Skip to main content
SpringerLink
Log in

High-Order Accurate Methods for the Numerical Analysis of a Levitation Device

  • Original Paper
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

This paper establishes different axisymmetric and two-dimensional models for a levitation device. Therein, the Maxwell equations are combined with the balance of linear momentum. Different possible formulations to describe the Maxwell equations are presented and compared and discussed in the example. A high order finite element discretization using Galerkin’s method in space and the generalized Newmark\(-\alpha\) method in time are developed for the electro-magneto-mechanical approach. Several studies on spatial and temporal discretization with respect to convergence will be investigated. In addition, the boundary influences and the domain size with respect to the levitation device are also examined.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  1. Gleim T, Kuhl D, Schleiting M, Wetzel A, Middendorf B (2019) High-order numerical methods for the thermal activation of SMA fibers. PAMM 17(1):509

    Article  Google Scholar 

  2. Karl H, Fetzer J, Kurz S, Lehner G, Rucker WM (1997) Description of team workshop problem 28: an electrodynamic levitation. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.124.1240

  3. Demkowicz L (2007) Computing with hp-adaptive finite elements: volume 1: one and two dimensional elliptic and Maxwell problems. In: Applied mathematics and nonlinear science series. Chapman & Hall/CRC

  4. Bastos J, Sadowski N (2003) Electromagnetic modeling by finite element methods. Marcel Dekker Inc, New York

    Book  Google Scholar 

  5. Jin J (2002) The finite element method in electromagnetics, vol 2. Wiley, Hoboken

    MATH  Google Scholar 

  6. Hughes T (2000) The finite element method. linear static and dynamic finite element analysis. Dover Publications, New York

    MATH  Google Scholar 

  7. Zienkiewicz O, Taylor R (2000) The finite element method. Volume 1. The basis. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  8. Steinle F (2005) Explorative Experimente: Ampère, Faraday und die Ursprünge der Elektrodynamik, Boethius, vol 50. Franz Steiner Verlag, Stuttgart

    Google Scholar 

  9. Rapetti F (2010) An overlapping mortar element approach to coupled magneto-mechanical problems. Math Comput Simul 80(8):1647 ESCO 2008 Conference

    Article  MathSciNet  Google Scholar 

  10. Schiesser W (1991) The numerical method of lines: integration of partial differential equations. Academic Press, Cambridge

    MATH  Google Scholar 

  11. Beck R, Deuflhard P, Hiptmair R, Hopper R, Wohlmuth B (1995) Adaptive multilevel methods for edge element discretizations of Maxwell’s equations. Surv Math Ind 8:1–31

    MathSciNet  Google Scholar 

  12. Ciarlet P (2005) Augmented formulations for solving Maxwell equations. Comput Methods Appl Mech Eng 194:559

    Article  MathSciNet  Google Scholar 

  13. Hoffman J (2000) Adaptive finite element methods for the unsteady Maxwell’s equations. Chalmers University of Technology, Göteborg

    Google Scholar 

  14. Elsherbeni A, Demir V (2008) The finite difference time domain for electromagnetics: with Matlab Simulations. SciTech Publishing Incorporated, Chennai

    Google Scholar 

  15. Morgan M (2013) Finite element and finite difference methods in electromagnetic scattering. In: Progress in electromagnetics research. Elsevier Science

  16. Newmark N (1952) Earthquake and blast effects on structures. Earthquake Engineering Research, Institute University of California, Los Angeles, pp 114–129

    Google Scholar 

  17. Chung J, Hulbert G (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha\) method. J Appl Mech 60:371

    Article  MathSciNet  Google Scholar 

  18. Kuhl D, Ramm E (1999) Generalized energy-momentum method for non-linear adaptive shell dynamics. Comput Methods Appl Mech Eng 178:343

    Article  MathSciNet  Google Scholar 

  19. Faraday M (1832) Experimental researches in electricity. R. Taylor, pp 125–162. No. 122 in Philosophical Transactions of the Royal Society of London: Giving Some Accounts of the Present Undertakings, Studies, and Labours, of the Ingenious, in Many Considerable Parts of the World

  20. Maxwell J (1873) A treatise on electricity and magnetism. Clarendon Press, Oxford

    MATH  Google Scholar 

  21. Gauss C (1877) Carl Friedrich Gauss Werke. No. Bd. 5 in Carl Friedrich Gauss Werke. Königlichen Gesellschaft der wissenschaften

  22. Maxwell J (1865) A dynamical theory of the electromagnetic. Philos Trans R Soc Lond 155:459

    Google Scholar 

  23. Assous F, Ciarlet P, Labrunie S, Segré J (2003) Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: The singular complement method. J Comput Phys 191:147–176

