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arxiv: 2407.11703 · v1 · pith:BPBF4IP4new · submitted 2024-07-16 · 🧮 math.OC · cs.NA· math.NA

Numerical Eigenvalue Optimization by Shape-Variations for Maxwell's Eigenvalue Problem

Christine Herter , Sebastian Sch\"ops , Winnifried Wollner This is my paper

Pith reviewed 2026-05-23 22:54 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords Maxwell eigenvalue problemshape optimizationadjoint methoddomain variationsmixed finite elementsBFGS optimizationelectromagnetic resonance
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The pith

Normalization of eigenfunctions yields adjoint formulas for eigenvalue derivatives under domain variations, enabling shape optimization of Maxwell eigenvalues via damped inverse BFGS.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives adjoint-based sensitivities for Maxwell eigenvalues with respect to domain shape changes. It starts from the mixed variational formulation on a reference domain using Piola-type mappings, then normalizes eigenfunctions to secure local uniqueness. This normalization permits explicit adjoint expressions for the derivatives that feed into a damped inverse BFGS optimizer. The method is discretized with mixed finite elements and tested on a numerical example that drives an eigenvalue to a prescribed target. A reader would care because resonant frequencies in electromagnetic devices are directly controlled by geometry, and the approach supplies a concrete computational route for that control.

Core claim

After the mixed formulation of the Maxwell eigenvalue problem is pulled back to a fixed reference domain via suitable transformations, normalization of the eigenfunctions produces local uniqueness and thereby allows derivation of adjoint formulas for the eigenvalue derivatives with respect to domain variations; these formulas are inserted into a damped inverse BFGS algorithm whose positive-definiteness and line-search properties are retained, and the resulting optimization problem is discretized by mixed finite elements.

What carries the argument

Adjoint formulas for the derivatives of the eigenvalues with respect to domain variations, obtained after normalization of the eigenfunctions in the mixed formulation.

If this is right

  • Target eigenvalue values are reached by repeated small deformations of the computational domain.
  • The optimization stays on a fixed reference mesh because all quantities are pulled back via the domain mapping.
  • The damped inverse BFGS iteration preserves positive definiteness while simplifying the line search.
  • Mixed finite-element discretization of the pulled-back problem produces the discrete sensitivities used in each step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same normalization-plus-adjoint pattern could be applied to other Maxwell-type problems once an analogous mixed formulation exists.
  • Convergence of the optimizer may slow when the normalization constant approaches zero near eigenvalue crossings.
  • Extending the single-eigenvalue objective to a weighted sum of several eigenvalues would require only additional adjoint solves.

Load-bearing premise

Normalization of the eigenfunctions produces local uniqueness of the solution and thereby permits the adjoint formulas to be derived.

What would settle it

A concrete domain deformation for which the eigenvalue derivative computed from the adjoint formula differs from the value obtained by finite-difference perturbation of the same normalized eigenpair.

Figures

Figures reproduced from arXiv: 2407.11703 by Christine Herter, Sebastian Sch\"ops, Winnifried Wollner.

Figure 1
Figure 1. Figure 1: Electric field of the TM010, also called π-mode The analytic sensitives of eigenpairs of Maxwell’s eigenvalue problem have been considered in isogeometric analysis, e.g., in work of [54], where the sensitivities of the eigenpair have been computed with respect to a scalar parameter. The therein parameter optimization has been done with a simplified formulation of Maxwell’s eigenvalue problem excluding the … view at source ↗
Figure 2
Figure 2. Figure 2: Geometry of a Cavity [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mesh of the 5-cell Cavity BFGS method converges after 16 iterations. For a setting with 5438 DoFs, it takes 12. The values of Jq are slightly smaller than 1, which means that we obtain a small shrinkage of the domain. The deformation of a 5-cell cavity after optimization is shown in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Deformation of a 5-cell cavity after optimization of the first eigenvalue to target value λ∗ = 6017 using the BFGS method from algorithm 1 with regularization parameters α = 100 and β = 10−6 . The chosen refinement level is 2, the number of DoFs is 86723. Lagrange elements of order 1 and lowest order Nédélec elements are used. order to solve this optimization problem, we discussed a damped inverse BFGS met… view at source ↗
read the original abstract

