IGA BEM for Maxwell Eigenvalue Problems RICAM Workshop Analysis and - - PowerPoint PPT Presentation

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IGA BEM for Maxwell Eigenvalue Problems RICAM Workshop Analysis and - - PowerPoint PPT Presentation

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IGA BEM for Maxwell Eigenvalue Problems RICAM Workshop Analysis and Numerics of Acoustic and Electromagnetic Problems Stefan Kurz, Sebastian Schps, Felix Wolf Computational Electromagnetics Laboratory and Graduate School Computational

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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

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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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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

!

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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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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

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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

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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

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IGA BEM Definition of B-Splines (1)

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

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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
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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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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

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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

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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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Convergence Approximation Property (1)

Consider

  • single patch domain Γ, quasi-uniform knot vector
  • spline space of minimal degree p
  • Then
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Convergence Approximation Property (2)

Consider

  • f 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 Error DOFs 102 103 104 105 10- 6 10- 4 10- 2 100 L2 Error deg p = 2 deg p = 3 B-Splines Raviart-Thomas x-3 x-4 Save ~ 61.000 DOFs

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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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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

1

  • 1

1

  • 1

1

  • 1

1

  • 1

1

  • 1

1

  • BEM solution is

expensive

  • Solve sloppily,

with few contour points

  • Limited accuracy
  • Increase number
  • f contour points
  • Expensive
  • Accuracy

saturates

  • Adaptive method
  • Compute distance
  • f points
  • Solve for disjoint

domains, containing

  • nly one EV1) each

1) Eigenvalue

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Contour Integral Method Adaptive Method: Performance

100 150 200 250 300 350 10-15 10-13 10-11 10-9 direct adaptive Error Evaluations of polynomial

  • Matrix-valued

polynomial,

  • rder m = 60,

polynomial degree 5

  • Octave‘s

polyeig as reference

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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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„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

  • utperforms

diagonal preconditioning

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„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

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  • 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.

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Outline

  • Motivation
  • IGA BEM
  • Spaces
  • Convergence
  • Contour Integral Method
  • „Fast Methods”
  • Conclusions and Outlook
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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 Design 2)Computer-Aided Engineering

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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