Abstract

We present the Zernike–Galerkin method, a tool for the discretization of partial differential equations (PDEs) on thin membranes in polar coordinates. The use of a truncated Zernike series as ansatz yields a semianalytical compact and parametric solution of the PDE. We demonstrate its use for the solution of the Poisson equation in polar coordinates, which is the equation of governing a thin strained membrane’s deformation, or the flow of heat, both of which are important influences for the deformable membrane design. The obtained solution is directly expressed in terms of the components of the wavefront error, which highly facilitates the formulation of design questions. The method is computational highly efficient due to the sparsity and recursivity of the ansatz, is applicable to other PDEs, and can be efficiently combined with geometric optical and optimization methods. Its application to model a pressure-driven adaptive lens membrane is demonstrated.

© 2011 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1989).
    [PubMed]
  2. M. Peschka, F. Blechinger, H. Gross, and H. Zügge, Handbook of Optical Systems (Wiley-VCH, 2007), Vol.  3.
  3. F. Zernike, “Beugungstheorie des Schneidenver-fahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [CrossRef]
  4. W. Rueckert, “Methode zur Minimierung von Formabweichung einer elastischen Optik variabler Brennweite,” Kautschuk Gummi Kunststoffe 12, 540–547 (2010).
  5. F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Optimization of an adaptive PDMS-membrane lens with an integrated actuator,” Sens. Actuators A 154, 316–321 (2009).
    [CrossRef]
  6. F. Schneider, “Adaptive Silikon-Membransinsen mit integriertem Piezo-Aktor,” Ph.D thesis (IMTEK, University of Freiburg, 2009).
  7. B. Berge, “Lens with variable focus,” U.S. reissued patent: US RE39,874 E (9 October 2007).
  8. J. Peseux and B. Berge, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159–163 (2000).
    [CrossRef]
  9. Umesh A. Korde, “Large-displacement electrostatic actuation of membrane reflectors through mechanical control of electrode-membrane gap,” J. Intell. Mater. Syst. Struct. 21, 61–82 (2010).
    [CrossRef]
  10. Comsol Multiphysics, http://www.comsol.de (2010).
  11. Optima Research, ZEMAX, http://www.optima-research.com(2010).
  12. L. A. Carcalho, “Accuracy of Zernike polynomials in characterizing optical aberrations and the corneal surface of the eye,” Invest. Ophthalmol. Vis. Sci. 46, 1915–1926 (2005).
    [CrossRef]
  13. J. P. Boyd, and F. Yu, “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions,” J. Comput. Phys. 230, 1408–1436 (2011).
    [CrossRef]
  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  15. E. W. Weisstein, “Zernike polynomial, MathWorld—A Wolfram Web Resource,” http://mathworld.wolfram.com/ZernikePolynomial.html (2010).
  16. H.-G. Roos and C. Grossmann, Numerische Behandlung partieller Differentialgelichungen (Teubner, 1992).
  17. L. D. Landau and E. M. Lifschitz, Theoretical Physics, Theory of Elasticity, Vol.  7, 3rd ed. (Butterworth-Heinemann, 2002).
  18. C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, 1984).
  19. W. Gui and I. Babuška, “The h, p and h-p versions of the finite element method in 1 dimension,” Numer. Math. 49, 577–612(1986).
    [CrossRef]
  20. S. Timoshenko, The Theory of Plates and Shells (McGraw-Hill Higher Education, 1964).
  21. D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

2011 (1)

J. P. Boyd, and F. Yu, “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions,” J. Comput. Phys. 230, 1408–1436 (2011).
[CrossRef]

2010 (2)

W. Rueckert, “Methode zur Minimierung von Formabweichung einer elastischen Optik variabler Brennweite,” Kautschuk Gummi Kunststoffe 12, 540–547 (2010).