In this paper we consider the free-form optimization of eigenvalues in electromagnetic systems by means of shape-variations with respect to small deformations. The objective is to optimize a particular eigenvalue to a target value. We introduce the mixed variational formulation of the Maxwell eigenvalue problem introduced by Kikuchi (1987) in function spaces of (H(\operatorname{curl}; \Omega)) and (H^1(\Omega)). To handle this formulation, suitable transformations of these spaces are utilized, e.g., of Piola-type for the space of (H(\operatorname{curl}; \Omega)). This allows for a formulation of the problem on a fixed reference domain together with a domain mapping. Local uniqueness of the solution is obtained by a normalization of the the eigenfunctions. This allows us to derive adjoint formulas for the derivatives of the eigenvalues with respect to domain variations. For the solution of the resulting optimization problem, we develop a particular damped inverse BFGS method that allows for an easy line search procedure while retaining positive definiteness of the inverse Hessian approximation. The infinite dimensional problem is discretized by mixed finite elements and a numerical example shows the efficiency of the proposed approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper develops a shape-optimization method for Maxwell eigenvalues via domain variations. It employs Kikuchi's mixed variational formulation in (H(curl), H^1) spaces, applies Piola-type transformations to map to a fixed reference domain, invokes normalization of eigenfunctions to obtain local uniqueness, derives adjoint formulas for eigenvalue derivatives with respect to the domain, solves the resulting problem with a damped inverse BFGS algorithm, discretizes via mixed finite elements, and demonstrates the approach on a numerical example.

Significance. If the adjoint derivation holds, the work supplies a concrete, implementable pipeline that combines established domain-mapping and adjoint techniques with a specialized quasi-Newton solver for electromagnetic eigenvalue shape optimization. The numerical example provides evidence of practical efficiency in the mixed-FEM setting.

major comments (1)
  1. [Abstract, §3] Abstract and §3: The claim that normalization of the eigenfunctions produces local uniqueness permitting derivation of the adjoint formulas for dλ/dΩ does not address multiplicity. When the target eigenvalue has multiplicity ≥2, the normalization (typically an L2-type constraint) leaves a circle of equivalent eigenfunctions; the shape derivative becomes set-valued and the adjoint construction in the mixed (H(curl),H^1) setting does not automatically select a unique representative. No assumption that the eigenvalue remains simple under the domain variations, nor any selection mechanism (phase fixing, orthogonalization), is stated.
minor comments (2)
  1. [Abstract] Abstract contains the repeated article 'the the eigenfunctions'.
  2. [§4] The description of the damped inverse BFGS line-search procedure and the precise form of the inverse-Hessian update should be expanded for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comment. We address the major point on eigenvalue multiplicity below.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3: The claim that normalization of the eigenfunctions produces local uniqueness permitting derivation of the adjoint formulas for dλ/dΩ does not address multiplicity. When the target eigenvalue has multiplicity ≥2, the normalization (typically an L2-type constraint) leaves a circle of equivalent eigenfunctions; the shape derivative becomes set-valued and the adjoint construction in the mixed (H(curl),H^1) setting does not automatically select a unique representative. No assumption that the eigenvalue remains simple under the domain variations, nor any selection mechanism (phase fixing, orthogonalization), is stated.

