Abstract

We present an improved adaptive mesh process that allows the accurate control of the numerical solution of interest derived from the solution of the partial differential equation. In the cases of two-dimensional studies, such an adaptive meshing is applied to compute phenomenon involving high field gradients in near-field (electric intensity, Poynting’s vector, optical forces,…). We show, that this improved scheme permits to decrease drastically the computationnal time and the memory requirements.

© 2007 Optical Society of America

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  1. J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
    [CrossRef]
  2. D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B,  84, 55-60 (2006).
    [CrossRef]
  3. D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
    [CrossRef]
  4. B. Guizal, D. Barchiesi and D. Felbacq, "Electromagnetic beam diffraction by a finite lamellar structure," J. Opt. Soc. Am. A 20, 2274-2280 (2003).
    [CrossRef]
  5. T. Grosges, A. Vial and D. Barchiesi, "Models of near-field spectroscopic studies: comparison between Finite-Element and Finite-Difference methods," Opt. Express 13, 8483-8497 (2005).
    [CrossRef] [PubMed]
  6. M. Born, and E. Wolf, Principle of Optics (Pergamon Press, Oxford, 1993).
  7. J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York, 1993).
  8. P. Ingelström and A. Bondeson, "Goal-oriented error estimation and h-adaptivity for Maxwell’s equations," Comput. Methods Appl. Mech. Eng. 192, 2597-2616 (2003).
    [CrossRef]
  9. P. Houston, I. Perugia, and D. Schotzau, "Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator," Comput. Methods Appl. Mech. Eng. 194, 499-510 (2005).
    [CrossRef]
  10. D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005).
    [CrossRef]
  11. D. Xue and L. Demkowicz, "Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method," Finite Elem. Anal. Design 42, 570-579 (2006).
    [CrossRef]
  12. H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005).
    [CrossRef]
  13. M. Berzins, "Mesh quality: a function of geometry, error estimates or both?" Eng. Comput. 15, 236-247 (1999).
    [CrossRef]
  14. M. Ainsworth and J. T. Oden, "A posteriori error estimation in finite element analysis," Comput. Methods Appl. Mech. Eng. 142, 1-88 (1997).
    [CrossRef]
  15. R. Radovitzky and M. Ortiz, "Error estimation and adaptive meshing in strongly non-linear dynamic problems," Comput. Methods Appl. Mech. Eng. 172, 203-240 (1999).
    [CrossRef]
  16. P. Laug and H. Borouchaki, "BL2D-V2: mailleur bidimensionnel adaptatif," Report INRIA RT-0275http://www-rocq1.inria.fr/gamma/cdrom/www/bl2d-v2/INDEX.html (2003).
  17. G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908).
    [CrossRef]
  18. H. Du, "Mie-scattering calculation," Appl. Opt. 43, 1951-1956 (2004).
    [CrossRef] [PubMed]
  19. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
    [CrossRef] [PubMed]
  20. T. A. Davis and I. S. Duff, "A combined unifrontal multifrontal method for unsymmetric sparse matrices," ACM Trans.Math Softw. 25, 1-20 (1999).
    [CrossRef]

2006 (2)

D. Xue and L. Demkowicz, "Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method," Finite Elem. Anal. Design 42, 570-579 (2006).
[CrossRef]

D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B,  84, 55-60 (2006).
[CrossRef]

2005 (5)

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005).
[CrossRef]

P. Houston, I. Perugia, and D. Schotzau, "Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator," Comput. Methods Appl. Mech. Eng. 194, 499-510 (2005).
[CrossRef]

D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005).
[CrossRef]

T. Grosges, A. Vial and D. Barchiesi, "Models of near-field spectroscopic studies: comparison between Finite-Element and Finite-Difference methods," Opt. Express 13, 8483-8497 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

B. Guizal, D. Barchiesi and D. Felbacq, "Electromagnetic beam diffraction by a finite lamellar structure," J. Opt. Soc. Am. A 20, 2274-2280 (2003).
[CrossRef]

P. Ingelström and A. Bondeson, "Goal-oriented error estimation and h-adaptivity for Maxwell’s equations," Comput. Methods Appl. Mech. Eng. 192, 2597-2616 (2003).
[CrossRef]

