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Bernstein–Bézier \(H({\text {curl}})\)-Conforming Finite Elements for Time-Harmonic Electromagnetic Scattering Problems

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Abstract

This paper deals with a high-order \(H({\text {curl}})\)-conforming Bernstein–Bézier finite element method (BBFEM) to accurately solve time-harmonic Maxwell short wave problems on unstructured triangular mesh grids. We suggest enhanced basis functions, defined on the reference triangle and tetrahedron, aiming to reduce the condition number of the resulting global matrix. Moreover, element-level static condensation of the interior degrees of freedom is performed in order to reduce memory requirements. The performance of BBFEM is assessed using several benchmark tests. A preliminary analysis is first conducted to highlight the advantage of the suggested basis functions in improving the conditioning. Numerical results dealing with the electromagnetic scattering from a perfect electric conductor demonstrate the effectiveness of BBFEM in mitigating the pollution effect and its efficiency in capturing high-order evanescent wave modes. Electromagnetic wave scattering by a circular dielectric, with high wave speed contrast, is also investigated. The interior curved interface between layers is accurately described based on a linear blending map to avoid numerical errors due to geometry description. The achieved results support our expectations for highly accurate and efficient BBFEM for time harmonic wave problems.

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References

  1. Nédélec, J.C.: Mixed finite elements in \({\mathbb{R} }^{3}\). Numer. Math. 35(3), 315–341 (1980)

    MathSciNet  MATH  Google Scholar 

  2. Nédélec, J.C.: A new family of mixed finite elements in \({\mathbb{R} }^{3}\). Numer. Math. 50(1), 57–81 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Barton, M.L., Cendes, Z.J.: New vector finite elements for three-dimensional magnetic field computation. J. Appl. Phys. 61(8), 3919–3921 (1987)

    Google Scholar 

  4. Bespalov, A.N.: Finite element method for the eigenmode problem of a RF cavity resonator (1988)

  5. Lee, J.F., Sun, D.K., Cendes, Z.J.: Tangential vector finite elements for electromagnetic field computation. IEEE Trans. Magn. 27(5), 4032–4035 (1991)

    Google Scholar 

  6. Ahagon, A., Fujiwara, K., Nakata, T.: Comparison of various kinds of edge elements for electromagnetic field analysis. IEEE Trans. Magn. 32(3), 898–901 (1996)

    Google Scholar 

  7. Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)

    MATH  Google Scholar 

  8. Bossavit, A., Verite, J.C.: A mixed FEM-BIEM method to solve 3-D eddy-current problems. IEEE Trans. Magn. 18(2), 431–435 (1982)

    Google Scholar 

  9. Bossavit, A., Verite, J.C.: The TRIFOU code: Solving the 3-D eddy-currents problem by using H as state variable. IEEE Trans. Magn. 19(6), 2465–2470 (1983)

    Google Scholar 

  10. Bossavit, A.: Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEE Proc. A Phys. Sci. Meas. Instrum. Manag. Educ. Rev. 135(8), 493–500 (1988)

  11. Bossavit, A.: A rationale for edge-elements in 3-D fields computations. IEEE Trans. Magn. 24(1), 74–79 (1988)

    Google Scholar 

  12. Bossavit, A., Mayergoyz, I.: Edge elements for scattering problems. IEEE Trans. Magn. 25(4), 2816–2821 (1989)

    Google Scholar 

  13. Mur, G., De Hoop, A.: A finite element method for computing three-dimensional electromagnetic fields in inhomogeneous media. IEEE Trans. Magn. 21(6), 2188–2191 (1985)

    Google Scholar 

  14. Cendes, Z.J.: Vector finite elements for electromagnetic field computation. IEEE Trans. Magn. 27(5), 3958–3966 (1991)

    Google Scholar 

  15. Webb, J.P., Forgahani, B.: Hierarchal scalar and vector tetrahedra. IEEE Trans. Magn. 29(2), 1495–1498 (1993)

    Google Scholar 

  16. Geuzaine, C., Meys, B., Dular, P., Legros, W.: Convergence of high order curl-conforming finite elements [for EM field calculations]. IEEE Trans. Magn. 35(3), 1442–1445 (1999)

    Google Scholar 

  17. Graglia, R.D., Wilton, D.R., Peterson, A.F.: Higher-order interpolatory vector bases for computational electromagnetics. IEEE Trans. Antennas Propag. 45(3), 329–342 (1997)

