Chair of Electromagnetic Theory

Full Space-Time Adaptive Numerical Computation Methods for Transient Magnetic Fields

Project title (German)
Vollständige raum-zeit-adaptive Verfahren für die numerische Berechnung von transienten Magnetfeldern

Project leader
Prof. Dr. rer. nat. Markus Clemens

Project member
Georg Wimmer (BUW),
Prof. Dr. Jens Lang (TU Darmstadt)
Dr. Delia Teleaga

Deutsche Forschungsgemeinschaft (DFG) under grant number CL 143/3-1/-2

Project description

Based on two simulation codes for full space-time adaptive magnetic field computations, MEQSICO (Magneto-/Electro-Quasistatic SImulation COde) and KARDOS, which have been newly developed and further developed respectively, and simulations for typical transient benchmark problems, such as the TEAM 7 problem and the SRC disk read-write head, the following results were obtained:

  1. The full space-time adaptive methods developed in this research project allow error-estimator controlled simulation of transient nonlinear magnetic fields, automatically resolving zones of local ferromagnetic saturation and locally confined eddy current layers efficiently. The grid refinements and coarsenings resolve the problems dynamics as finely as necessary in respect to error tolerances and as coarsely as possible regarding the minimum number of unknowns. This relieves potential users of such methods in terms of quality control of simulation results. Prior knowledge of the application problem and subsequent complex parameter studies for spatial and temporal resolution are thus no longer required.
  2. Linear-implicit time integration methods have proven to be very reliable. The control of a Newton-Raphson method is omitted due to the linearized structures. These methods have high consistency and very good stability properties, especially for differential-algebraic systems of index 1. These methods also immediately provide the time derivation for each time step, which can be used to calculate the eddy currents with high accuracy. Novel linear-implicit methods in the functional space have been derived for electro-quasistatic problems with nonlinear ferroelectrics. The one-step nature allows fast adaptation to the temporal evolution by rapidly varying the time step lengths and using different grid resolutions in a time step without additional adjustments.
  3. Hierarchical-based error estimators and Zienkiewicz-Zhu gradient reconstruction methods have proven to be effective for controlling both interpolation errors and spatial solution resolution.
  4. Improvements have been made in understanding geometric multigrid methods by a physically motivated interpretation of intergrid operators and special smoothing techniques for the curl-curl equations according to the Finite Integration Technique (FIT).
  5. Start value generation methods have proven to be an interesting way to reduce the effort of solving linear systems of equations of multilevel implicit time integration methods. Here, thematic cross-connections with methods of model order reduction have been shown.
  6. The research code MEQSICO (Magneto-/Electro-Quasistatic SImulation COde) is based on WFEM with lowest order basis functions, resulting in discrete magneto-quasi-static formulations in 2D and 3D analogous to the finite integration technique even when using unstructured triangular and tetrahedral grids. These discrete formulations have been extended with gradient reconstruction error estimators and adaptive grid generation techniques. The simulation code KARDOS has been extended with hierarchical Whitney elements and equipped with a new, improved linear implicit time integrator ROS3PL. Both simulation tools thus represent software prototypes for full space-time adaptive magnetic field computations that have been successfully validated on benchmark problems based on real engineering applications in energy and information technology.

With these results, an important step has been made towards robust and efficient time-domain simulations for complex two- and three-dimensional magnetic field problems. A subsequent transfer of the published empirical values into future simulation tools is possible and allows there a user-friendly simulation of transient magnetic fields.