2000 (1)

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

1999 (3)

T. A. Davis and I. S. Duff, "A combined unifrontal multifrontal method for unsymmetric sparse matrices," ACM Trans.Math Softw. 25, 1-20 (1999).
[CrossRef]

R. Radovitzky and M. Ortiz, "Error estimation and adaptive meshing in strongly non-linear dynamic problems," Comput. Methods Appl. Mech. Eng. 172, 203-240 (1999).
[CrossRef]

M. Berzins, "Mesh quality: a function of geometry, error estimates or both?" Eng. Comput. 15, 236-247 (1999).
[CrossRef]

1997 (1)

M. Ainsworth and J. T. Oden, "A posteriori error estimation in finite element analysis," Comput. Methods Appl. Mech. Eng. 142, 1-88 (1997).
[CrossRef]

1996 (1)

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

1908 (1)

G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Ainsworth, M.

M. Ainsworth and J. T. Oden, "A posteriori error estimation in finite element analysis," Comput. Methods Appl. Mech. Eng. 142, 1-88 (1997).
[CrossRef]

Barchiesi, D.

D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B,  84, 55-60 (2006).
[CrossRef]

T. Grosges, A. Vial and D. Barchiesi, "Models of near-field spectroscopic studies: comparison between Finite-Element and Finite-Difference methods," Opt. Express 13, 8483-8497 (2005).
[CrossRef] [PubMed]

B. Guizal, D. Barchiesi and D. Felbacq, "Electromagnetic beam diffraction by a finite lamellar structure," J. Opt. Soc. Am. A 20, 2274-2280 (2003).
[CrossRef]

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

Berzins, M.

M. Berzins, "Mesh quality: a function of geometry, error estimates or both?" Eng. Comput. 15, 236-247 (1999).
[CrossRef]

Bondeson, A.

P. Ingelström and A. Bondeson, "Goal-oriented error estimation and h-adaptivity for Maxwell’s equations," Comput. Methods Appl. Mech. Eng. 192, 2597-2616 (2003).
[CrossRef]

Borouchaki, H.

H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005).
[CrossRef]

Cherouat, A.

H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005).
[CrossRef]

Courjon, D.

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

Davis, T. A.

T. A. Davis and I. S. Duff, "A combined unifrontal multifrontal method for unsymmetric sparse matrices," ACM Trans.Math Softw. 25, 1-20 (1999).
[CrossRef]

Demkowicz, L.

D. Xue and L. Demkowicz, "Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method," Finite Elem. Anal. Design 42, 570-579 (2006).
[CrossRef]

D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005).
[CrossRef]

Du, H.

Duff, I. S.

T. A. Davis and I. S. Duff, "A combined unifrontal multifrontal method for unsymmetric sparse matrices," ACM Trans.Math Softw. 25, 1-20 (1999).
[CrossRef]

Felbacq, D.

Girard, C.

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

Grosges, T.

D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B,  84, 55-60 (2006).
[CrossRef]

T. Grosges, A. Vial and D. Barchiesi, "Models of near-field spectroscopic studies: comparison between Finite-Element and Finite-Difference methods," Opt. Express 13, 8483-8497 (2005).
[CrossRef] [PubMed]

Guizal, B.

D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B,  84, 55-60 (2006).
[CrossRef]

B. Guizal, D. Barchiesi and D. Felbacq, "Electromagnetic beam diffraction by a finite lamellar structure," J. Opt. Soc. Am. A 20, 2274-2280 (2003).
[CrossRef]

Houston, P.

P. Houston, I. Perugia, and D. Schotzau, "Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator," Comput. Methods Appl. Mech. Eng. 194, 499-510 (2005).
[CrossRef]

Ingelström, P.

P. Ingelström and A. Bondeson, "Goal-oriented error estimation and h-adaptivity for Maxwell’s equations," Comput. Methods Appl. Mech. Eng. 192, 2597-2616 (2003).
[CrossRef]

Kim, D. S.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Kim, J.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Kottmann, J. P.

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

Lang, P.

H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005).
[CrossRef]

Lienau, C.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Martin, O. J. F.

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

Mie, G.

G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Oden, J. T.