Umesh A. Korde, “Large-displacement electrostatic actuation of membrane reflectors through mechanical control of electrode-membrane gap,” J. Intell. Mater. Syst. Struct. 21, 61–82 (2010).
[CrossRef]

2009 (1)

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Optimization of an adaptive PDMS-membrane lens with an integrated actuator,” Sens. Actuators A 154, 316–321 (2009).
[CrossRef]

2005 (1)

L. A. Carcalho, “Accuracy of Zernike polynomials in characterizing optical aberrations and the corneal surface of the eye,” Invest. Ophthalmol. Vis. Sci. 46, 1915–1926 (2005).
[CrossRef]

2000 (1)

J. Peseux and B. Berge, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159–163 (2000).
[CrossRef]

1986 (1)

W. Gui and I. Babuška, “The h, p and h-p versions of the finite element method in 1 dimension,” Numer. Math. 49, 577–612(1986).
[CrossRef]

1976 (1)

1934 (1)

F. Zernike, “Beugungstheorie des Schneidenver-fahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Babuška, I.

W. Gui and I. Babuška, “The h, p and h-p versions of the finite element method in 1 dimension,” Numer. Math. 49, 577–612(1986).
[CrossRef]

Berge, B.

J. Peseux and B. Berge, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159–163 (2000).
[CrossRef]

B. Berge, “Lens with variable focus,” U.S. reissued patent: US RE39,874 E (9 October 2007).

Blechinger, F.

M. Peschka, F. Blechinger, H. Gross, and H. Zügge, Handbook of Optical Systems (Wiley-VCH, 2007), Vol.  3.

Born, M.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1989).
[PubMed]

Boyd, J. P.

J. P. Boyd, and F. Yu, “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions,” J. Comput. Phys. 230, 1408–1436 (2011).
[CrossRef]

Carcalho, L. A.

L. A. Carcalho, “Accuracy of Zernike polynomials in characterizing optical aberrations and the corneal surface of the eye,” Invest. Ophthalmol. Vis. Sci. 46, 1915–1926 (2005).
[CrossRef]

Domaszewski, M.

D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

Draheim, J.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Optimization of an adaptive PDMS-membrane lens with an integrated actuator,” Sens. Actuators A 154, 316–321 (2009).
[CrossRef]

D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

Fletcher, C. A. J.

C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, 1984).

Greiner, A.

D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

Gross, H.

M. Peschka, F. Blechinger, H. Gross, and H. Zügge, Handbook of Optical Systems (Wiley-VCH, 2007), Vol.  3.

Grossmann, C.

H.-G. Roos and C. Grossmann, Numerische Behandlung partieller Differentialgelichungen (Teubner, 1992).

Gui, W.

W. Gui and I. Babuška, “The h, p and h-p versions of the finite element method in 1 dimension,” Numer. Math. 49, 577–612(1986).
[CrossRef]

Kamberger, R.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Optimization of an adaptive PDMS-membrane lens with an integrated actuator,” Sens. Actuators A 154, 316–321 (2009).
[CrossRef]

Korde, Umesh A.

Umesh A. Korde, “Large-displacement electrostatic actuation of membrane reflectors through mechanical control of electrode-membrane gap,” J. Intell. Mater. Syst. Struct. 21, 61–82 (2010).
[CrossRef]

Korvink, J. G.

D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

Landau, L. D.

L. D. Landau and E. M. Lifschitz, Theoretical Physics, Theory of Elasticity, Vol.  7, 3rd ed. (Butterworth-Heinemann, 2002).

Lifschitz, E. M.

L. D. Landau and E. M. Lifschitz, Theoretical Physics, Theory of Elasticity, Vol.  7, 3rd ed. (Butterworth-Heinemann, 2002).

Noll, R. J.

Peschka, M.

M. Peschka, F. Blechinger, H. Gross, and H. Zügge, Handbook of Optical Systems (Wiley-VCH, 2007), Vol.  3.

Peseux, J.

J. Peseux and B. Berge, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159–163 (2000).
[CrossRef]

Roos, H.-G.

H.-G. Roos and C. Grossmann, Numerische Behandlung partieller Differentialgelichungen (Teubner, 1992).

Rueckert, W.

W. Rueckert, “Methode zur Minimierung von Formabweichung einer elastischen Optik variabler Brennweite,” Kautschuk Gummi Kunststoffe 12, 540–547 (2010).

Schneider, F.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Optimization of an adaptive PDMS-membrane lens with an integrated actuator,” Sens. Actuators A 154, 316–321 (2009).
[CrossRef]

F. Schneider, “Adaptive Silikon-Membransinsen mit integriertem Piezo-Aktor,” Ph.D thesis (IMTEK, University of Freiburg, 2009).

Strohmeier, D.

D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

Timoshenko, S.

S. Timoshenko, The Theory of Plates and Shells (McGraw-Hill Higher Education, 1964).

Wallrabe, U.

