zbMATH — the first resource for mathematics

Structure-preserving mesh coupling based on the Buffa-Christiansen complex. (English) Zbl 1359.78020
Summary: The state of the art for mesh coupling at nonconforming interfaces is presented and reviewed. Mesh coupling is frequently applied to the modeling and simulation of motion in electromagnetic actuators and machines. The paper exploits Whitney elements to present the main ideas. Both interpolation- and projection-based methods are considered. In addition to accuracy and efficiency, we emphasize the question whether the schemes preserve the structure of the de Rham complex, which underlies Maxwell’s equations. As a new contribution, a structure-preserving projection method is presented, in which Lagrange multiplier spaces are chosen from the Buffa-Christiansen complex. Its performance is compared with a straightforward interpolation based on Whitney and de Rham maps, and with Galerkin projection.
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
14F40 de Rham cohomology and algebraic geometry
Full Text: DOI
[1] Andriulli, Francesco P.; Cools, Kristof; Ba\ugci, Hakan; Olyslager, Femke; Buffa, Annalisa; Christiansen, Snorre; Michielssen, Eric, A multiplicative Calderon preconditioner for the electric field integral equation, IEEE Trans. Antennas and Propagation, 56, 8, 2398-2412, (2008) · Zbl 1369.78872
[2] Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.), 47, 2, 281-354, (2010) · Zbl 1207.65134
[3] Ben Belgacem, F.; Buffa, A.; Maday, Y., The mortar finite element method for 3D Maxwell equations: first results, SIAM J. Numer. Anal., 39, 3, 880-901 (electronic), (2001) · Zbl 1001.65123
[4] Bernardi, Christine; Maday, Yvon; Rapetti, Francesca, Basics and some applications of the mortar element method, GAMM-Mitt., 28, 2, 97-123, (2005) · Zbl 1177.65178
[5] Bossavit, Alain, Computational electromagnetism, Electromagnetism, xx+352 pp., (1998), Academic Press, Inc., San Diego, CA · Zbl 0945.78001
[6] Bouillault, F.; Buffa, A.; Maday, Y.; Rapetti, F., The mortar edge element method in three dimensions: application to magnetostatics, SIAM J. Sci. Comput., 24, 4, 1303-1327, (2003) · Zbl 1031.78013
[7] Buffa, Annalisa; Christiansen, Snorre H., A dual finite element complex on the barycentric refinement, Math. Comp., 76, 260, 1743-1769 (electronic), (2007) · Zbl 1130.65108
[8] Buffa, Annalisa; Hiptmair, Ralf, Galerkin boundary element methods for electromagnetic scattering. Topics in Computational Wave Propagation, Lect. Notes Comput. Sci. Eng. 31, 83-124, (2003), Springer, Berlin · Zbl 1055.78013
[9] Buffa, Annalisa; Maday, Yvon; Rapetti, Francesca, Applications of the mortar element method to 3D electromagnetic moving structures. Computational Electromagnetics, Kiel, 2001, Lect. Notes Comput. Sci. Eng. 28, 35-50, (2003), Springer, Berlin · Zbl 1065.78017
[10] Burtscher2013 A. Burtscher et al., \emph LehrFEM – a 2D finite element toolbox., Tech. report, Seminar for Applied Mathematics, ETH Z\"urich, 2013.
[11] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods, Scientific Computation, xxx+596 pp., (2007), Springer, Berlin · Zbl 1121.76001
[12] Clemens2004a M. Clemens, S. Feigh, and T. Weiland, \emph Geometric multigrid algorithms using the conformal Finite Integration Technique, IEEE Transactions on Magnetics <span class=”textbf”>4</span>0 (2004), no. 2, 1065–1068.
[13] Flemisch2007 B. Flemisch, \emph Nonmatching triangulations of curvilinear interfaces applied to electro-mechanics and elasto-acoustics, Ph.D. thesis, Institut f\`‘ur angewandte Analysis und Numerische Simulation, Universit\'’at Stuttgart, 2007.
[14] Gander, Martin J.; Japhet, Caroline, Algorithm 932: PANG: software for nonmatching grid projections in 2D and 3D with linear complexity, ACM Trans. Math. Software, 40, 1, Art. 6, 25 pp., (2013) · Zbl 1295.65120
[15] Hollaus2010 K. Hollaus, D. Feldengut, J. Sch\"oberl, M. Wabro, and D. Omeragic, \emph Nitsche-type mortaring for Maxwell’s equations, Progress In Electromagnetics Research Symposium Proceedings (Cambridge, USA), July 2010, pp. 397–402.
[16] Hoppe, R. H. W., Mortar edge element methods in \(\mathbf{R}^3\), East-West J. Numer. Math., 7, 3, 159-173, (1999) · Zbl 0944.65118
[17] Hu, Qiya; Shu, Shi; Zou, Jun, A mortar edge element method with nearly optimal convergence for three-dimensional Maxwell’s equations, Math. Comp., 77, 263, 1333-1353, (2008) · Zbl 1230.78028
[18] Journeaux2014 A. A. Journeaux, F. Bouillaut, and J. Y. Roger, \emph Reducing the cost of mesh-to-mesh data transfer, IEEE Transactions on Magnetics <span class=”textbf”>5</span>0 (2014), no. 2, 437–440.
[19] Journeaux2013a A. A. Journeaux, N. Nemitz, and O. Moreau, \emph Locally conservative projection methods: benchmarking and practical implementation, COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering <span class=”textbf”>3</span>3 (2014), no. 1/2, 663–687.
[20] Kurz, Stefan; Auchmann, Bernhard, Differential forms and boundary integral equations for Maxwell-type problems. Fast Boundary Element Methods in Engineering and Industrial Applications, Lect. Notes Appl. Comput. Mech. 63, 1-62, (2012), Springer, Heidelberg · Zbl 1250.78009
[21] Lange2012 E. Lange, F. Henrotte, and K. Hameyer, \emph Biorthogonal shape functions for nonconforming sliding interfaces in 3-D electrical machine FE models with motion, IEEE Transactions on Magnetics <span class=”textbf”>4</span>8 (2012), no. 2, 855–858.
[22] Niu2012 S. Niu, S. L. Ho, W. N. Fu, and J. Zhu, \emph A convenient mesh rotation method of finite element analysis using sub-matrix transformation approach, IEEE Transactions on Magnetics <span class=”textbf”>4</span>8 (2012), no. 2, 303–306.
[23] Rapetti2000 F. Rapetti, A. Buffa, F. Bouillaut, and Y. Maday, \emph Simulation of a coupled magneto-mechanical system through the sliding-mesh mortar element method, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering <span class=”textbf”>1</span>9 (2000), no. 2, 332–340. · Zbl 0966.78012
[24] Rodger1990 D. Rodger, H. C. Lai, and P. J. Leonard, \emph Coupled elements for problems involving movement, IEEE Transactions on Magnetics <span class=”textbf”>2</span>6 (1990), no. 2, 548–550.
[25] Shi2008 Xiaodong Shi, Y. Le Menach, J.-P. Ducreux, and F. Piriou, \emph Comparison between the mortar element method and the polynomial interpolation method to model movement in the Finite Element Method, IEEE Transactions on Magnetics <span class=”textbf”>4</span>4 (2008), no. 6, 1314–1317.
[26] Smirnova2013 Y. Smirnova, \emph Calder\'on preconditioning for higher order boundary element method, Master’s thesis, Swiss Federal Institute of Technology, Seminar for Applied Mathematics, July 2013.
[27] Sonneveld, Peter, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10, 1, 36-52, (1989) · Zbl 0666.65029
[28] Wohlmuth, Barbara I., A comparison of dual Lagrange multiplier spaces for mortar finite element discretizations, M2AN Math. Model. Numer. Anal., 36, 6, 995-1012 (2003), (2002) · Zbl 1024.65111
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.