    Google Scholar 

  18. Webb, J.P.: Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE Trans. Antennas Propag. 47(8), 1244–1253 (1999)

    MathSciNet  MATH  Google Scholar 

  19. Díaz-Morcillo, A., Jin, J. M.: A comparison of hierarchical and interpolatory basis functions in the finite element analysis of waveguiding structures. In: IEEE Antennas and Propagation Society International Symposium (IEEE Cat. No. 02CH37313), vol. 4, pp. 710–713. IEEE (2002)

  20. Ainsworth, M., Coyle, J.: Hierarchic \(hp\)-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Eng. 190(49–50), 6709–6733 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Ainsworth, M., Coyle, J.: Hierarchic finite element bases on unstructured tetrahedral meshes. Int. J. Numer. Methods Eng. 58(14), 2103–2130 (2003)

    MathSciNet  MATH  Google Scholar 

  22. Ingelstrom, P.: A new set of H (curl)-conforming hierarchical basis functions for tetrahedral meshes. IEEE Trans. Microw. Theory Tech. 54(1), 106–114 (2006)

    Google Scholar 

  23. Jorgensen, E., Volakis, J.L., Meincke, P., Breinbjerg, O.: Higher-order hierarchical Legendre basis functions for electromagnetic modeling. IEEE Trans. Antennas Propag. 52(11), 2985–2995 (2004)

    MathSciNet  MATH  Google Scholar 

  24. Rachowicz, W., Demkowicz, L.F.: An \(hp\)-adaptive finite element method for electromagnetics-part II: a 3D implementation. Int. J. Numer. Methods Eng. 53(1), 147–180 (2002)

    MATH  Google Scholar 

  25. Schöberl, J., Zaglmayr, S.: High-order Nédélec elements with local complete sequence properties. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 24(2), 374–384 (2005)

    MATH  Google Scholar 

  26. Sun, D.K., Lee, J.F., Cendes, Z.J.: Construction of nearly orthogonal Nédélec bases for rapid convergence with multilevel preconditioned solvers. SIAM J. Sci. Comput. 23(4), 1053–1076 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Abdul-Rahman, R., Kasper, M.: Orthogonal hierarchical Nédélec elements. IEEE Trans. Magn. 44(6), 1210–1213 (2008)

    Google Scholar 

  28. Graglia, R.D., Peterson, A.F., Andriulli, F.P.: Curl-conforming hierarchical vector bases for triangles and tetrahedra. IEEE Trans. Antennas Propag. 59(3), 950–959 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Perugia, I., Schötzau, D., Monk, P.: Stabilized interior penalty methods for the time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Eng. 191(41–42), 4675–4697 (2002)

    MathSciNet  MATH  Google Scholar 

  30. Hesthaven, J.S., Warburton, T.: High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362(1816), 493–524 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator: the indefinite case. ESAIM Math. Model. Numer. Anal. 39(4), 727–753 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100(3), 485–518 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Buffa, A., Perugia, I.: Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J. Numer. Anal. 44(5), 2198–2226 (2006)

    MathSciNet  MATH  Google Scholar 

  34. Warburton, T., Embree, M.: The role of the penalty in the local discontinuous Galerkin method for Maxwell’s eigenvalue problem. Comput. Methods Appl. Mech. Eng. 195(25–28), 3205–3223 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Lohrengel, S., Nicaise, S.: A discontinuous Galerkin method on refined meshes for the two-dimensional time-harmonic Maxwell equations in composite materials. J. Comput. Appl. Math. 206(1), 27–54 (2007)

    MathSciNet  MATH  Google Scholar 

  36. Dolean, V., Fol, H., Lanteri, S., Perrussel, R.: Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods. J. Comput. Appl. Math. 218(2), 435–445 (2008)

    MathSciNet  MATH  Google Scholar 

  37. El Bouajaji, M., Lanteri, S.: High order discontinuous Galerkin method for the solution of 2D time-harmonic Maxwell’s equations. Appl. Math. Comput. 219(13), 7241–7251 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Hiptmair, R., Moiola, A., Perugia, I.: Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82(281), 247–268 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations. J. Comput. Phys. 230(19), 7151–7175 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Li, L., Lanteri, S., Perrussel, R.: Numerical investigation of a high order hybridizable discontinuous Galerkin method for 2D time-harmonic Maxwell’s equations. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. (2013)

