Abstract

In this paper, a digital speckle correlation method for coefficient of thermal expansion (CTE) measurement of film is developed, in which CTE is the intrinsic parameter and direct variable. Deformation pattern governed by the CTE and temperature is used to affine transform the image captured after the film is heated. If the values of CTE are properly chosen, the image after affine transformation will have a highest similarity to the original image. This turns CTE measurement into a purely numerical search of an optimal trial CTE. Results of CTEs from this method and conventional DIC methods are compared with the actual CTE, showing an improved accuracy.

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References

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  1. W. Fang and J. A. Wickert, “Determining mean and gradient residual stress in thin films using micromachined cantilevers,” J. Micromech. Microeng. 6(3), 301–309 (1996).
    [CrossRef]
  2. M. B. David and V. M. Bright, “Design and performance of a double hot arm polysilicon thermal actuator,” Proc. SPIE, Micromacined devices and components III 3224, 296–306(1997).
  3. J. W. Suh, S. F. Glander, R. B. Darling, and C. W. Storment, “Organic thermal and electrostatic ciliary microactuator array for object manipulation,” Sens. Act. A: Physical 58, 51–60 (1997).
    [CrossRef]
  4. H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
    [CrossRef]
  5. W. L. Fang, H. C. Tsai, and C. Y. Lo, “Determining thermal expansion coefficients of thin films using micromachined cantilevers,” Sens. Act. A: Physical 77, 21–27 (1999).
    [CrossRef]
  6. H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
    [CrossRef]
  7. P. H. Townsend, D. M. Barnett, and T. A. Brunner, “Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate,” J. Appl. Phys. 62(11), 4438–4444 (1987).
    [CrossRef]
  8. C. C. Lee, C. L. Tien, W. S. Sheu, and C. C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
    [CrossRef]
  9. C. Dudescu, J. Naumann, M. Stockmann, and S. Nebel, “Characterisation of thermal expansion coefficient of anisotropic materials by electronic speckle pattern interferometry,” Strain 42(3), 197–205 (2006).
    [CrossRef]
  10. J. B. Zhang and T. C. Chong, “Fiber electronic speckle pattern interferometry and its applications in residual stress measurements,” Appl. Opt. 37(28), 6707–6715 (1998).
    [CrossRef] [PubMed]
  11. W. H. Peter and W. F. Ranson, “Digital imaging technique in experimental stress analysis,” Opt. Eng. 21, 427–431 (1982).
  12. B. Pan, H. M. Xie, T. Hua, and A. Anand, “Measurement of coefficient of thermal expansion of films using digital image correlation method,” Polym. Test. 28(1), 75–83 (2009).
    [CrossRef]
  13. J. X. Gao and H. X. Shang, “Deformation-pattern-based digital image correlation method and its application to residual stress measurement,” Appl. Opt. 48(7), 1371–1381 (2009).
    [CrossRef] [PubMed]
  14. F. P. Zhu, W. W. Liu, H. J. Shi, and X. Y. He, “Accurate 3D measurement system and calibration for speckle projection method,” Opt. Lasers Eng. 48(11), 1132–1139 (2010).
    [CrossRef]
  15. Ultem* 1000B Film, Product Datasheet, http://www.tekra.com/products/polycarbonate/Ultem-1000B.pdf
  16. DuPont Kapton® HN, polyimide film Technical Data Sheet, http://www2.dupont.com/Kapton/en_US/assets/downloads/pdf/HN_datasheet.pdf

2010 (1)

F. P. Zhu, W. W. Liu, H. J. Shi, and X. Y. He, “Accurate 3D measurement system and calibration for speckle projection method,” Opt. Lasers Eng. 48(11), 1132–1139 (2010).
[CrossRef]

2009 (2)

B. Pan, H. M. Xie, T. Hua, and A. Anand, “Measurement of coefficient of thermal expansion of films using digital image correlation method,” Polym. Test. 28(1), 75–83 (2009).
[CrossRef]

J. X. Gao and H. X. Shang, “Deformation-pattern-based digital image correlation method and its application to residual stress measurement,” Appl. Opt. 48(7), 1371–1381 (2009).
[CrossRef] [PubMed]

2006 (1)

C. Dudescu, J. Naumann, M. Stockmann, and S. Nebel, “Characterisation of thermal expansion coefficient of anisotropic materials by electronic speckle pattern interferometry,” Strain 42(3), 197–205 (2006).
[CrossRef]

