IGA BEM for Maxwell Eigenvalue Problems - Ricam · IGA BEM for Maxwell Eigenvalue Problems Stefan...

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


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence

• Contour Integral Method

• „Fast Methods”

• Conclusions and Outlook


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence





Motivation

Particle Accelerators

LHC: 27 km

Source:

CERN

Aerial image of Geneva region

with LHC ring indicated in red


Motivation

Superconducting Radiofrequency Cavity

TESLA

9-cell cavity

Source:

Fermilab

Nice photograph of TESLA 9-cell cavity


Motivation

Fields and Design of a TESLA 9-Cell Cavity


Motivation

Maxwell Eigenvalue Problem

Curvilinear

Lipschitz

polyhedron

(at least)

Relative accuracy 10−9 for the resonance

frequency of the accelerating mode required!


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence





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)


IGA BEM

Electric Field Integral Equation

Find and

such that


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


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


IGA BEM

Definition of B-Splines (1)

deg p = 1, dim k = 4 deg p = 2, dim k = 7


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


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence





Spaces

The Hilbert-de Rham Complex


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


Spaces

Mapping to the Physical Domain

• Pullbacks for single-patch domain:

• Extension to multi-patch domain:

Piola

NURBS


Spaces

The Buffa Spline Complex (1)


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


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence





Convergence

Approximation Property (1)

Consider

• single patch domain Γ, quasi-uniform knot vector

• spline space of minimal degree p

•

Then


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


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


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence





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!



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



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



Beyn’s Method (cont’d)

contour points # eigenvalues in contour



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



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


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence





„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


„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


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.


Outline

• Motivation

• IGA BEM

• Spaces

• Convergence





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


Outlook:

• Implement and investigate integration with fast methods

• Complete numerical analysis for multi-patch domains

1) Computer-Aided Design2) Computer-Aided Engineering


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

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