    Authors: We agree that the manuscript does not explicitly discuss the case of multiple eigenvalues. The derivation of the adjoint sensitivity formulas in Section 3 relies on the assumption that the eigenvalue λ is simple. Under this assumption, the normalization condition provides the necessary local uniqueness to differentiate the eigenvalue with respect to the domain. For eigenvalues with multiplicity greater than one, the shape derivative is indeed set-valued, and the adjoint approach as presented would require additional selection criteria, such as phase fixing or orthogonalization to a reference eigenfunction. In our numerical example, the optimized eigenvalue remains simple throughout the iterations. We will revise the abstract and Section 3 to explicitly state the simplicity assumption and briefly note the limitation for multiple eigenvalues. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation from external Kikuchi formulation and standard domain-mapping arguments

full rationale

The paper's central steps derive adjoint formulas for eigenvalue shape derivatives from the mixed variational formulation (Kikuchi 1987, external citation) after normalization to obtain local uniqueness, followed by domain mapping to a reference domain and discretization. No step reduces by the paper's own equations to a fitted input renamed as prediction, self-definition of a quantity in terms of its output, or a load-bearing self-citation chain. The normalization step is presented as enabling the adjoint derivation without the result being tautological or forced by prior author work. This is the common case of an independent derivation resting on cited external foundations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the pre-existing Kikuchi mixed formulation and standard assumptions of shape calculus; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The mixed variational formulation of the Maxwell eigenvalue problem in H(curl; Ω) and H¹(Ω) introduced by Kikuchi (1987).
    Cited directly as the starting point for the formulation and subsequent transformations.

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Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Allaire and F

    G. Allaire and F. Jouve. Minimum stress optimal design with the level set method. Engineering Analysis with Boundary Elements, 32(11):909–918, 2008. doi:10.1016/j.enganabound.2007.05.007

    work page doi:10.1016/j.enganabound.2007.05.007 2008

  2. [2]

    Allaire, F

    G. Allaire, F. Jouve, and A.-M. Toader. Structural optimization using sensi- tivity analysis and a level-set method.Journal of Computational Physics, 194 (1):363–393, 2004. doi:10.1016/j.jcp.2003.09.032

    work page doi:10.1016/j.jcp.2003.09.032 2004

  3. [3]

    Allaire, F

    G. Allaire, F. Jouve, and N. Van Goethem. Damage and fracture evolution in brittle materials by shape optimization methods.Journal of Computational Physics, 230(12):5010–5044, 2011. doi:10.1016/j.jcp.2011.03.024. 14 CHRISTINE HERTER, SEBASTIAN SCHÖPS, AND WINNIFRIED WOLLNER

    work page doi:10.1016/j.jcp.2011.03.024 2011

  4. [4]

    Allaire, C

    G. Allaire, C. Dapogny, and P. Frey. Shape optimization with a level set based mesh evolution method.Computer Methods in Applied Mechanics and Engineering, 282:22–53, 2014. doi:10.1016/j.cma.2014.08.028

    work page doi:10.1016/j.cma.2014.08.028 2014

  5. [5]

    A. L. Andrew, K.-W. E. Chu, and P. Lancaster. Derivatives of eigenvalues and eigenvectors of matrix functions.SIAM J. Matrix Anal. Appl., 14(4):903–926,

    work page

  6. [6]

    Arndt, W

    D. Arndt, W. Bangerth, T. C. Clevenger, D. Davydov, M. Fehling, D. Garcia- Sanchez, G. Harper, T. Heister, L. Heltai, M. Kronbichler, R. M. Kynch, M. Maier, J.-P. Pelteret, B. Turcksin, and D. Wells. The deal.II li- brary, version 9.1. Journal of Numerical Mathematics, 27(4):203–213, 2019. doi:10.1515/jnma-2019-0064

    work page doi:10.1515/jnma-2019-0064 2019

  7. [7]

    Arndt, W

    D. Arndt, W. Bangerth, B. Blais, T. C. Clevenger, M. Fehling, A. V. Grayver, T. Heister, L. Heltai, M. Kronbichler, M. Maier, P. Munch, J.-P. Pelteret, R. Rastak, I. Thomas, B. Turcksin, Z. Wang, and D. Wells. Thedeal.II library, version 9.2.Journal of Numerical Mathematics, 28(3):131–146, 2020. doi:10.1515/jnma-2020-0043

    work page doi:10.1515/jnma-2020-0043 2020

  8. [8]