2001 (1)

C. C. Lee, C. L. Tien, W. S. Sheu, and C. C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

2000 (2)

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

1999 (1)

W. L. Fang, H. C. Tsai, and C. Y. Lo, “Determining thermal expansion coefficients of thin films using micromachined cantilevers,” Sens. Act. A: Physical 77, 21–27 (1999).
[CrossRef]

1998 (1)

1997 (1)

J. W. Suh, S. F. Glander, R. B. Darling, and C. W. Storment, “Organic thermal and electrostatic ciliary microactuator array for object manipulation,” Sens. Act. A: Physical 58, 51–60 (1997).
[CrossRef]

1996 (1)

W. Fang and J. A. Wickert, “Determining mean and gradient residual stress in thin films using micromachined cantilevers,” J. Micromech. Microeng. 6(3), 301–309 (1996).
[CrossRef]

1987 (1)

P. H. Townsend, D. M. Barnett, and T. A. Brunner, “Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate,” J. Appl. Phys. 62(11), 4438–4444 (1987).
[CrossRef]

1982 (1)

W. H. Peter and W. F. Ranson, “Digital imaging technique in experimental stress analysis,” Opt. Eng. 21, 427–431 (1982).

Anand, A.

B. Pan, H. M. Xie, T. Hua, and A. Anand, “Measurement of coefficient of thermal expansion of films using digital image correlation method,” Polym. Test. 28(1), 75–83 (2009).
[CrossRef]

Barnett, D. M.

P. H. Townsend, D. M. Barnett, and T. A. Brunner, “Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate,” J. Appl. Phys. 62(11), 4438–4444 (1987).
[CrossRef]

Brunner, T. A.

P. H. Townsend, D. M. Barnett, and T. A. Brunner, “Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate,” J. Appl. Phys. 62(11), 4438–4444 (1987).
[CrossRef]

Chong, T. C.

Darling, R. B.

J. W. Suh, S. F. Glander, R. B. Darling, and C. W. Storment, “Organic thermal and electrostatic ciliary microactuator array for object manipulation,” Sens. Act. A: Physical 58, 51–60 (1997).
[CrossRef]

Dudescu, C.

C. Dudescu, J. Naumann, M. Stockmann, and S. Nebel, “Characterisation of thermal expansion coefficient of anisotropic materials by electronic speckle pattern interferometry,” Strain 42(3), 197–205 (2006).
[CrossRef]

Fang, W.

W. Fang and J. A. Wickert, “Determining mean and gradient residual stress in thin films using micromachined cantilevers,” J. Micromech. Microeng. 6(3), 301–309 (1996).
[CrossRef]

Fang, W. L.

W. L. Fang, H. C. Tsai, and C. Y. Lo, “Determining thermal expansion coefficients of thin films using micromachined cantilevers,” Sens. Act. A: Physical 77, 21–27 (1999).
[CrossRef]

Gao, J. X.

Glander, S. F.

J. W. Suh, S. F. Glander, R. B. Darling, and C. W. Storment, “Organic thermal and electrostatic ciliary microactuator array for object manipulation,” Sens. Act. A: Physical 58, 51–60 (1997).
[CrossRef]

He, X. Y.

F. P. Zhu, W. W. Liu, H. J. Shi, and X. Y. He, “Accurate 3D measurement system and calibration for speckle projection method,” Opt. Lasers Eng. 48(11), 1132–1139 (2010).
[CrossRef]

Hua, T.

B. Pan, H. M. Xie, T. Hua, and A. Anand, “Measurement of coefficient of thermal expansion of films using digital image correlation method,” Polym. Test. 28(1), 75–83 (2009).
[CrossRef]

Jaing, C. C.

C. C. Lee, C. L. Tien, W. S. Sheu, and C. C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Kumpel, A. E.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

Lathrop, R. E.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

Lee, C. C.

C. C. Lee, C. L. Tien, W. S. Sheu, and C. C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Liu, W. W.

F. P. Zhu, W. W. Liu, H. J. Shi, and X. Y. He, “Accurate 3D measurement system and calibration for speckle projection method,” Opt. Lasers Eng. 48(11), 1132–1139 (2010).
[CrossRef]

Lo, C. Y.