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Optimization of an adaptive PDMS-membrane lens with an integrated actuator,” Sens. Actuators A 154, 316–321 (2009).
[CrossRef]

D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

Weisstein, E. W.

E. W. Weisstein, “Zernike polynomial, MathWorld—A Wolfram Web Resource,” http://mathworld.wolfram.com/ZernikePolynomial.html (2010).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1989).
[PubMed]

Yu, F.

J. P. Boyd, and F. Yu, “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions,” J. Comput. Phys. 230, 1408–1436 (2011).
[CrossRef]

Zernike, F.

F. Zernike, “Beugungstheorie des Schneidenver-fahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Zügge, H.

M. Peschka, F. Blechinger, H. Gross, and H. Zügge, Handbook of Optical Systems (Wiley-VCH, 2007), Vol.  3.

Eur. Phys. J. E (1)

J. Peseux and B. Berge, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159–163 (2000).
[CrossRef]

Invest. Ophthalmol. Vis. Sci. (1)

L. A. Carcalho, “Accuracy of Zernike polynomials in characterizing optical aberrations and the corneal surface of the eye,” Invest. Ophthalmol. Vis. Sci. 46, 1915–1926 (2005).
[CrossRef]

J. Comput. Phys. (1)

J. P. Boyd, and F. Yu, “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions,” J. Comput. Phys. 230, 1408–1436 (2011).
[CrossRef]

J. Intell. Mater. Syst. Struct. (1)

Umesh A. Korde, “Large-displacement electrostatic actuation of membrane reflectors through mechanical control of electrode-membrane gap,” J. Intell. Mater. Syst. Struct. 21, 61–82 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

Kautschuk Gummi Kunststoffe (1)

W. Rueckert, “Methode zur Minimierung von Formabweichung einer elastischen Optik variabler Brennweite,” Kautschuk Gummi Kunststoffe 12, 540–547 (2010).

Numer. Math. (1)

W. Gui and I. Babuška, “The h, p and h-p versions of the finite element method in 1 dimension,” Numer. Math. 49, 577–612(1986).
[CrossRef]

Physica (1)

F. Zernike, “Beugungstheorie des Schneidenver-fahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Sens. Actuators A (1)

F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, “Optimization of an adaptive PDMS-membrane lens with an integrated actuator,” Sens. Actuators A 154, 316–321 (2009).
[CrossRef]

Other (12)

F. Schneider, “Adaptive Silikon-Membransinsen mit integriertem Piezo-Aktor,” Ph.D thesis (IMTEK, University of Freiburg, 2009).

B. Berge, “Lens with variable focus,” U.S. reissued patent: US RE39,874 E (9 October 2007).

Comsol Multiphysics, http://www.comsol.de (2010).

Optima Research, ZEMAX, http://www.optima-research.com(2010).

S. Timoshenko, The Theory of Plates and Shells (McGraw-Hill Higher Education, 1964).

D. Strohmeier, J. Draheim, M. Domaszewski, A. Greiner, U. Wallrabe, and J. G. Korvink, “Particle swarm optimization on a new parametric model of a deformable membrane lens,” in IEEE 2011 International Conference on Optical MEMS and Nanophotonics (IEEE, 2011), pp. 201–202.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1989).
[PubMed]

M. Peschka, F. Blechinger, H. Gross, and H. Zügge, Handbook of Optical Systems (Wiley-VCH, 2007), Vol.  3.

E. W. Weisstein, “Zernike polynomial, MathWorld—A Wolfram Web Resource,” http://mathworld.wolfram.com/ZernikePolynomial.html (2010).

H.-G. Roos and C. Grossmann, Numerische Behandlung partieller Differentialgelichungen (Teubner, 1992).

L. D. Landau and E. M. Lifschitz, Theoretical Physics, Theory of Elasticity, Vol.  7, 3rd ed. (Butterworth-Heinemann, 2002).

C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, 1984).

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Figures (6)

Fig. 1
Fig. 1

Configuration of the unit disk ( 0 < ρ 1 , 0 < φ 2 π ), Dirichlet boundary u ( 1 , φ ) = 0 , and source function F ( ρ , φ ) with maximum at ρ 0 = 1 / e and φ 0 = π / e .