M. Ainsworth and J. T. Oden, "A posteriori error estimation in finite element analysis," Comput. Methods Appl. Mech. Eng. 142, 1-88 (1997).
[CrossRef]

Ortiz, M.

R. Radovitzky and M. Ortiz, "Error estimation and adaptive meshing in strongly non-linear dynamic problems," Comput. Methods Appl. Mech. Eng. 172, 203-240 (1999).
[CrossRef]

Pardo, D.

D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005).
[CrossRef]

Park, D. J.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Perugia, I.

P. Houston, I. Perugia, and D. Schotzau, "Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator," Comput. Methods Appl. Mech. Eng. 194, 499-510 (2005).
[CrossRef]

Radovitzky, R.

R. Radovitzky and M. Ortiz, "Error estimation and adaptive meshing in strongly non-linear dynamic problems," Comput. Methods Appl. Mech. Eng. 172, 203-240 (1999).
[CrossRef]

Ropers, C.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Saanouni, K.

H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005).
[CrossRef]

Schotzau, D.

P. Houston, I. Perugia, and D. Schotzau, "Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator," Comput. Methods Appl. Mech. Eng. 194, 499-510 (2005).
[CrossRef]

Steinmeyer, G.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Stibenz, G.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Tabarovsky, L.

D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005).
[CrossRef]

Torre-Verdìn, C.

D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005).
[CrossRef]

Van Labeke, D.

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

Vial, A.

Xue, D.

D. Xue and L. Demkowicz, "Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method," Finite Elem. Anal. Design 42, 570-579 (2006).
[CrossRef]

Ann. Phys. (1)

G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

D. Barchiesi, B. Guizal and T. Grosges, "Accuracy of local field enhancement models: toward predictive models?" Appl. Phys. B,  84, 55-60 (2006).
[CrossRef]

Comput. Methods Appl. Mech. Eng. (4)

P. Ingelström and A. Bondeson, "Goal-oriented error estimation and h-adaptivity for Maxwell’s equations," Comput. Methods Appl. Mech. Eng. 192, 2597-2616 (2003).
[CrossRef]

P. Houston, I. Perugia, and D. Schotzau, "Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator," Comput. Methods Appl. Mech. Eng. 194, 499-510 (2005).
[CrossRef]

M. Ainsworth and J. T. Oden, "A posteriori error estimation in finite element analysis," Comput. Methods Appl. Mech. Eng. 142, 1-88 (1997).
[CrossRef]

R. Radovitzky and M. Ortiz, "Error estimation and adaptive meshing in strongly non-linear dynamic problems," Comput. Methods Appl. Mech. Eng. 172, 203-240 (1999).
[CrossRef]

Eng. Comput. (1)

M. Berzins, "Mesh quality: a function of geometry, error estimates or both?" Eng. Comput. 15, 236-247 (1999).
[CrossRef]

Finite Elem. Anal. Design (1)

D. Xue and L. Demkowicz, "Modeling of electromagnetic absorption/scattering problems on curvilinear geometries using hp finite/infinite element method," Finite Elem. Anal. Design 42, 570-579 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

Int. J. Numer. Methods Eng. (2)

H. Borouchaki, P. Lang, A. Cherouat and K. Saanouni, "Adaptive remeshing in large plastic strain with damage," Int. J. Numer. Methods Eng. 63, 1-36 (2005).
[CrossRef]

D. Pardo, L. Demkowicz, C. Torre-Verdìn and L. Tabarovsky, "A goal-oriented hp-adaptive nite element method with electromagnetic applications. Part I: Electrostatics," Int. J. Numer. Methods Eng. 65, 1269-1309 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Math Softw. (1)

T. A. Davis and I. S. Duff, "A combined unifrontal multifrontal method for unsymmetric sparse matrices," ACM Trans.Math Softw. 25, 1-20 (1999).
[CrossRef]

Opt. Express (1)

Phys. Rev. E (1)

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke and D. Courjon, "Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods," Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, "Femtosecond light transmission and subradiant damping in Plasmonic Crystals," Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Other (3)

M. Born, and E. Wolf, Principle of Optics (Pergamon Press, Oxford, 1993).

J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York, 1993).