W. L. Fang, H. C. Tsai, and C. Y. Lo, “Determining thermal expansion coefficients of thin films using micromachined cantilevers,” Sens. Act. A: Physical 77, 21–27 (1999).
[CrossRef]

Miaoulis, I. N.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

Naumann, J.

C. Dudescu, J. Naumann, M. Stockmann, and S. Nebel, “Characterisation of thermal expansion coefficient of anisotropic materials by electronic speckle pattern interferometry,” Strain 42(3), 197–205 (2006).
[CrossRef]

Nebel, S.

C. Dudescu, J. Naumann, M. Stockmann, and S. Nebel, “Characterisation of thermal expansion coefficient of anisotropic materials by electronic speckle pattern interferometry,” Strain 42(3), 197–205 (2006).
[CrossRef]

Nieva, P.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

Pan, B.

B. Pan, H. M. Xie, T. Hua, and A. Anand, “Measurement of coefficient of thermal expansion of films using digital image correlation method,” Polym. Test. 28(1), 75–83 (2009).
[CrossRef]

Peter, W. H.

W. H. Peter and W. F. Ranson, “Digital imaging technique in experimental stress analysis,” Opt. Eng. 21, 427–431 (1982).

Ranson, W. F.

W. H. Peter and W. F. Ranson, “Digital imaging technique in experimental stress analysis,” Opt. Eng. 21, 427–431 (1982).

Shang, H. X.

Sheu, W. S.

C. C. Lee, C. L. Tien, W. S. Sheu, and C. C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Shi, H. J.

F. P. Zhu, W. W. Liu, H. J. Shi, and X. Y. He, “Accurate 3D measurement system and calibration for speckle projection method,” Opt. Lasers Eng. 48(11), 1132–1139 (2010).
[CrossRef]

Slanina, J. B.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

Stockmann, M.

C. Dudescu, J. Naumann, M. Stockmann, and S. Nebel, “Characterisation of thermal expansion coefficient of anisotropic materials by electronic speckle pattern interferometry,” Strain 42(3), 197–205 (2006).
[CrossRef]

Storment, C. W.

J. W. Suh, S. F. Glander, R. B. Darling, and C. W. Storment, “Organic thermal and electrostatic ciliary microactuator array for object manipulation,” Sens. Act. A: Physical 58, 51–60 (1997).
[CrossRef]

Suh, J. W.

J. W. Suh, S. F. Glander, R. B. Darling, and C. W. Storment, “Organic thermal and electrostatic ciliary microactuator array for object manipulation,” Sens. Act. A: Physical 58, 51–60 (1997).
[CrossRef]

Tada, H.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

Tien, C. L.

C. C. Lee, C. L. Tien, W. S. Sheu, and C. C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Townsend, P. H.

P. H. Townsend, D. M. Barnett, and T. A. Brunner, “Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate,” J. Appl. Phys. 62(11), 4438–4444 (1987).
[CrossRef]

Tsai, H. C.

W. L. Fang, H. C. Tsai, and C. Y. Lo, “Determining thermal expansion coefficients of thin films using micromachined cantilevers,” Sens. Act. A: Physical 77, 21–27 (1999).
[CrossRef]

Wickert, J. A.

W. Fang and J. A. Wickert, “Determining mean and gradient residual stress in thin films using micromachined cantilevers,” J. Micromech. Microeng. 6(3), 301–309 (1996).
[CrossRef]

Wong, P. Y.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

Xie, H. M.

B. Pan, H. M. Xie, T. Hua, and A. Anand, “Measurement of coefficient of thermal expansion of films using digital image correlation method,” Polym. Test. 28(1), 75–83 (2009).
[CrossRef]

Zavracky, P.

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

Zhang, J. B.

Zhu, F. P.

F. P. Zhu, W. W. Liu, H. J. Shi, and X. Y. He, “Accurate 3D measurement system and calibration for speckle projection method,” Opt. Lasers Eng. 48(11), 1132–1139 (2010).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (2)

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Thermal expansion coefficient of polycrystalline silicon and silicon dioxide thin films at high temperatures,” J. Appl. Phys. 87(9), 4189–4194 (2000).
[CrossRef]

P. H. Townsend, D. M. Barnett, and T. A. Brunner, “Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate,” J. Appl. Phys. 62(11), 4438–4444 (1987).
[CrossRef]