    B. Aune, R. Bandelmann, D. Bloess, B. Bonin, A. Bosotti, M. Champion, C. Crawford, G. Deppe, B. Dwersteg, D. A. Edwards, H. T. Edwards, M. Fer- rario, M. Fouaidy, P.-D. Gall, A. Gamp, A. Gössel, J. Graber, D. Hu- bert, M. Hüning, M. Juillard, T. Junquera, H. Kaiser, G. Kreps, M. Kuch- nir, R. Lange, M. Leenen, M. Liepe, L. Lilje, A. Matheisen, W.-D. Mölle...

    work page doi:10.1103/physrevstab.3.092001 2000

  9. [9]

    Balay, W

    S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith. Efficient management of parallelism in object oriented numerical software libraries. In E. Arge, A. M. Bruaset, and H. P. Langtangen, editors,Modern Software Tools in Scientific Computing, pages 163–202. Birkhauser Press, 1997. doi:10.1007/978-1-4612- 1986-6_8

    work page doi:10.1007/978-1-4612- 1997

  10. [10]

    Balay, S

    S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang. PETSc users manual. Technical Report ANL-95/11 - Revision 3.15, Argonne National Laboratory, 2021

    work page 2021

  11. [11]

    Balay, S

    S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang. PETSc Web page, 2021. URL https://petsc.org/

    work page 2021

  12. [12]

    Becker, D

    R. Becker, D. Meidner, and B. Vexler. Efficient numerical solution of parabolic optimization problems by finite element methods.Optim. Methods Softw., 22 (5):813–833, 2007. doi:10.1080/10556780701228532

    work page doi:10.1080/10556780701228532 2007

  13. [13]

    Bezbaruah, M

    M. Bezbaruah, M. Maier, and W. Wollner. Shape optimization of opti- cal microscale inclusions. SIAM J. Sci. Comput., 46(4):B377–B402, 2024. doi:10.1137/23M158262

    work page doi:10.1137/23m158262 2024

  14. [14]

    D. Boffi. Finite element approximation of eigenvalue problems.Acta Numer., 19:1–120, 2010. doi:10.1017/S0962492910000012

    work page doi:10.1017/s0962492910000012 2010

  15. [15]

    J. Cea. Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût.RAIRO - Modélisation mathématique NUMERICAL SHAPE EIGENV ALUE OPTIMIZATION 15 et analyse numérique, 20(3):371–402, 1986. doi:10.1051/m2an/1986200303711

    work page doi:10.1051/m2an/1986200303711 1986

  16. [16]

    G. Cohen. Higher-Order Numerical Methods for Transient Wave Equations. Scientific computation. Springer, 2002. doi:10.1007/978-3-662-04823-8

    work page doi:10.1007/978-3-662-04823-8 2002

  17. [17]

    J. Corno. Numerical Methods for the Estimation of the Impact of Geomet- ric Uncertainties on the Performance of Electromagnetic Devices. PhD the- sis, Technische Universität, Darmstadt, 2017. URL http://tuprints.ulb. tu-darmstadt.de/7038/

    work page 2017

  18. [18]

    M. C. Delfour and J. P. Zolésio.Shapes and Geometries. Society for Industrial and Applied Mathematics, second edition, 2011. doi:10.1137/1.9780898719826

    work page doi:10.1137/1.9780898719826 2011

  19. [19]

    Fischer, F

    M. Fischer, F. Lindemann, M. Ulbrich, and S. Ulbrich. Fréchet differ- entiability of unsteady incompressible Navier–Stokes flow with respect to domain variations of low regularity by using a general analytical frame- work. SIAM Journal on Control and Optimization, 55(5):3226–3257, 2017. doi:10.1137/16M1089563

    work page doi:10.1137/16m1089563 2017

  20. [20]