  41. Li, L., Lanteri, S., Perrussel, R.: A class of locally well-posed hybridizable discontinuous Galerkin methods for the solution of time-harmonic Maxwell’s equations. Comput. Phys. Commun. 192, 23–31 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Lu, P., Chen, H., Qiu, W.: An absolutely stable \(hp\)-HDG method for the time-harmonic Maxwell equations with high wave number. Math. Comput. 86(306), 1553–1577 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Zhu, L., Huang, T.Z., Li, L.: A hybrid-mesh hybridizable discontinuous Galerkin method for solving the time-harmonic Maxwell’s equations. Appl. Math. Lett. 68, 109–116 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Agullo, E., Giraud, L., Gobé, A., Kuhn, M., Lanteri, S., Moya, L.: High order HDG method and domain decomposition solvers for frequency-domain electromagnetics. Int. J. Numer. Model. Electron. Networks Devices Fields 33(2), e2678 (2020)

    Google Scholar 

  45. Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations, vol. 110. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  46. Ainsworth, M., Andriamaro, M.G., Davydov, O.: Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures. SIAM J. Sci. Comput. 33(6), 3087–3109 (2011)

    MathSciNet  MATH  Google Scholar 

  47. Lohmann, C., Kuzmin, D., Shadid, J.N., Mabuza, S.: Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements. J. Comput. Phys. 344, 151–186 (2017)

    MathSciNet  MATH  Google Scholar 

  48. El Kacimi, A., Laghrouche, O., Mohamed, M.S., Trevelyan, J.: Bernstein-Bézier based finite elements for efficient solution of short wave problems. Comput. Methods Appl. Mech. Eng. 343, 166–185 (2019)

    MATH  Google Scholar 

  49. El Kacimi, A., Laghrouche, O., Ouazar, D., Mohamed, M.S., Seaid, M., Trevelyan, J.: Enhanced conformal perfectly matched layers for Bernstein–Bézier finite element modelling of short wave scattering. Comput. Methods Appl. Mech. Eng. 355, 614–638 (2019)

    MATH  Google Scholar 

  50. Peng, X., Xu, G., Zhou, A., Yang, Y., Ma, Z.: An adaptive Bernstein–Bézier finite element method for heat transfer analysis in welding. Adv. Eng. Softw. 148, 102–855 (2020)

    Google Scholar 

  51. Benatia, N., El Kacimi, A., Laghrouche, O., El Alaoui Talibi, M., Trevelyan, J.: Frequency domain Bernstein–Bézier finite element solver for modelling short waves in elastodynamics. Appl. Math. Model. 102, 115–136 (2022)

    MathSciNet  MATH  Google Scholar 

  52. Engvall, L., Evans, J.A.: Isogeometric unstructured tetrahedral and mixed-element Bernstein–Bézier discretizations. Comput. Methods Appl. Mech. Eng. 319, 83–123 (2017)

    MATH  Google Scholar 

  53. Farouki, R., Goodman, T.: On the optimal stability of the Bernstein basis. Math. Comput. 65(216), 1553–1566 (1996)

    MathSciNet  MATH  Google Scholar 

  54. Farouki, R.T.: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geometric Des. 29(6), 379–419 (2012)

    MathSciNet  MATH  Google Scholar 

  55. Kirby, R.C., Thinh, K.T.: Fast simplicial quadrature-based finite element operators using Bernstein polynomials. Numer. Math. 121(2), 261–279 (2012)

    MathSciNet  MATH  Google Scholar 

  56. Kirby, R.C.: Low-complexity finite element algorithms for the de Rham complex on simplices. SIAM J. Sci. Comput. 36(2), A846–A868 (2014)

    MathSciNet  Google Scholar 

  57. Ainsworth, M., Andriamaro, M.G., Davydov, O.: A Bernstein-Bézier basis for arbitrary order Raviart–Thomas finite elements. Constr. Approx. 41(1), 1–22 (2015)

    MathSciNet  MATH  Google Scholar 

  58. Ainsworth, M., Fu, G.: Bernstein-Bézier bases for tetrahedral finite elements. Comput. Methods Appl. Mech. Eng. 340, 178–201 (2018)

    MATH  Google Scholar 

  59. Arnold, D.N., Falk, R.S., Winther, R.: Geometric decompositions and local bases for spaces of finite element differential forms. Comput. Methods Appl. Mech. Eng. 198(21–26), 1660–1672 (2009)

    MathSciNet  MATH  Google Scholar 

  60. Gordon, W.J., Hall, C.A.: Construction of curvilinear coordinate systems and applications to mesh generation. Int. J. Numer. Methods Eng. 7(4), 461–477 (1973)