Project-related publications

G. Wimmer, T. Steinmetz and M. Clemens, "Reuse, Recycle, Reduce (3R) - Strategies for the Calculation of Transient Magnetic Fields", Journal of Applied Numerical Mathematics, (Presented at 11th Seminar "NUMDIFF" on Numerical Solution of Differential and Differential-Algebraic Equations, Halle, Germany, 04.-08.09.2006.), vol. 59, no. 3-4, pp. 830-844, Mä. 2009.
M. Clemens, J. Lang, D. Teleaga and G. Wimmer, "Adaptivity in Space and Time for Magnetoquasistatics", Journal of Computational Mathematics, vol. 27, no. 5, pp. 642-656, 2009.
J. Lang and D. Teleaga, "Towards a Fully Space-Time Adaptive FEM for Magnetoquasistatics", IEEE Transactions on Magnetics, vol. 44, no. 6, pp. 1238 - 1241, 05 2008. IEEE.
G. Wimmer, M. Clemens and J. Lang, "Calculation of Magnetic Fields with Finite Elements", Buchbeitrag in: From Nano-Scale to Space, Applied Mathematics Inspired by Roland Bulirsch, Hrsg.: M. Breitner, G. Denk, P. Rentrop, Springer-Verlag Berlin Heidelberg, 2008.
M. Clemens, M. Helias, T. Steinmetz and G. Wimmer, "Multiple Righthand Side Techniques for the Numerical Simulation of Transient Quasistatic Electric and Magnetic Fields", Journal of Computational and Applied Mathematics, vol. 215, pp. 328-338, 2008.
M. Clemens, G. Wimmer, D. Teleaga and J. Lang, "Space and Time Adaptive Calculation of Transient 3D Magnetic Fields", Numerical Mathematics and Advanced Applications, Proceedings of the ENUMATH 2007, 7th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, 10.-14.09.2007, K. Kunisch, G. Of, O-Steinbach; Eds., Springer, Berlin, Heidelberg, pp. 109-116, 2008.

ISBN: 978-3-540-69776-3

T. Steinmetz, G. Wimmer and M. Clemens, "Adaptive Linear-Implicit Time Integration Using Subspace Projection Techniques for Electroquasistatic and Thermodynamic Field Simulations", IEEE Transactions on Magnetics, vol. 43, no. 4, pp. 1273-1276, Apr. 2007. IEEE.
B. Gottermeier, "Rosenbrock-Verfahren für elektro-quasistatische Gleichungen", 03 2007.
A. Gerisch, J. Lang, H. Podhaisky and R. Weiner, "High-order linearly implicit twostep peer - finite element methods for time-dependent PDEs" , 03 2007.
D. Teleaga and J. Lang, "Programming Aspects of Adaptive Whitney FEM for Magnetoquasistatics" , 2007.
J. Lang and K. Debrabant, "On Global Error Estimation and Control for Parabolic Equations" , 2007.
J. Lang and J. G. Verwer, "On global error estimation and control for initial value problems", SIAM Journal on Scientific Computing, vol. 29, no. 4, pp. 1460-1475, 2007. Society for Industrial and Applied Mathematics.
M. Clemens, T. Steinmetz and G. Wimmer, "Error Control Schemes for Adaptive Time Integration of Magnetodynamic Systems with Variable Spatial Mesh Resolution", COMPEL, vol. 26, no. 3, pp. 758-772, 2007.
A. Skarlatos, T. Weiland and M. Clemens, "Start Vector Generation for Implicit Newmark Time Integration of the Wave Equation", IEEE Transactions on Magnetics, vol. 42, no. 4, pp. 631-634, Apr. 2006.
M. Clemens, S. Feigh and T. Weiland, "Construction Principles of Multigrid Smoothers for Curl-Curl Equations", IEEE Transactions on Magnetics, vol. 41, no. 5, pp. 1680-1683, Mai 2005.
M. Clemens, "Large Systems of Equations in a Discrete Electromagnetism: Formulations and Numerical Algorithms", IEE Proceedings - Science, Measurement and Technology, vol. 152, no. 2, pp. 50-72, Mä. 2005.
M. Clemens, S. Feigh and T. Weiland, "Geometric Multigrid Algorithms Using the Conformal Finite Integration Technique", IEEE Transactions on Magnetics, vol. 40, no. 2, pp. 1065-1068, Mä. 2004.
M. Clemens, M. Wilke, R. Schuhmann and T. Weiland, "Subspace Projection Extrapolation Scheme for Transient Field Simulations", IEEE Transactions on Magnetics, vol. 40, no. 2, pp. 934-937, Mä. 2004.
M. Clemens, S. Feigh, M. Wilke and T. Weiland, "Non-Nested Geometric Multigrid Method Using Consistency Error Correction for Discrete Magnetic Curl-Curl Formulations", COMPEL, vol. 23, no. 4, pp. 913-922, 2004.
M. Clemens, M. Wilke and T. Weiland, "Linear-Implicit Time Integration Schemes for Error-Controlled Transient Nonlinear Magnetic Field Simulations", IEEE Transactions on Magnetics, vol. 39, no. 3, pp. 1175-1178, Mai 2003.

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