagnetics: B-spline and T-
spline discretizations, 2014
3. A. Buffa et al., Approximation estimates for isogeometric spaces in multi-patch geometries,
2015
4. G. Unger, Convergence orders of iterative methods for nonlinear eigenvalue problems, 2013
5. G. Unger, Numerical analysis of boundary element methods for time-harmonic Maxwell’s
eigenvalue problems, 2016
6. C. Wieners & J. Xin, Boundary element approximation for Maxwell's eigenvalue problem,
2013
7. J. Xiao et al., Solving large‐scale nonlinear eigenvalue problems by rational interpolation and
resolvent sampling based Rayleigh‐Ritz method, 2016