    Georg, W

    N. Georg, W. Ackermann, J. Corno, and S. Schöps. Uncertainty quantification for Maxwell’s eigenproblem based on isogeometric analysis and mode tracking. Computer Methods in Applied Mechanics and Engineering, 350:228–244, 2019. doi:10.1016/j.cma.2019.03.002

    work page doi:10.1016/j.cma.2019.03.002 2019

  21. [21]

    Geuzaine and J.-F

    C. Geuzaine and J.-F. Remacle. Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering , 79(11):1309–1331, 2009. doi:10.1002/nme.2579

    work page doi:10.1002/nme.2579 2009

  22. [22]

    C. Goll, T. Wick, and W. Wollner. DOpElib: Differential equations and op- timization environment; A goal oriented software library for solving pdes and optimization problems with pdes. Archive of Numerical Software, 5(2):1–14,

    work page

  23. [23]

    doi:10.11588/ans.2017.2.11815

    work page doi:10.11588/ans.2017.2.11815 2017

  24. [24]

    Griewank

    A. Griewank. The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert spaces. SIAM J. Numer. Anal., 24(3):684–705, 1987. doi:10.1137/07240

    work page doi:10.1137/07240 1987

  25. [25]

    Introduction to Shape Optimization

    J.HaslingerandR.A.E.Mäkinen. Introduction to Shape Optimization. Society for Industrial and Applied Mathematics, 2003. doi:10.1137/1.9780898718690

    work page doi:10.1137/1.9780898718690 2003

  26. [26]

    Heners, L

    J. Heners, L. Radtke, M. Hinze, and A. Düster. Adjoint shape optimization for fluid–structure interaction of ducted flows.Computational Mechanics, 61 (3):259–276, 2018. doi:10.1007/s00466-017-1465-5

    work page doi:10.1007/s00466-017-1465-5 2018

  27. [27]

    Hernandez, J

    V. Hernandez, J. E. Roman, and V. Vidal. Slepc: Scalable library for eigen- value problem computations. In J. M. L. M. Palma, A. A. Sousa, J. Dongarra, and V. Hernández, editors,High Performance Computing for Computational Science — VECPAR 2002, volume 2565 ofLecture Notes in Computer Science, pages 377–391. Springer, 2003. doi:10.1007/3-540-36569-9_25

    work page doi:10.1007/3-540-36569-9_25 2002

  28. [28]

    Hernandez, J

    V. Hernandez, J. E. Roman, and V. Vidal. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems.ACM Trans. Math. Software, 31(3):351–362, 2005. doi:10.1145/1089014.108901

    work page doi:10.1145/1089014.108901 2005

  29. [29]

    Herter, S

    C. Herter, S. Schöps, and W. Wollner. Eigenvalue optimization with respect to shape-variations in electromagnetic cavities.PAMM, 22(1):e202200122, 2023. doi:10.1002/pamm.202200122

    work page doi:10.1002/pamm.202200122 2023

  30. [30]

    Heuveline and R

    V. Heuveline and R. Rannacher. A posteriori error control for finite approx- imations of elliptic eigenvalue problems. Adv. Comput. Math., 15:107–138,

    work page

  31. [31]

    doi:10.1023/A:1014291224961

    work page doi:10.1023/a:1014291224961

  32. [32]

    Hinze, R

    M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE Constraints, volume 23 ofMathematical Modelling: Theory and Applications. Springer, 2009. doi:10.1007/978-1-4020-8839-1. 16 CHRISTINE HERTER, SEBASTIAN SCHÖPS, AND WINNIFRIED WOLLNER

    work page doi:10.1007/978-1-4020-8839-1 2009

  33. [33]

    Jorkowski and R

    P. Jorkowski and R. Schuhmann. Mode tracking for parametrized eigenvalue problems in computational electromagnetics. In 2018 International Applied Computational Electromagnetics Society Symposium (ACES), pages 1–2, 2018. doi:10.23919/ROPACES.2018.8364147

    work page doi:10.23919/ropaces.2018.8364147 2018

  34. [34]