Fig. 2
Fig. 2

Exponential decay of the projection parameters of the source function [Eq. (35)] with respect to the rising Zernike index i (exponential decay, lower curve) and the linearly growing sum over the maximum amplitude of the Zernike functions j = 1 i max ( Z j ( ρ , φ ) ) over the disk (linear growth, upper curve). The absolute values of the coefficients decrease exponentially, which indicates convergence of the projection.

Fig. 3
Fig. 3

(a) Pointwise logarithmic difference diff ( u ( x , y ) ) of the results computed with the FEM u ¯ FEM ( x , y ) (number of degrees of freedom n dof 11 · 10 3 ) and the ZGM method u ¯ ZGM ( x , y ) ( n dof = 903 ). b) Solution of the modeled problem with a cross section through the maximum. Note that the FEM and ZGM solutions coincide favorably.

Fig. 4
Fig. 4

Difference between the FEM solution and Zernike–Galerkin solution [Eq. (36)] with respect to the number of nodes M and the number of polynomials N, log 10 δ u ( N , M ) for fixed M. No change in accuracy is found in the Zernike–Galerkin solution by increasing the truncation size beyond N = 800 .

Fig. 5
Fig. 5

Time of computation of a shape evaluation of a parametric model obtained by the ZGM (dashed line with circles) and the repeated FEM simulation (solid line with squares), which are necessary to compute solutions for finding optimal parameters for a optimized membrane. Note that only the time needed to compute the deformation is compared.

Fig. 6
Fig. 6

Many of the coefficient values b i ( N ) / b i ( ) of solutions with rising number of polynomials N do not change with further increases in the size of the solution space. This diagram indicates which coefficients are new and have to be computed (dotted), which are computed but not necessarily converged (vertical striped), and which coefficient values are converged (waved).

Equations (39)

