In the recent years, reformulating the mathematical description of physical laws in an algebraic form using tools from algebraic topology gained popularity in computational physics. Physical variables are defined as fluxes or circulations on oriented geometric elements of a pair of dual interlocked cell complexes, while physical laws are expressed in a metric-free fashion with incidence matrices. The metric and the material information are encoded in the discrete counterpart of the constitutive laws of materials, also referred to as constitutive or material matrices. The stability and consistency of the method is guaranteed by precise properties (symmetry, positive definiteness, consistency) that material matrices have to fulfill. The main advantage of this approach is that material matrices, even for arbitrary star-shaped polyhedral elements, can be geometrically defined, by simple closed-form expressions, in terms of the geometric elements of the primal and dual grids. That is why this original technique may be considered as a “Discrete Geometric Approach” (DGA) to computational physics. This paper first details the set of vector basis functions associated with the edges and faces of a polyhedral primal grid or of a dual grid. Then, it extends the construction of constitutive matrices for bianisotropic media.
Mots-clés : Discrete Geometric Approach (DGA), discrete constitutive equations, discrete hodge star operator, non-orthogonal polyhedral dual grids, bianisotropic media
@article{M2AN_2016__50_3_677_0,
author = {Codecasa, Lorenzo and Specogna, Ruben and Trevisan, Francesco},
title = {Geometrically defined basis functions for polyhedral elements with applications to computational electromagnetics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {677--698},
publisher = {EDP-Sciences},
volume = {50},
number = {3},
year = {2016},
doi = {10.1051/m2an/2015077},
zbl = {1347.78010},
mrnumber = {3507269},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015077/}
}
TY - JOUR AU - Codecasa, Lorenzo AU - Specogna, Ruben AU - Trevisan, Francesco TI - Geometrically defined basis functions for polyhedral elements with applications to computational electromagnetics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 677 EP - 698 VL - 50 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015077/ DO - 10.1051/m2an/2015077 LA - en ID - M2AN_2016__50_3_677_0 ER -
%0 Journal Article %A Codecasa, Lorenzo %A Specogna, Ruben %A Trevisan, Francesco %T Geometrically defined basis functions for polyhedral elements with applications to computational electromagnetics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 677-698 %V 50 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015077/ %R 10.1051/m2an/2015077 %G en %F M2AN_2016__50_3_677_0
Codecasa, Lorenzo; Specogna, Ruben; Trevisan, Francesco. Geometrically defined basis functions for polyhedral elements with applications to computational electromagnetics. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 677-698. doi : 10.1051/m2an/2015077. http://archive.numdam.org/articles/10.1051/m2an/2015077/
and , A FIT Formulation of Bianisotropic Materials Over Polyhedral Grids. IEEE Trans. Magn. 50 (2014) 7008504. | DOI
, A residual based error estimator for the Mimetic Finite Difference method. Numer. Math. 108 (2008) 387–406. | DOI | MR | Zbl
, , and , Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comput. Phys. 228 (2009) 7215–7232. | DOI | MR | Zbl
and , Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes. ESAIM: M2AN 48 (2014) 553–581. | DOI | Numdam | MR | Zbl
, and , Low-order reconstruction operators on polyhedral meshes: Application to Compatible Discrete Operator schemes. Comput. Aid. Geom. Des. 35–36 (2015) 27–41. | DOI | MR | Zbl
, On the geometry of electromagnetism. 4: Maxwell’s house. J. Japan Soc. Appl. Electromagn. Mech. 6 (1998) 318–326.
, Computational electromagnetism and geometry. 5: The Galerkin hodge. J. Japan Soc. Appl. Electromagn. Mech. 2 (2000) 203–209.
A. Bossavit, Generalized Finite Differences in Computational Electromagnetics. Progress in Electromagnetics Research. Vol. 32 of PIER 32, edited by F.L. Teixeira. EMW, Cambridge, Ma (2001) 45–64.
and , Yee-like schemes on staggered cellular Grids: A synthesis between FIT and FEM approaches. IEEE Trans. Mag. 36 (2000) 861–867. | DOI
and , Innovative mimetic discretizations for electromagnetic problems. J. Comput. Appl. Math. 234 (2010) 1980–1987. | DOI | MR | Zbl
, A construction of spaces of compatible differential forms on cellular complexes. Math. Models Methods Appl. Sci. 18 (2008) 739–757. | DOI | MR | Zbl
and , A Tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172 (2001) 739–765. | DOI | MR | Zbl
M. Clemens and T. Weiland, Discrete Electromagnetism with the Finite Integration Technique. Vol. 32 of PIER, edited by F.L. Teixeira. EMW Publishing, Cambridge, Massachusetts, USA (2001) 65–87.
and , Piecewise uniform bases and energetic approach for discrete constitutive matrices in electromagnetic problems. Int. J. Numer. Meth. Eng. 65 (2006) 548–565. | DOI | MR | Zbl
and , Constitutive equations for discrete electromagnetic problems over polyhedral grids. J. Comput. Phys. 225 (2007) 1894–1918. | DOI | MR | Zbl
, and , Base functions and discrete constitutive relations for staggered polyhedral grids. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1117–1123. | DOI | MR | Zbl
, and , Symmetric Positive-Definite Constitutive Matrices for Discrete Eddy-Current Problems. IEEE Trans. Mag. 43 (2007) 510–515. | DOI
, and , Discrete constitutive equations over hexahedral grids for eddy-current problems. CMES 1 (2008) 1–14. | MR | Zbl
, and , Subgridding to solving magnetostatics within Discrete Geometric Approach. IEEE Trans. Magn. 45 (2009) 1024–1027. | DOI
, and , A new set of basis functions for the Discrete Geometric Approach. J. Comput. Phys. 229 (2010) 7401–7410. | DOI | MR | Zbl
and , Efficient generalized source field computation for h-oriented magnetostatic formulations. Eur. Phys. J. Appl. Phys. 53 (2011) 20801. | DOI
T. Euler, Consistent Discretization of Maxwell’s Equations on Polyhedral Grids. Ph. D. thesis, TU Darmstadt, Darmstadt, Germany (2007).
, and , Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043. | DOI | MR | Zbl
, and , Reinterpretation of the nodal force method within Discrete Geometric Approaches. IEEE Trans. Magn. 44 (2008) 690–693. | DOI
Y. Kuznetsov and S. Repin, Mixed Finite Element Method on Polygonal and Polyhedral Meshes. Numer. math. Adv. Appl. Springer, Berlin (2004) 615–622. | MR | Zbl
I. Lindell, Methods for Electromaghnetic Field Analysis. IEEE Press, Piscataway, NJ, USA (1992).
, Properties of Constitutive Matrices for Electrostatic and Magnetostatic Problems. IEEE Trans. Mag. 40 (2004) 1516–1520. | DOI
, Mixed finite elements in . Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl
, A new family of mixed finite elements in R3. Numer. Math. 50 (1986) 57–81. | DOI | MR | Zbl
, and , Some realizations of a discrete Hodge operator: A reinterpretation of finite element techniques [for EM field analysis]. IEEE Trans. Magn. 35 (1999) 1494–1497. | DOI
, Finite Formulation of the Electromagnetic Field. IEEE Trans. Mag. 38 (2002) 333–336. | DOI
, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14 (1966) 302–307. | DOI | Zbl
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