Skip to main content
SpringerLink
Log in

Motivation: The Cavity Problem

  • Chapter
  • First Online:
Analysis and Implementation of Isogeometric Boundary Elements for Electromagnetism

Part of the book series: Springer Theses ((Springer Theses))

  • 226 Accesses

Abstract

This chapter gives a brief overview of the historircal context and the state-of-the-art relevant for the development of isogeometric boundary element methods for electromagnetic problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aune B, Bandelmann R, Bloess D, Bonin B, Bosotti A, Champion M, Crawford C, Deppe G, Dwersteg B, Edwards DA, Edwards HT, Ferrario M, Fouaidy M, Gall P-D, Gamp A, Gössel A, Graber J, Hubert D, Hüning M, Juillard M, Junquera T, Kaiser H, Kreps G, Kuchnir M, Lange R, Leenen M, Liepe M, Lilje L, Matheisen A, Möller W-D, Mosnier A, Padamsee H, Pagani C, Pekeler M, Peters H-B, Peters O, Proch D, Rehlich K, Reschke D, Safa H, Schilcher T, Schmuser P, Sekutowicz J, Simrock S, Singer W, Tigner M, Trines D, Twarowski K, Weichert G, Weisend J, Wojtkiewicz J, Wolff S, Zapfe K (2000) Superconducting TESLA cavities. Phys Rev Accel Beams 3(9):092001

    Article  Google Scholar 

  2. Wiedemann H (2007) Particle accelerator physics, 3rd edn. Springer, Berlin

    Google Scholar 

  3. Georg N, Ackermann W, Corno J, Schöps S (2019) Uncertainty quantification for Maxwell’s eigenproblem based on isogeometric analysis and mode tracking. Comput Methods Appl Mech Eng 350:228–244

    Article  MathSciNet  Google Scholar 

  4. Weiland T, Timm M, Munteanu I (2008) A practical guide to 3-D simulation. IEEE Microwave Mag 9(6):62–75

    Article  Google Scholar 

  5. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  Google Scholar 

  6. Beyn W-J (2012) An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl 436(10):3839–3863. Special Issue dedicated to Heinrich Voss’s 65th birthday

    Google Scholar 

  7. Cheng AH-D, Cheng DT (2005) Heritage and early history of the boundary element method. Eng Anal with Bound Elem 29(3):268–302

    Article  Google Scholar 

  8. Townsend A (2014) On the spline: a brief history of the computational curve. Int J Inter Arch + Spatial Design 2(3):48–59

    Google Scholar 

  9. Rautio J (2014) The long road to Maxwell’s equations. IEEE Spect 51(12):36–56

    Article  Google Scholar 

  10. Ritz W (1909) Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. Journal für reine und angewandte Mathematik 135:1–61

    MathSciNet  MATH  Google Scholar 

  11. Galerkin BG (1915) Rods and plates. Series occurring in various questions concerning the elasticequilibrium of rods and plates. Engineers Bulletin (Vestnik Inzhenerov) 19:897–908 In Russian

    Google Scholar 

  12. Sobolev SL (1936) Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales. Matematicheskiĭ Sbornik 1 43(1):39–72

    Google Scholar 

  13. Courant R (1943) Variational methods for the solution of problems of equilibrium and vibrations. Bull Amer Math Soc 49:1–23

    Article  MathSciNet  Google Scholar 

  14. Braess D (2007) Finite elemente. Springer, Berlin

    Book  Google Scholar 

  15. Green G (1828) An essay on the application of mathematical analysis to the theories of electricity and magnetism. George Green, Göteborg

    Google Scholar 

  16. Stratton JA (1941) Electromagnetic theory. IEEE Press, Piscataway

    MATH  Google Scholar 

  17. Rjasanow S (1998) The structure of the boundary element matrix for the threedimensional Dirichlet problem in elasticity. Numer Linear Algebra Appl 5(3):203–217

    Article  MathSciNet  Google Scholar 

  18. Greengard L, Rokhlin V (1987) A fast algorithm for particle simulations. J Comput Phys 73(2):325–348

    Article  MathSciNet  Google Scholar 

  19. Schoenberg IJ (1946) Contributions to the problem of approximation of equidistant data by analytic functions, Parts A and B. Appl Math 4:45–99

    Google Scholar 

  20. de Boor C (2001) A practical guide to splines, vol 27, Revised edn. Applied mathematical sciences. Springer, New York

    Google Scholar 

  21. de Casteljau PdF (1963) Courbes et surfaces à Pôles. Technical Report, Paris, Citröen

