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A Survey in Mathematics for Industry An efficient method for the numerical simulation of magneto-mechanical sensors and actuators

Published online by Cambridge University Press:  01 April 2007

M. SCHINNERL ,
M. KALTENBACHER ,
U. LANGER ,
R. LERCH  and
J. SCHÖBERL
M. SCHINNERL
Affiliation:
Special Research Program “Numerical and Symbolic Scientific Computing”, SFB F013, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria email: ulanger@numa.uni-linz.ac.at
M. KALTENBACHER
Affiliation:
Department of Sensor Technology, Friedrich-Alexander-University Erlangen-Nuremberg, Paul-Gordan-Strasse 3/5, D-91052 Erlangen, Germany
U. LANGER
Affiliation:
Special Research Program “Numerical and Symbolic Scientific Computing”, SFB F013, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria email: ulanger@numa.uni-linz.ac.at
R. LERCH
Affiliation:
Department of Sensor Technology, Friedrich-Alexander-University Erlangen-Nuremberg, Paul-Gordan-Strasse 3/5, D-91052 Erlangen, Germany
J. SCHÖBERL
Affiliation:
Special Research Program “Numerical and Symbolic Scientific Computing”, SFB F013, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria email: ulanger@numa.uni-linz.ac.at
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Abstract