P. Laug and H. Borouchaki, "BL2D-V2: mailleur bidimensionnel adaptatif," Report INRIA RT-0275http://www-rocq1.inria.fr/gamma/cdrom/www/bl2d-v2/INDEX.html (2003).

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Figures (10)

Fig. 1.
Fig. 1.

Remeshing of one element e 1 1 into ek 2 for (a) the basic adaptation (k = 1−2, with 3 nodes), (b) the adaptation with the h-method interpolating the error on the PDE solution (Hz ) and (c) the improved adaptive scheme. The adaptives scheme (b) and the improved one (c) using h-method produce more than two new elements (k = 1−4, with 5 nodes). Moreover, with the basic adaptation (a) and the a posteriori h-method (b), the error is estimated on the computed PDE solution Hi k . This contrast to the improved scheme (c) for which Sϕk i denotes the interpolation error of the solution of interest (ϕ = E or ϕ = P,‥) on the remeshing with respect to the threshold δϕi for the iterative step i.

Fig. 2.
Fig. 2.

Geometry of the study for the infinite circular-cylinder along the z-axis illuminated by a p-polarized incident wave of vector k.

Fig. 3.
Fig. 3.

Normalized intensity of the electric field ∣E2/∣E0 2 in the xy-plane computed by the FEM with the classical adaptive remeshing (a,c,e) and the improved adaptive scheme (b,d,f). The number of nodes are (a) Nnodes = 892, (b) Nnodes = 768, (c) Nnodes = 4818, (d) Nnodes = 5556, (e) Nnodes = 71260 and (f) Nnodes = 31669, respectively.

Fig. 4.
Fig. 4.

Computational mesh for the classical adaptive remeshing (a,c,e) and the improved adaptive scheme (b,d,f).

Fig. 5.
Fig. 5.

Last computational mesh for (a) the improved adaptive scheme and (b) the classical adaptive remeshing. The adaptation to the solution of interest (intensity of the electric field) clearly appears with the improved remeshing.

Fig. 6.
Fig. 6.

Normalized intensity of the magnetic field ∣H2/∣H0 2 in the xy-plane computed by the FEM with classical adaptive remeshing (a,c,e) and improved adaptive scheme (b,d,f). The number of nodes are (a) Nnodes = 892, (b) Nnodes = 768, (c) Nnodes = 4818, (d) Nnodes = 5556, (e) Nnodes = 71260 and (f) Nnodes = 31669, respectively.

Fig. 7.
Fig. 7.

Evolution of the normalized intensity of the electric and magnetic field as a function of the mesh refinement (a,c) and (b,d) the errors, relatively to Mie computation, as a function of the distance from the center of the nano-object (radius a = 15 nm) along the x-axis for successive mesh refinements with the classical and the improved adaptive remeshing.

Fig. 8.
Fig. 8.

Geometry of the study for the infinite square-cylinder along the z-axis.

Fig. 9.
Fig. 9.

Normalized amplitude of the Poynting’s vector ∣P∣/∣P0 ∣ computed by the FEM with improved adaptive mesh scheme and the mesh view for (a,b) Nnodes = 3371, (c,d) Nnodes = 13205 and (e,f) Nnodes = 78175, respectively.

Fig. 10.
Fig. 10.

Evolution of the intensity as a function of the distance along (a) the x-axis and (b) the y-axis, for various mesh refinements.

Tables (1)

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Table 1. Number of nodes and computational time (in s) for each mesh step for the classical remeshing and the improved adaptive remeshing process.

Equations (7)

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[ · ( 1 ε r ) + ω 2 c 2 ] H z = 0 in Ω ,
E x y = j ωε ( × H x y ) ,
P x y = 1 2 ( E x y × H * x y ) ,
η C η ˜
ψ u ν = P + ψ u u + ψ ν ν + 1 2 ( ψ uu ʺ u 2 + 2 ψ ʺ + ψ νν ʺ ν 2 ) + o ( u 2 + ν 2 ) e ̂
1 2 ( n ( P ) ψ uu ʺ u 2 + 2 n ( P ) ψ ʺ + n ( P ) ψ νν ʺ ) + o ( u 2 + ν 2 )
h ϕ ( w ) = δ ϕ η ( w , S ( w ) )

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