    Google Scholar 

  22. Piegl L, Tiller W (1997) The NURBS book, 2nd edn. Springer, Berlin

    Book  Google Scholar 

  23. Piegl L, Tiller W (1987) Curve and surface constructions using rational B-splines. Comput-Aided Design 19(9):485–498

    Article  Google Scholar 

  24. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, New York

    Book  Google Scholar 

  25. Harbrecht H (2011) Wavelet Galerkin schemes for the boundary element method in three dimensions. Dissertation. Technische Universität Chemnitz

    Google Scholar 

  26. Boggs PT, Althsuler A, Larzelere AR, Walsh EJ, Clay RL, Hardwick MF (2005) DART system analysis. Technical report, Sandia National Laboratories

    Google Scholar 

  27. Bontinck Z, Corno J, De Gersem H, Kurz S, Pels A, Schöps S, Wolf F, de Falco C, Dölz J, Vázquez R, Römer U (2017) Recent advances of isogeometric analysis in computational electromagnetics. In: ICS Newsletter (International Compumag Society) 3. arXiv:1709.06004

  28. Buffa A, Sangalli G, Vázquez R (2013) Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations. J Comput Phys 257(Part B.0):1291–1320

    Google Scholar 

  29. Corno J, de Falco C, De Gersem H, Schöps S (2016) Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities. Comput Phys Commun 201:1–7

    Article  MathSciNet  Google Scholar 

  30. Beer G, Mallardo V, Ruocco E, Marussig B, Zechner J, Dunser C, Fries T-P (2017) Isogeometric boundary element analysis with elasto-plastic inclusions. Part 2: 3-D problems. Comput Methods Appl Mech Eng 315(Supplement C):418–433

    Google Scholar 

  31. Simpson RN, Bordas S, Trevelyan J, Rabczuk T (2012) A two-dimensional isogeometric boundary element method for elastostatic analysis. Comput Methods Appl Mech Eng 209(212):87–100

    Article  MathSciNet  Google Scholar 

  32. Takahashi T, Matsumoto T (2012) An application of fast multipole method to isogeometric boundary element method for Laplace equation in two dimensions. Eng Anal Bound Elem 36(12):1766–1775

    Article  MathSciNet  Google Scholar 

  33. Monk P (2003) Finite element methods for Maxwell’s equations. Oxford University Press, Oxford

    Book  Google Scholar 

  34. Buffa A, Sangalli G, Vázquez R (2010) Isogeometric analysis in electromagnetics: B-splines approximation. Comp Methods Appl Mech Eng 199:1143–1152

    Article  MathSciNet  Google Scholar 

  35. Weggler L (2011) High order boundary element methods. Dissertation. Universitat des Saarlandes, Saarbrücken

    Google Scholar 

  36. Hiptmair R, Kielhorn L (2012) BETL-a generic boundary element template library. Technical Report, Seminar for Applied Mathematics, ETH Zurich, pp 2012–36

    Google Scholar 

  37. Śmigaj W, Betcke T, Arridge S, Phillips J, Schweiger M (2015) Solving boundary integral problems with BEM++. ACM Trans Math Softw 41(2):1–40

    Article  MathSciNet  Google Scholar 

  38. Li J, Dault D, Zhao R, Liu B, Tong Y, Shanker B (2015) Isogeometric analysis of integral equations using subdivision. In: 2015 IEEE international symposium on antennas and propagation USNC/URSI national radio science meeting, pp 153–154

    Google Scholar 

  39. Catmull E, Clark J (1978) Recursively generated B-spline surfaces on arbitrary topological meshes. Comput Aided Design 10(6):350–355

    Article  Google Scholar 

  40. Simpson R, Scott M, Taus M, Thomas D, Lian H (2014) Acoustic isogeometric boundary element analysis. Comput Methods Appl Mech Eng 269:265–290

    Article  MathSciNet  Google Scholar 

  41. Dölz J, Harbrecht H, Kurz S, Schöps S, Wolf F (2018) A fast isogeometric BEM for the three dimensional Laplace- and Helmholtz problems. Comput Methods Appl Mech Eng 330(Supplement C):83–101

    Google Scholar 

  42. Simpson R, Liu Z, Vázquez R, Evans J (2018) An isogeometric boundary element method for electromagnetic scattering with compatible B-spline discretizations. J Comput Phys 362:264–289

    Article  MathSciNet  Google Scholar 

  43. Dölz J, Kurz S, Schöps S, Wolf F (2019) Isogeometric boundary elements in electromagnetism: rigorous analysis, fast methods, and examples. SIAM J Sci Comput 41(5):B983–B1010

    Article  MathSciNet  Google Scholar 

  44. Kurz S, Schöps S, Wolf F (2019) Towards a spectral method of moments using computer aided design

    Google Scholar 

  45. Dölz J, Kurz S, Schöps S, Wolf F (2019) A numerical comparison of an isogeometric and a parametric higher order Raviart-Thomas approach to the electric field integral equation. IEEE Trans Antennas Propag 68(1):593–597