    C. T. Kelley and E. W. Sachs. A new proof of superlinear convergence for Broyden’s method in Hilbert space.SIAM J. Optim., 1(1):146–150, 1991. doi:10.1137/0801011

    work page doi:10.1137/0801011 1991

  35. [35]

    F. Kikuchi. Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Computer Methods in Ap- plied Mechanics and Engineering, 64(1):509–521, 1987. doi:10.1016/0045- 7825(87)90053-3

    work page doi:10.1016/0045- 1987

  36. [37]

    Kranjčević, S

    M. Kranjčević, S. Gorgi Zadeh, A. Adelmann, P. Arbenz, and U. van Rienen. Constrained multiobjective shape optimization of superconducting rf cavities considering robustness against geometric perturbations. Phys. Rev. Accel. Beams, 22:122001, 2019. doi:10.1103/PhysRevAccelBeams.22.122001

    work page doi:10.1103/physrevaccelbeams.22.122001 2019

  37. [38]

    Kranjčević, A

    M. Kranjčević, A. Adelmann, P. Arbenz, A. Citterio, and L. Stingelin. Multi- objective shape optimization of radio frequency cavities using an evolutionary algorithm. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 920:106– 114, 2019. ISSN 0168-9002. doi:10.1016/j.nima.2018.12.066

    work page doi:10.1016/j.nima.2018.12.066 2019

  38. [39]

    C. S. Kubrusly.Spectral Theorem, chapter 3, pages 55–89. Birkhäuser Boston, Boston, 2012. ISBN 978-0-8176-8328-3. doi:10.1007/978-0-8176-8328-3_3

    work page doi:10.1007/978-0-8176-8328-3_3 2012

  39. [40]

    Lancaster

    P. Lancaster. On eigenvalues of matrices dependent on a parameter. Nu- merische Mathematik, 6:377–387, 1964. doi:10.1007/BF01386087

    work page doi:10.1007/bf01386087 1964

  40. [41]

    A. S. Lewis. The mathematics of eigenvalue optimization.Mathematical Pro- gramming, 97:155–176, 2003. doi:10.1007/s10107-003-0441-3

    work page doi:10.1007/s10107-003-0441-3 2003

  41. [42]

    Oxford Univer- sity Press (2018).https://doi.org/10.1093/oso/9780198814788.001.0001

    P. Monk.Finite element methods for Maxwell’s equations. Numerical Mathe- matics and Scientific Computation. Oxford University Press, New York, 2003. doi:10.1093/acprof:oso/9780198508885.001.0001

    work page doi:10.1093/acprof:oso/9780198508885.001.0001 2003

  42. [43]

    Murat and J

    F. Murat and J. Simon. Sur le contrôle par un domaine géométrique. Laboratoire d’Analyse Numérique de l’Université de Paris VI, 1976. URL https://api.semanticscholar.org/CorpusID:124497750

    work page 1976

  43. [44]

    Murat and J

    F. Murat and J. Simon. Etude de problemes d’optimal design. In J. Cea, editor, Optimization Techniques Modeling and Optimization in the Service of Man Part 2, volume 41 ofLecture Notes in Computer Science, pages 54–62. Springer Berlin Heidelberg, 1976. doi:10.1007/3-540-07623-9_279

    work page doi:10.1007/3-540-07623-9_279 1976

  44. [45]

    In: Proc

    F.MuratandL.Tartar. Calculus of Variations and Homogenization, volume31 of Progress in Nonlinear Differential Equations and Their Applications, chap- ter 6, pages 139–173. Birkhäuser Boston, Boston, MA, 1997. doi:10.1007/978- 1-4612-2032-9_6

    work page doi:10.1007/978- 1997

  45. [46]

    Numerical Optimization

    J. Nocedal and S. J. Wright.Numerical Optimization. Springer, New York, NY, USA, 2e edition, 2006. doi:10.1007/978-0-387-40065-5

    work page doi:10.1007/978-0-387-40065-5 2006

  46. [47]