J. Micromech. Microeng. (1)

W. Fang and J. A. Wickert, “Determining mean and gradient residual stress in thin films using micromachined cantilevers,” J. Micromech. Microeng. 6(3), 301–309 (1996).
[CrossRef]

Opt. Eng. (1)

W. H. Peter and W. F. Ranson, “Digital imaging technique in experimental stress analysis,” Opt. Eng. 21, 427–431 (1982).

Opt. Lasers Eng. (1)

F. P. Zhu, W. W. Liu, H. J. Shi, and X. Y. He, “Accurate 3D measurement system and calibration for speckle projection method,” Opt. Lasers Eng. 48(11), 1132–1139 (2010).
[CrossRef]

Polym. Test. (1)

B. Pan, H. M. Xie, T. Hua, and A. Anand, “Measurement of coefficient of thermal expansion of films using digital image correlation method,” Polym. Test. 28(1), 75–83 (2009).
[CrossRef]

Rev. Sci. Instrum. (2)

C. C. Lee, C. L. Tien, W. S. Sheu, and C. C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

H. Tada, A. E. Kumpel, R. E. Lathrop, J. B. Slanina, P. Nieva, P. Zavracky, I. N. Miaoulis, and P. Y. Wong, “Novel imaging system for measuring microscale curvatures at high temperatures,” Rev. Sci. Instrum. 71(1), 161–167 (2000).
[CrossRef]

Sens. Act. A: Physical (2)

J. W. Suh, S. F. Glander, R. B. Darling, and C. W. Storment, “Organic thermal and electrostatic ciliary microactuator array for object manipulation,” Sens. Act. A: Physical 58, 51–60 (1997).
[CrossRef]

W. L. Fang, H. C. Tsai, and C. Y. Lo, “Determining thermal expansion coefficients of thin films using micromachined cantilevers,” Sens. Act. A: Physical 77, 21–27 (1999).
[CrossRef]

Strain (1)

C. Dudescu, J. Naumann, M. Stockmann, and S. Nebel, “Characterisation of thermal expansion coefficient of anisotropic materials by electronic speckle pattern interferometry,” Strain 42(3), 197–205 (2006).
[CrossRef]

Other (3)

M. B. David and V. M. Bright, “Design and performance of a double hot arm polysilicon thermal actuator,” Proc. SPIE, Micromacined devices and components III 3224, 296–306(1997).

Ultem* 1000B Film, Product Datasheet, http://www.tekra.com/products/polycarbonate/Ultem-1000B.pdf

DuPont Kapton® HN, polyimide film Technical Data Sheet, http://www2.dupont.com/Kapton/en_US/assets/downloads/pdf/HN_datasheet.pdf

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Figures (4)

Fig. 1
Fig. 1

speckles image of film surface

Fig. 2
Fig. 2

CTE calculating window of a polyetherimide film

Fig. 3
Fig. 3

thermal deformation field at the temperature difference range of 30-100°C,(1pixel = 0.019mm):(a) U field with rigid body motion, (b) V field with rigid body motion,(c) Resultant displacement with rotation, (d) pure thermal expansion deformation

Fig. 4
Fig. 4

CTE comparsion of different film materials using different methods: (a) CTE of Polyetherimide material, (b) CTE of Polyimide material

Tables (1)

Tables Icon

Table 1 CTE results using different methods

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

u ( r , θ ) = ε ( r , θ ) r = α Δ T r
F = { F ( x i , y j ) } , i = 1... M , j = 1... N G = { G ( x ' i , y ' j ) } , i = 1... M , j = 1... N
x ' ' = x ' u t ( x , y ) y ' ' = y ' v t ( x , y )
x ' ' = x ' u t ( x , y ) = x ' α x x ' Δ T y ' ' = y ' v t ( x , y ) = y ' α y y ' Δ T
G ' = { G ( x ' ' i , y ' ' j ) } , i = 1... M , j = 1... N
C = < F G ' > < F > < G ' > < ( F < F > ) 2 > < ( G ' < G ' > ) 2 >
C = C [ u ( α x ) , v ( α y ) ]
x ' ' = x ' u 0 + ω y w 0 x u t ( x , y ) = x ' u 0 + ω y w 0 x α x x ' Δ T y ' ' = y ' v 0 ω x w 0 y v t ( x , y ) = y ' v 0 ω x w 0 y α y y ' Δ T
C = C ( u 0 , v 0 , ω , w 0 , α x , α y )

Metrics