IGA BEM for Maxwell Eigenvalue Problems RICAM Workshop Analysis and Numerics of Acoustic and Electromagnetic 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. | 1
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. | 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
Motivation Particle Accelerators Aerial image of Geneva region with LHC ring indicated in red LHC: 27 km Source: CERN 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 4
Motivation Superconducting Radiofrequency Cavity Nice photograph of TESLA 9-cell cavity TESLA 9-cell cavity Source: Fermilab 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 5
Motivation Fields and Design of a TESLA 9-Cell Cavity 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 6
Motivation Maxwell Eigenvalue Problem Curvilinear Lipschitz Relative accuracy 10 −9 for the resonance polyhedron ! frequency of the accelerating mode required (at least) 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 7
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. | 8
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) 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 9
IGA BEM Electric Field Integral Equation Find and such that 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 10
IGA BEM Why Isogeometric Analysis (IGA)? + NURBS 1) mapping F → exact geometry + CAD systems use NURBS F + B-Splines efficient in terms of DOFs − Volumetric spline geometries need to be constructed manually FEM NURBS J. Corno et al., Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities , 2015 1) Non-Uniform Rational B-Splines 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 11
IGA BEM Why Isogeometric Analysis (IGA)? + NURBS 1) mapping F Error IGA FEM FEM Degree → exact geometry 1e-07 18 304 158 050 1 + CAD systems use NURBS 1e-08 47 520 381 036 1 + B-Splines efficient in terms of DOFs 1e-08 4 480 15 618 2 1e-10 30 628 135 246 2 − Volumetric spline geometries DOFs required for given accuracy need to be constructed manually J. Corno et al., Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities , 2015 1) Non-Uniform Rational B-Splines 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 12
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. | 13
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. | 14
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. | 16
Spaces The Hilbert-de Rham Complex 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 17
Spaces Conforming Discretization Spline spaces on the unit square removing first and last element A. Buffa & R. Vázquez, Isogeometric analysis for electromagnetic scattering problems, 2014 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 18
Spaces Mapping to the Physical Domain • Pullbacks for single-patch domain: NURBS Piola • Extension to multi-patch domain: 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 19
Spaces The Buffa Spline Complex (1) 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 20
Spaces The Buffa Spline Complex (2) The diagram quasiinterpolant commutes. single-patch domain L. Beirao da Veiga et al., Mathematical analysis of variational isogeometric methods , 2014 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 21
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. | 22
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. | 23
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 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 24
Convergence Numerical Test: Plane Wave on a Sphere B-Splines Raviart-Thomas 10 0 10 0 deg p = 2 deg p = 3 10 - 1 10 - 2 L 2 Error L 2 Error 10 - 2 10 - 4 10 - 3 10 - 4 10 - 6 x -3 x -4 10 5 10 2 10 3 10 4 10 2 10 3 10 4 10 5 Save ~ DOFs DOFs 61.000 DOFs 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 28
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. | 29
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 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 30
Contour Integral Method Beyn‘s Version of Keldysh’s Theorem T as before, holomorphic . Then # eigenvalues in contour with normalized left and right eigenvectors W.-J. Beyn, An integral method for solving nonlinear eigenvalue problems , 2012 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 31
Contour Integral Method Beyn‘s Method Construct a diagonalizable matrix B computable from T within D with same eigenvalues as 1. Find such that has maximal rank Compute SVD 1) of 2. 3. Compute 4. is given by 1) Singular Value Decomposition 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 32
Contour Integral Method Beyn’s Method (cont’d) contour points # eigenvalues in contour 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 33
Contour Integral Method Adaptive Method: Introduction 1 1 1 0 0 0 - 1 - 1 - 1 - 1 - 1 - 1 0 1 0 1 0 1 • • • BEM solution is Increase number Adaptive method • expensive of contour points Compute distance • • Solve sloppily, Expensive of points • • with few contour Accuracy Solve for disjoint points saturates domains, containing only one EV 1) each • Limited accuracy 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 37 1) Eigenvalue
Contour Integral Method Adaptive Method: Performance • Error Matrix-valued direct polynomial, 10 -9 adaptive order m = 60, polynomial 10 -11 degree 5 10 -13 • Octave‘s polyeig 10 -15 100 150 200 250 300 350 as reference Evaluations of polynomial 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 38
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. | 39
„Fast Methods“ Calderón Preconditioning • Matrix T rapidly ill-conditioned for Fig. 20 from Li2016: EFIE 1) operator preconditions itself, • Relative eigenvalues accumulate around -1/4 residual versus number of • Need a Gram matrix to link domain and range of GMRES Discrete div- and curl-conforming spaces in stable L 2 duality • iterations. • Classical BEM: Raviart- Thomas ↔ Buffa -Christiansen Calderón vastly • IGA BEM: Suitable B-Spline spaces under investigation outperforms diagonal preconditioning Li et al., Subdivision based isogeometric analysis technique for electric field integral equations for simply connected structures , 2016, Fig. 20 19.10.2016 | Fachbereich ETIT | Institut TEMF | S. Kurz et al. | 40 1) Electric Field Integral Equation
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