    M. J. D. Powell. A fast algorithm for nonlinearly constrained optimization calculations. In Numerical analysis (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977), volume 630 ofLecture Notes in Math., pages 144–157, 1978. doi:10.1007/BFb0067703

    work page doi:10.1007/bfb0067703 1977

  47. [48]

    Putek, S

    P. Putek, S. G. Zadeh, and U. van Rienen. Multi-objective shape op- timization of TESLA-like cavities: Addressing stochastic Maxwell’s eigen- problem constraints. Journal of Computational Physics, 513:113125, 2024. NUMERICAL SHAPE EIGENV ALUE OPTIMIZATION 17 doi:10.1016/j.jcp.2024.113125

    work page doi:10.1016/j.jcp.2024.113125 2024

  48. [49]

    Rannacher, A

    R. Rannacher, A. Westenberger, and W. Wollner. Adaptive finite ele- ment solution of eigenvalue problems: Balancing of discretization and it- eration error. Journal of Numerical Mathematics , 18(4):303–327, 2010. doi:10.1515/JNUM.2010.015

    work page doi:10.1515/jnum.2010.015 2010

  49. [50]

    Shemelin

    V. Shemelin. Optimal choice of cell geometry for a multicell su- perconducting cavity. Phys. Rev. ST Accel. Beams , 12:114701, 2009. doi:10.1103/PhysRevSTAB.12.114701

    work page doi:10.1103/physrevstab.12.114701 2009

  50. [51]

    Sokolowski and J.-P

    J. Sokolowski and J.-P. Zolesio.Introduction to shape optimization. Springer Berlin Heidelberg, Berlin, Heidelberg, 1992. doi:10.1007/978-3-642-58106-9_1

    work page doi:10.1007/978-3-642-58106-9_1 1992

  51. [52]

    Toader and C

    A.-M. Toader and C. Barbarosie. 6. Optimization of eigenvalues and eigenmodes by using the adjoint method. In M. Bergounioux, Édouard Oudet, M. Rumpf, G. Carlier, T. Champion, and F. Santambrogio, edi- tors, Topological Optimization and Optimal Transport, volume 17 of Radon Ser. Comput. Appl. Math., pages 142–158. De Gruyter, Berlin, Boston, 2017. doi:10....

    work page doi:10.1515/9783110430417-006 2017

  52. [53]

    Tröltzsch

    F. Tröltzsch. Optimal control of partial differential equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Provi- dence, RI, 2010. doi:10.1090/gsm/112

    work page doi:10.1090/gsm/112 2010

  53. [54]

    Udongwo, S

    S.-A. Udongwo, S. G. Zadeh, R. Calaga, and U. van Rienen. Design and opti- misation of an 800 MHz 5-cell elliptical SRF cavity for T¯t working point of the future circular Electron-Positron Collide. InProc. IPAC’23, number 14 in InternationalParticleAcceleratorConference, pages764–767.JACoWPublish- ing, Geneva, Switzerland, 2023. doi:10.18429/JACoW-IPAC20...

    work page doi:10.18429/jacow-ipac2023-mopl089 2023

  54. [55]

    Ulbrich and S

    M. Ulbrich and S. Ulbrich.Nichtlineare Optimierung. Mathematik Kompakt. Birkhäuser Basel, 2012. doi:10.1007/978-3-0346-0654-7

    work page doi:10.1007/978-3-0346-0654-7 2012

  55. [56]

    Ziegler, M

    A. Ziegler, M. Merkel, P. Gangl, and S. Schöps. On the computa- tion of analytic sensitivities of eigenpairs in isogeometric analysis. Com- puter Methods in Applied Mechanics and Engineering , 409:115961, 2023. doi:10.1016/j.cma.2023.115961. Universität Hamburg, F achbereich Mathematik, Bundesstr. 55, 20146 Hamburg Email address: christine.herter@uni-hamb...

    work page doi:10.1016/j.cma.2023.115961 2023