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Z n m ( ρ , φ ) = z n m ( ρ ) a m ( φ ) ,
z n m ( ρ ) = l = 0 ( n m ) / 2 ( 1 ) l ( n l ) ! l ! [ 1 2 ( n + m ) l ] ! [ 1 2 ( n m ) l ] ! ρ n 2 l ,
z n m ( ρ ) ρ = n ( z n 1 m 1 ( ρ ) + z n 1 m + 1 ( ρ ) ) + z n 2 m ( ρ ) ρ .
z n m ( ρ ) ρ = 2 n n + m z n 1 m 1 ( ρ ) n m n + m z n 2 m ( ρ ) ρ .
a m ( φ ) = { sin ( m φ ) m > 0 cos ( m φ ) m 0 ,
k a m ( φ ) φ k = m k a m ( φ + k π 2 ) for     k N 0 .
j ( n , m ) = n ( n + 1 ) 2 + n m 2 + 1.
Z 0 0 = Z 1 Z 1 1 = Z 2 Z 1 1 = Z 3 Z 2 2 = Z 4 Z 2 0 = Z 5 Z 2 2 = Z 6 Z 3 3 = Z 7 Z 3 1 = Z 8 Z 3 1 = Z 9 Z 3 3 = Z 10 Z 4 4 = Z 11 etc. .
n ( j ) = ( 1 + 8 j ) / 2 ,
m ( j ) = n ( j ) ( n ( j ) + 2 ) + 2 2 j .
Z j ( ρ φ ) ρ = a j ( φ ) ( n ( j ) ( z ( j n ( j ) ) ( ρ ) + z ( j n ( j ) 1 ) ( ρ ) ) + z ( j 2 n ( j ) ) ( ρ ) ρ ) .
Z = ( Z 1 ( ρ , φ ) , Z 2 ( ρ , φ ) , , Z j ( ρ , φ ) , , Z ( N + 1 ) ( N + 2 ) 2 ( ρ , φ ) ) ,
Z j ( ρ , φ ) = a j ( φ ) z j ( ρ ) .
Z j ( ρ , φ ) ρ = a j ( φ ) i = 1 j 1 γ j i z i ( ρ ) .
γ j i = ( n ( i ) + 1 ) δ 1 | m ( j ) m ( i ) | δ 1 sign ( i j ) with δ j i = { 1 j = i 0 otherwise and with sign ( x ) = { + 1 x > 0 1 x < 0 0 x = 0 ,
z ( ρ ) ρ = Γ ( z 1 ( ρ ) , z 2 ( ρ ) , , z i ( ρ ) , , z N ( ρ ) ) T .
κ Δ u ( ρ , φ ) = F ( ρ , φ ) .
u ( ρ , φ ) = j = 1 b j Z j ( ρ , φ ) ,
u ( ρ , φ ) u ¯ ( ρ , φ ) = j = 1 N b j Z j ( ρ , φ ) = j = 1 N b j a j ( φ ) z j ( ρ ) .
κ Δ u ¯ ( ρ , φ ) = F ( ρ , φ ) + R ( ρ , φ ) .
Ω ( Δ u ¯ ( ρ , φ ) F ( ρ , φ ) ) ψ i ( ρ , φ ) d Ω = Ω R ( ρ , φ ) ψ i ( ρ , φ ) d Ω .
0 1 0 2 π ( Δ ( j = 1 N b j Z j ( ρ , φ ) ) F ( ρ , φ ) ) Z i ( ρ , φ ) ρ d ρ d φ = 0.
0 1 0 2 π ( j = 1 N b j Z i ( ρ , φ ) Δ Z j ( ρ , φ ) ) ρ d ρ d φ 0 1 0 2 π ( Z i ( ρ , φ ) F ( ρ , φ ) ) ρ d ρ d φ = 0.
Ω ( 2 v ( r ) ) w ( r ) d Ω + Ω ( v ( r ) ) ( w ( r ) ) d Ω = Ω ( v ( r ) ) | ρ = 1 w ( r ) | ρ = 1 d Ω ,
j = 1 N ( Ω b j Z i ( ρ , φ ) ρ Z j ( ρ , φ ) | ρ = 1 d Ω I Ω b j Ω ( Z j ( ρ , φ ) ) ( Z i ( ρ , φ ) ) d Ω I Ω ) + Ω Z i ( ρ , φ ) F ( ρ , φ ) d Ω = 0.
u ¯ ( ρ , φ ) = β ( φ ) z ( ρ ) .
u ¯ ( ρ , φ ) = ( e ^ ρ ρ + e ^ φ ρ φ ) u ¯ ( ρ , φ ) = e ^ ρ β ( φ ) Γ z ( ρ ) e ^ φ ρ M β ( φ + π 2 ) z ( ρ ) .
j = 1 N b j 0 1 0 2 π ( a j ( φ ) a i ( φ ) z j ( ρ ) ρ z i ( ρ ) ρ + z j ( ρ ) ρ z i ( ρ ) ρ a j ( φ ) φ a i ( φ ) φ ) ρ d ρ d φ .
ϑ j i = 0 2 π a m ( j ) ( φ ) a m ( i ) ( φ ) d φ = π δ m ( j ) m ( i ) + π δ 0 m ( j ) δ 0 m ( i ) ,
j , k , l = 1 N b j [ 0 1 ϑ j i γ j k z k ( ρ ) γ i l z l ( ρ ) ρ d ρ + 0 1 ϑ j i m ( i ) m ( j ) z i ( ρ ) ρ z j ( ρ ) ρ ρ d ρ ] .
z j ( ρ ) ρ = i = 1 N ϵ j i z i ( ρ ) .
j , k , l = 1 N b j [ 0 1 ϑ j i γ j k z k ( ρ ) γ i l z l ( ρ ) ρ d ρ + 0 1 ϑ j i m ( i ) m ( j ) ϵ j k z k ( ρ ) ϵ i l z l ( ρ ) ρ d ρ ] .
Φ i j = 0 1 z j ( ρ ) z i ( ρ ) ρ d ρ = 1 2 ( n ( j ) + 1 ) δ n ( j ) n ( i ) .
j , k , l = 1 N b j [ ϑ j i γ j k γ i l Φ k l + ϑ j i m ( j ) m ( i ) ϵ j k ϵ i l Φ i l ] .
F ( ρ , φ ) = F 0 e [ ( ρ * ) 2 cos 2 ( φ * ) + ( ρ * ) 2 sin 2 ( φ * ) ] / ( 2 σ 2 ) .
δ u ¯ ( N , M ) = 1 M i = 1 M ( u ¯ ZGM N ( ρ i , φ i ) u ¯ FEM ( ρ i , φ i ) ) 2 ,
u ¯ ( ρ , φ , v , P ) = j = 1 N b j ( v , P ) Z j ( ρ , φ ) ,
D Δ Δ u ¯ ( ρ , φ ) = F ,
D ( d ) Δ Δ j = 1 N b j ( E , d , F ) Z i ( ρ , φ ) = F ,

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