    Article  Google Scholar 

  46. Dölz J, Harbrecht H, Kurz S, Multerer M, Schöps S, Wolf F (2019) Bembel. The C++ boundary element engineering library. Official website, www.bembel.eu. Accessed 20 Aug 2019

  47. Dölz J, Harbrecht H, Kurz S, Multerer M, Schops S, Wolf F (2019) Bembel. The C++ boundary element engineering library. Archived Version 0.9. Zenodo. https://doi.org/10.5281/zenodo.2671596

  48. Dölz J, Harbrecht H, Kurz S, Multerer M, Schöps S, Wolf F (2020) Bembel: the fast isogeometric boundary element C++ library for laplace, Helmholtz, and Electric Wave Equation. In: SoftwareX 11, p 10476

    Google Scholar 

  49. Schobert DT, Eibert TF (2010) A multilevel interpolating fast integral solver with fast Fourier transform acceleration. In: 2010 URSI international symposium on electromagnetic theory, pp 520–523

    Google Scholar 

  50. Schobert DT, Eibert TF (2012) Fast integral equation solution by multilevel Green’s function interpolation combined with multilevel fast multipole method. IEEE Trans Antennas Propag 60(9):4458–4463

    Article  MathSciNet  Google Scholar 

  51. Dölz J, Harbrecht H, Peters M (2016) An interpolation-based fast multipole method for higher-order boundary elements on parametric surfaces. Int J Numer Methods Eng 108(13):1705–1728

    Article  MathSciNet  Google Scholar 

  52. Hackbusch W (2015) Hierarchical matrices: algorithms and analysis. Springer, Berlin

    Book  Google Scholar 

  53. Harbrecht H, Peters M (2013) Comparison of fast boundary element methods on parametric surfaces. Comput Methods Appl Mech Eng 261:39–55

    Article  MathSciNet  Google Scholar 

  54. Marussig B, Zechner J, Beer G, Fries T-P (2015) Fast isogeometric boundary element method based on independent field approximation. Comput Methods Appl Mech Eng 284:458–488. Isogeometric Analysis Special Issue

    Google Scholar 

  55. Jensen TK, Christensen O, Pedersen M (2003) On adaptive wavelet-based methods for the Maxwell equations. Technical University of Denmark, Kongens Lyngby, Denmark

    Google Scholar 

  56. Dahmen W, Harbrecht H, Schneider R (2006) Compression techniques for boundary integral equations. Asymptotically optimal complexity estimates. SIAM J Numer Anal 43(6):2251–2271

    Google Scholar 

  57. Buffa A, Vázquez RH, Sangalli G, da Veiga LB (2015) Approximation estimates for isogeometric spaces in multipatch geometries. Numer Methods Partial Diff Equ 31(2):422–438

    Article  MathSciNet  Google Scholar 

  58. Buffa A, Dölz J, Kurz S, Schöps S, Vázquez R, Wolf F (2020) Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis. Numerische Mathematik 144:201–236

    Article  MathSciNet  Google Scholar 

  59. Xin J (2011) Boundary element approximation for Maxwell’s eigenvalue problem. Dissertation. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Google Scholar 

  60. Mehrmann V, Voss H (2004) Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods. GAMM-Mitteilungen 27(2):121–152

    Article  MathSciNet  Google Scholar 

  61. Asakura J, Sakurai T, Tadano H, Ikegami T, Kimura K (2009) A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett 1:52–55

    Article  MathSciNet  Google Scholar 

  62. Imakura A, Du L, Sakurai T (2016) Relationships among contour integral-based methods for solving generalized eigenvalue problems. Jpn J Ind Appl Math 33(3):721–750

    Article  MathSciNet  Google Scholar 

  63. Elasmi M (2017) Boundary element method for Maxwell’s eigenvalue problems in TESLA cavities using the contour integral method. Master’s Thesis. Universität des Saarlandes, Saarbrücken

    Google Scholar 

  64. Unger G (2017) Convergence analysis of a Galerkin boundary element method for electromagnetic eigenvalue problems. Technical Report 2017/2, Institute of Computational Mathematics, Graz University of Technology

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute TEMF and Centre for Computational Engineering, Computational Electromagnetics Technische, Technical University of Darmstadt, Darmstadt, Germany

    Dr. Felix Wolf

Authors
  1. Dr. Felix Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Felix Wolf .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Wolf, F. (2021). Motivation: The Cavity Problem. In: Analysis and Implementation of Isogeometric Boundary Elements for Electromagnetism . Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-61939-8_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-61939-8_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-61938-1

  • Online ISBN: 978-3-030-61939-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation