Abstract
In this paper we propose different multi-field variational formulations for electrostatics and magnetostatics, which can provide optimal discrete approximation of any particular vector field. The proposed formulations are constructed by appealing to mechanics point of view amenable to using general constitutive equations, which is quite different from electrostatics and magnetostatics formulations typical of physics and electrical engineering focusing on the corresponding global form suitable only for linear case. In particular, the formulations we propose can be combined with mixed discrete approximations that can ensure the continuity of tangential component of electric or magnetic field and normal component of electric displacement and magnetic flux even for low order interpolations. The choice of this kind is quite different from currently favorite choice of high order finite element interpolations used for coupling electromagnetism with mechanics. The discrete approximation is based upon Whitney’s interpolations representing the vector fields in terms of corresponding differential forms, with electric and magnetic fields as one-form and electric displacement and magnetic flux as two-form. The implementation of interpolations of this kind is made for 3D tetrahedron elements with non-standard approximation parameters defined not only at vertices (for zero-form), but at edges (for one-form) and at facets (for two-form). The results of several numerical simulations are presented to illustrate the performance of different formulations proposed herein.
Similar content being viewed by others
References
Albanese R, Rubinacci G (1997) Finite element methods for the solution of 3d eddy current problems. In: Advances in imaging and electron physics, vol 102, pp 1–86. Elsevier
Alotto P, Freschi F, Repetto M, Rosso C (2013) The cell method for electrical engineering and multiphysics problems: an introduction, vol 230. Springer, Berlin
Angoshtari A, Shojaei MF, Yavari A (2017) Compatible-strain mixed finite element methods for 2d compressible nonlinear elasticity. Comput Methods Appl Mech Eng 313:596–631
Arnold DN, Falk RS, Winther R (2006) Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15:1–155
Balanis CA (1999) Advanced engineering electromagnetics. Wiley, Hoboken
Bamberg P, Sternberg S (1991) A course in mathematics for students of physics, vol 2. Cambridge University Press, Cambridge
Bochev PB, Robinson AC (2002) Matching algorithms with physics: exact sequences of finite element spaces. Collected Lectures on Preservation of Stability Under Discretization, pp 145–166
Bossavit A (1988) A rationale for ‘edge-elements’ in 3-d fields computations. IEEE Trans Magn 24(1):74–79
Bossavit A (1988) Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEE Proc 135(8):493–500
Bossavit A (2010) Discrete magneto-elasticity: a geometrical approach. IEEE Trans Magn 46(8):3485–3491
Desbrun M, Kanso E, Tong Y (2008) Discrete differential forms for computational modeling. In: Discrete differential geometry, pp 287–324. Springer
Dular P, Geuzaine C (1997) Getdp: a general environment for the treatment of discrete problems
Dular P, Hody J-Y, Nicolet A, Genon A, Legros W (1994) Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms. IEEE Trans Magn 30(5):2980–2983
Felippa CA, Park KC, Farhat C (2001) Partitioned analysis of coupled mechanical systems. Comput Methods Appl Mech Eng 190(24–25):3247–3270
Gil AJ, Ortigosa R (2016) A new framework for large strain electromechanics based on convex multi-variable strain energies: variational formulation and material characterisation. Comput Methods Appl Mech Eng 302:293–328
Golias NA, Tsiboukis TD, Bossavit A (1994) Constitutive inconsistency: rigorous solution of maxwell equations based on a dual approach. IEEE Trans Magn 30(5):3586–3589
Gradinaru V, Hiptmair R (1999) Whitney elements on pyramids. Electron Trans Numer Anal 8:154–168
Hale HW (1961) A logic for identifying the trees of a graph. Trans Am Inst Electr Eng Part III Power Apparatus Syst 80(3):195–197
Hammond P (2013) Electromagnetism for engineers: an introductory course. Elsevier, Amsterdam
Hammond P, Penman J (1976) Calculation of inductance and capacitance by means of dual energy principles. In: Proceedings of the Institution of Electrical Engineers, vol 123, pp 554–559. IET
Hammond P, Tsiboukis TD (1983) Dual finite-element calculations for static electric and magnetic fields. IEE Proc A (Phys Sci Measurement Instrum Manag Educ Rev) 130(3):105–111
Ibrahimbegovic A (2009) Nonlinear solid mechanics: theoretical formulations and finite element solution methods, vol 160. Springer, Dordrecht
Jackson JD (1999) Classical electrodynamics. Wiley, New York
Kameari A (1989) Three dimensional eddy current calculation using edge elements for magnetic vector potential. In: Applied electromagnetics in materials, pp 225–236. Elsevier
Kassiotis C, Ibrahimbegovic A, Niekamp R, Matthies HG (2011) Nonlinear fluid–structure interaction problem. Part I: implicit partitioned algorithm, nonlinear stability proof and validation examples. Comput Mech 47(3):305–323
Kassiotis C, Ibrahimbegovic A, Niekamp R, Matthies HG (2011) Nonlinear fluid–structure interaction problem. Part II: space discretization, implementation aspects, nested parallelization and application examples. Comput Mech 47(3):335–357
Keip Marc-André, Steinmann Paul, Schröder Jörg (2014) Two-scale computational homogenization of electro-elasticity at finite strains. Comput Methods Appl Mech Eng 278:62–79
Ladevèze P, Pelle J-P (2005) Mastering calculations in linear and nonlinear mechanics, vol 171. Springer, New York
Macneal RH, Harder Robert L (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1(1):3–20
Manges J, Cendes Z (1996) Generation of tangential vector finite elements. Int Compumag Soc Newsl 3(1):4–10
Mitchell BS (2004) An introduction to materials engineering and science for chemical and materials engineers. Wiley, Hoboken
Moreno-Navarro P, Ibrahimbegovic A, Pérez-Aparicio JL (2017) Plasticity coupled with thermo-electric fields: thermodynamics framework and finite element method computations. Comput Methods Appl Mech Eng 315:50–72
Moreno-Navarro P, Ibrahimbegovic A, Pérez-Aparicio JL (2018) Linear elastic mechanical system interacting with coupled thermo-electro-magnetic fields. Coupled Syst Mech 7(1):5–25
Nédélec J-C (1980) Mixed finite elements in \(\mathbb{R}^3\). Numer Math 35(3):315–341
Penman J, Fraser J (1982) Complementary and dual energy finite element principles in magnetostatics. IEEE Trans Magn 18(2):319–324
Razek A (1995) Computation of 3d electrostatic local fields and forces using complementarity of dual formulations. Application for capacitance and torque calculations in micromotors
Ren Z (1995) A 3d vector potential formulation using edge element for electrostatic field computation. IEEE Trans Magn 31(3):1520–1523
Ren Z (2009) On the complementarity of dual formulations on dual meshes. IEEE Trans Magn 45(3):1284–1287
Repetto M, Trevisan F (2004) Global formulation of 3d magnetostatics using flux and gauged potentials. Int J Numer Meth Eng 60(4):755–772
Schröder J, Keip M-A (2012) Two-scale homogenization of electromechanically coupled boundary value problems. Comput Mech 50(2):229–244
Stark S, Semenov AS, Balke H (2015) On the boundary conditions for the vector potential formulation in electrostatics. Int J Numer Meth Eng 102(11):1704–1732
Taylor RL (2012) Feap–a finite element analysis program. Version 8.4 Theory Manual
Tonti E (2013) The mathematical structure of classical and relativistic physics. Springer, New York
Vogel F, Bustamante R, Steinmann P (2012) On some mixed variational principles in electro-elastostatics. Int J Nonlinear Mech 47(2):341–354
Wang J-S, Ida N (1993) Curvilinear and higher order ‘edge’ finite elements in electromagnetic field computation. IEEE Trans Magn 29(2):1491–1494
Webb JP, Forgahani B (1993) Hierarchal scalar and vector tetrahedra. IEEE Trans Magn 29(2):1495–1498
Yioultsis TV, Tsiboukis TD (1996) The mystery and magic of Whitney elements—an insight in their properties and construction. ICS Newsl 3:1389–1392
Zienkiewicz OC, Taylor RL (1977) The finite element method, vol 36. McGraw-Hill, London
Acknowledgements
This work was supported jointly by Haut-de-France Region (CR Picardie) (120-2015-RDISTRUCT-000010 and RDISTRUCT-000010) and EU funding (FEDER) for Chaire-de-Mécanique (120-2015-RDISTRUCTF-000010 and RDISTRUCTI-000004). AI was also supported by IUF.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Moreno-Navarro, P., Ibrahimbegovic, A. & Ospina, A. Multi-field variational formulations and mixed finite element approximations for electrostatics and magnetostatics. Comput Mech 65, 41–59 (2020). https://doi.org/10.1007/s00466-019-01751-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-019-01751-x