The dynamic behaviour of magneto-mechanical sensors and actuators can be completely described by Maxwell's and Navier-Lamé's partial differential equations (PDEs) with appropriate coupling terms reflecting the interactions of these fields and with the corresponding initial, boundary and interface conditions. Neglecting the displacement currents, which can be done for the classes of problems considered in this paper, and introducing the vector potential for the magnetic field, we arrive at a system of degenerate parabolic PDEs for the vector potential coupled with the hyperbolic PDEs for the displacements.Usually the computational domain, the finite element discretization, the time integration, and the solver are different for the magnetic and mechanical parts. For instance, the vector potential is approximated by edge elements whereas the finite element discretization of the displacements is based on nodal elements on different meshes. The most time consuming modules in the solution procedure are the solvers for both, the magnetical and the mechanical finite element equations arising at each step of the time integration procedure. We use geometrical multigrid solvers which are different for both parts. These multigrid solvers enable us to solve quite efficiently not only academic test problems, but also transient 3D technical magneto-mechanical systems of high complexity such as solenoid valves and electro-magnetic-acoustic transducers. The results of the computer simulation are in very good agreement with the experimental data.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 2 , April 2007 , pp. 233 - 271
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Arnold, D. N., Falk, R. S. & Winther, R. (2000) Multigrid in H(div) and H(curl). Numerische Mathematik 85, 197–218.CrossRefGoogle Scholar
[2]Axelsson, O. (1996) Iterative solution methods (2nd edn). London. Cambridge University Press.Google Scholar
[3]Bachinger, F., Langer, U. & Schöberl, J. (2005) Numerical analysis of nonlinear multiharmonic eddy current problems. Numerische Mathematik 100, 593–616.CrossRefGoogle Scholar
[4]Ballanyi, (2000) Handbuch Weicheisenprüfer WP-Cs).Google Scholar
[5]Bathe, K. J. (1996) Finite element procedures. New Jersey. Prentice-Hall.Google Scholar
[6]Beitz, W. & Küttner, K.-H. (1990) Dubbel, Taschenbuch für den Maschinenbau (17th edn). Berlin. Springer-Verlag.CrossRefGoogle Scholar
[7]Costabel, M. & Dauge, M. (2002) Weighted regularization of Maxwell equations in polyhedral domains. Numerische Mathematik 93, 239–278.CrossRefGoogle Scholar
[8]Douglas, C., Haase, G. & Langer, U. (2003) A tutorial on elliptic PDE solvers and their parallelization. Philadelphia. SIAM.CrossRefGoogle Scholar
[9]Ettinger, K. (1998) Numerical simulation of electromagnetic acoustic transducers. Master thesis. Linz. Johannes Kepler University Linz.Google Scholar
[10]Haase, G., Kuhn, M. & Langer, U. (2001) Parallel multigrid 3d Maxwell solvers. Parallel Computing 27, 761–775.CrossRefGoogle Scholar
[11]Haase, G. & Langer, U. (2002) Multigrid methods: From geometrical to algebraic versions. In Bourlioux, A. and Gander, M.J., editors, Modern Methods in Scientific Computing and Applications Volume 75 of NATO Science Ser. II, Mathematics, Physics and Chemistry. Pages 103–153. Dordrecht. Kluwer Academic.CrossRefGoogle Scholar
[12]Hackbusch, W. (1985) Multi-grid methods and applications. Berlin. Springer-Verlag.CrossRefGoogle Scholar
[13]Hewlett-Packard. Handbuch Impedance/Gain Phase Analyzer 4194 A.Google Scholar
[14]Hiptmair, R. (1999) Multigrid methods for Maxwell's equations. SIAM Journal on Numerical Analysis 36, 204–225.CrossRefGoogle Scholar
[15]Hofer, M. (1999) Simulation und Messung der Abstrahleigenschaften von Elektromagnetisch – Akustischen Transducern. Master thesis. Linz. Johannes Kepler University Linz.Google Scholar
[16]Hughes, T. J. R. (1987) Finite element method. New Jersey. Prentice-Hall.Google Scholar
[17]Jung, M. & Langer, U. (1991) Applications of multilevel methods to practical problems. Surveys on Mathematics for Industry 1, 217–257.Google Scholar
[18]Kallenbach, E., Eick, R. & Quendt, P. (1994) Elektromagnete. Stuttgart. Teubner-Verlag.CrossRefGoogle Scholar
[19]Kaltenbacher, M. (2004) Numerical simulation of mechatronic sensors and actuators. Berlin, Heidelberg, New York. Springer-Verlag.CrossRefGoogle Scholar
[20]Kaltenbacher, M., Landes, H. & Lerch, R. (1997) An efficient calculation scheme for the numerical simulation of coupled magnetomechanical systems. IEEE Transactions on Magnetics 33, 1646–1649.CrossRefGoogle Scholar
[21]Kaltenbacher, M., Landes, H. & Lerch, R. (1999) CAPA Verification Manual, Release 3. Linz. Johannes Kepler University Linz.Google Scholar
[22]Kino, G. S. (1987) Acoustic waves – devices, imaging & analog signal processing. New Jersey. Prentice-Hall.Google Scholar
[23]Krautkrämer, J. & Krautkrämer, H. (1986) Werkstoffprüfung mit Ultraschall (5th edn). Berlin. Springer-Verlag.CrossRefGoogle Scholar
[24]Kurz, S., Fetzer, J., Lehner, G. & Rucker, W. (1998) A novel formulation for 3d eddy current problems with moving bodies using a Lagrangian description and bem-fem coupling. IEEE Transactions on Magnetics 34, 3068–3073.CrossRefGoogle Scholar
[25]Lerch, R. (2004) Elektrische Messtechnik. Berlin. Springer-Verlag.Google Scholar
[26]Ludwig, R. & Dai, X.-W. (1991) Numerical simulation of electromagnetic acoustic transducer in the time domain. J. Appl. Phys. 69, 89–98.CrossRefGoogle Scholar
[27]Morgan, D. P. (1991) SAW devices and signal processing. Amsterdam. Elsevier.Google Scholar
[28]Nédélec, J. (1980) Mixed finite elements in R 3. Numerische Mathematik 35, 315–341.CrossRefGoogle Scholar
[29]Polytec (1999) Handbuch Polytec Laser Doppler Vibrometer.Google Scholar
[30]Reitzinger, S. & Schöberl, J. (2002) An algebraic multigrid method for finite element discretizations with edge elements. Numerical Linear Algebra with Applications 31, 223–238.CrossRefGoogle Scholar
[31]Ren, Z. & Razek, A. (1994) A strong coupled model for analysing dynamic behaviours of non-linear electromagnetic systems. IEEE Transactions on Magnetics 30, 3252–3255.CrossRefGoogle Scholar
[32]RITEC (1997) Operation manual for advanced measurement system model RAM-0.25-17.5.Google Scholar
[33]Schinnerl, M. (2001) Numerische Berechnung magneto-mechanischer Systeme mit Mehrgitterverfahren. PhD thesis. Erlangen-Nürnberg. University of Erlangen-Nürnberg.Google Scholar
[34]Schmitz, K. P. (1988) Entwicklung und Untersuchung einer schnellschaltenden elektromagnetischen Stelleinheit. PhD thesis. Aachen. RWTH Aachen.Google Scholar
[35]Silvester, P. S. & Ferrari, R. L. (1996) Finite elements for electrical engineers (3rd edn). London. Cambridge University Press.CrossRefGoogle Scholar
[36]Stratton, J. A. (1941) Electromagnetic theory. McGraw-Hill, Inc.Google Scholar
[37]Wunsch, G. & Schulz, H. (1996) Finite elements for electrical engineers (2nd edn). Berlin. Verlag Technik.Google Scholar
[38]Ziegler, F. (1991) Mechanics of Solids and Fluids. Vienna, Springer-Verlag.CrossRefGoogle Scholar