    Article  MathSciNet  Google Scholar 

  24. Faraday M (1834) Experimental researches in electricity. Seventh series. Philos Trans R Soc Lond 124:77

    Google Scholar 

  25. Lenz E (1834) Über die Bestimmung der Richtung der durch elektrodynamische Verteilung erregten galvanischen Ströme. Ann Phys 107(31):483

    Article  Google Scholar 

  26. Gleim T, Schröder B, Kuhl D (2015) Nonlinear thermo-electromagnetic analysis of inductive heating processes. Arch Appl Mech 85(8):1055

    Article  Google Scholar 

  27. Gleim T (2016) Simulation of manufacturing sequences of functionally graded structures. Ph.D. Thesis, Schriftenreihe Fachgebiet Baumechanik/Baudynamik, Universität Kassel, Kassel

  28. Gleim T, Kuhl D (2018) Electromagnetic analysis using high-order numerical schemes in space and time. Arch Comput Methods Eng 26(2):405

    Article  MathSciNet  Google Scholar 

  29. Biro O, Richter K (1969) CAD in electromagnetism. Adv Electron Electron Phys 82:1

    Google Scholar 

  30. Klingbeil H (2010) Elektromagnetische Feldtheorie: ein Lehr- und Übungsbuch. Springer, Berlin

    Google Scholar 

  31. Silvester P, Ferrari R (1996) Finite elements for electrical engineers. Cambridge University Press, Cambridge

    Book  Google Scholar 

  32. Abali B (2016) Computational reality: solving nonlinear and coupled problems in continuum mechanics. In: Advanced structured materials. Springer, Singapore

  33. Motoasca E (2003) Electrodynamics in deformable solids for electromagnetic forming. Ph.D. Thesis, Delft University of Technology, Delft

  34. Pao YH (1978) In: Nemat-Nasser S (ed) Mechanics Today. Pergamon, pp 209–305. https://doi.org/10.1016/B978-0-08-021792-5.50012-4

  35. Rinaldi C, Brenner H (2002) Body versus surface forces in continuum mechanics: is the Maxwell stress tensor a physically objective Cauchy stress? Phys Rev E 65:036615

    Article  MathSciNet  Google Scholar 

  36. Wall W (1999) Fluid-struktur-interaktion mit stabilisierten finiten elementen. Ph.D. Thesis, Universität Stuttgart, Stuttgart

  37. Kurz S, Fetzer J, Lehner G, Rucker WM (1998) A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM–FEM coupling. IEEE Trans Magn 34(5):3068

    Article  Google Scholar 

  38. Gleim T, Kuhl D (2013) Higher order accurate discontinuous and continuous p-Galerkin methods for linear elastodynamics. Z Angew Math Mech 93:177–194

    Article  MathSciNet  Google Scholar 

  39. Wood W, Bossak M, Zienkiewicz O (1981) An alpha modification of Newmark’s method. Int J Numer Methods Eng 15:1562

    Article  MathSciNet  Google Scholar 

  40. Hilber H, Hughes T, Taylor R (1977) Improved numerical dissipation for the time integration algorithms in structural dynamics. Earthq Eng Struct Dyn 5:283

    Article  Google Scholar 

  41. Hoff C, Pahl P (1988) Development of an implicit method with numerical dissipation from generalized single step algorithm for structural dynamics. Comput Methods Appl Mech Eng 67:367

    Article  MathSciNet  Google Scholar 

  42. Hughes T, Caughey T, Liu W (1978) Finite-element methods for nonlinear elastodynamics which conserve energ. Trans ASME J Appl Mech 45:366

    Article  Google Scholar 

  43. Kuhl D, Meschke G (2007) Numerical analysis of dissolution processes in cementitious materials using discontinuous and continuous Galerkin time integration schemes. Int J Numer Methods Eng 69:1775–1803

    Article  Google Scholar 

  44. Jackson JD (1999) Classical electrodynamics, 3rd edn. Wiley, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Mechanics and Dynamics, University of Kassel, Mönchebergstr. 7, 34125, Kassel, Germany

    Hefeng Chen & Tobias Gleim

  2. Safety of Transport Containers, Federal Institute for Materials Research and Testing, Unter den Eichen 44-46, 12203, Berlin, Germany

    Tobias Gleim

Authors
  1. Hefeng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobias Gleim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tobias Gleim.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, H., Gleim, T. High-Order Accurate Methods for the Numerical Analysis of a Levitation Device. Arch Computat Methods Eng 28, 1517–1543 (2021). https://doi.org/10.1007/s11831-020-09427-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-020-09427-z

Search

Navigation