    MATH  Google Scholar 

  61. Gordon, W.J., Hall, C.A.: Transfinite element methods: blending-function interpolation over arbitrary curved element domains. Numer. Math. 21(2), 109–129 (1973)

    MathSciNet  MATH  Google Scholar 

  62. Hiptmair, R., Moiola, A., Perugia, I.: Stability results for the time-harmonic Maxwell equations with impedance boundary conditions. Math. Models Methods Appl. Sci. 21(11), 2263–2287 (2011)

    MathSciNet  MATH  Google Scholar 

  63. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93. Springer Nature, Berlin (2019)

    MATH  Google Scholar 

  64. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  65. Demkowicz, L.F.: Finite element methods for Maxwell’s equations. Encyclopedia of Computational Mechanics, 2nd edn., pp. 1–20 (2017)

  66. Nicaise, S., Tomezyk, J.: Convergence analysis of a \(hp\)-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary. Numer. Methods Partial Differ. Equ. 36(6), 1868–1903 (2020)

    MathSciNet  Google Scholar 

  67. Kirsch, A., Monk, P.: A finite element/spectral method for approximating the time-harmonic Maxwell system in \({\mathbb{R} }^{3}\). SIAM J. Appl. Math. 55(5), 1324–1344 (1995)

    MathSciNet  MATH  Google Scholar 

  68. Monk, P.: A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63(1), 243–261 (1992)

    MathSciNet  MATH  Google Scholar 

  69. Rognes, M.E., Kirby, R.C., Logg, A.: Efficient assembly of H(div) and H(curl) conforming finite elements. SIAM J. Sci. Comput. 31(6), 4130–4151 (2010)

    MathSciNet  MATH  Google Scholar 

  70. P. Ŝolín, Segeth, K., Dolezel, I.: Higher-Order Finite Element Methods. CRC Press, Cambridge (2003)

  71. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications, vol. 44. Springer, Berlin (2013)

    MATH  Google Scholar 

  72. Andriamaro, M.G.: Bernstein–Bézier techniques and optimal algorithms in finite element analysis. Ph.D. thesis, University of Strathclyde (2013)

  73. Gopalakrishnan, J., García-Castillo, L.E., Demkowicz, L.F.: Nédélec spaces in affine coordinates. Comput. Math. Appl. 49(7), 1285–1294 (2005)

    MathSciNet  MATH  Google Scholar 

  74. Szabó, B., Babuška, I.: Finite Element Analysis. Wiley, New York (1991)

    MATH  Google Scholar 

  75. Amestoy, P.R., Duff, I.S., L’excellent, J.Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184(2–4), 501–520 (2000)

    MATH  Google Scholar 

  76. Jin, J.M.: Theory and Computation of Electromagnetic Fields. Wiley, New York (2015)

    Google Scholar 

  77. Dolean, V., Gander, M.J., Gerardo-Giorda, L.: Optimized Schwarz methods for Maxwell’s equations. SIAM J. Sci. Comput. 31(3), 2193–2213 (2009)

    MathSciNet  MATH  Google Scholar 

  78. Rawat, V., Lee, J.F.: Nonoverlapping domain decomposition with second order transmission condition for the time-harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32(6), 3584–3603 (2010)

    MathSciNet  MATH  Google Scholar 

  79. El Bouajaji, M., Thierry, B., Antoine, X., Geuzaine, C.: A quasi-optimal domain decomposition algorithm for the time-harmonic Maxwell’s equations. J. Comput. Phys. 294, 38–57 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the Moroccan Ministry of Higher Education, Scientific Research and Innovation and the OCP Foundation who funded this work through the APRD research program.

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Authors and Affiliations

  1. LMC, FPS, Cadi Ayyad University, Marrakech, Morocco

    Nawfel Benatia & Abdellah El Kacimi

  2. Institute for Infrastructure and Environment, School of Energy, Geoscience, Infrastructure and Society, Heriot-Watt University, Edinburgh, UK

    Omar Laghrouche

  3. MSDA, Mohammed VI Polytechnic University, Benguerir, Morocco

    Ahmed Ratnani

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  1. Nawfel Benatia
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  2. Abdellah El Kacimi
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  3. Omar Laghrouche
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  4. Ahmed Ratnani
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Benatia, N., El Kacimi, A., Laghrouche, O. et al. Bernstein–Bézier \(H({\text {curl}})\)-Conforming Finite Elements for Time-Harmonic Electromagnetic Scattering Problems. J Sci Comput 97, 69 (2023). https://doi.org/10.1007/s10915-023-02381-5

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