Abstract

The low velocity impacts (LVIs) monitoring based on optical fiber Bragg grating (FBG) sensors have attracted more attention in recent years. The center wavelength migrations of FBG sensors were determined by strain and residual strain during and after LVI on composite material structure. We presented a method to discriminate the energy characters of LVI response signals related to LVI locations. By analyzing the wavelet packet energy spectra of LVI response signals monitored by FBG sensors, the sixth node’s energy was found to be sensitive to LVI location. Thus, the sixth node’s energies as LVI feature values, were used to predict the LVI locations by the method of support vector regression (SVR). By optimization of the SVR models’ free parameters, predicting accuracy was 4.62% in the work.

© 2013 Optical Society of America

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References

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  1. E. Kirkby, R. D. Oliveria, V. Michaud, and J. A. Manson, “Impact localization with FBG for a self-healing carbon fibre composite structure,” Compos. Struct. 94, 8–14(2011).
    [CrossRef]
  2. A. Fender, E. J. Rigg, R. R. J. Maier, W. N. MacPherson, J. S. Barton, A. J. Moore, J. D. C. Jones, D. Zhao, L. Zhang, I. Bennion, S. McCulloch, and B. J. S. Jones, “Dynamic two-axis curvature measurement using multicore fiber Bragg gratings interrogated by arrayed waveguide gratings,” Appl. Opt. 45, 9041–9048 (2006).
    [CrossRef]
  3. I. C. Song, S. K. Lee, S. H. Jeong, and B. H. Lee, “Absolute strain measurements made with fiber Bragg grating sensors,” Appl. Opt. 43, 1337–1341 (2004).
    [CrossRef]
  4. C. Doyle, “In-situ process and condition monitoring of advanced fibre-reinforced composite materials using optical fibre sensors,” Smart Mater. Struc. 7, 145–158(1998).
    [CrossRef]
  5. S. Takeda, S. Minakuchi, Y. Okabe, and N. Takeda, “Delamination monitoring of laminated composites subjected to low-velocity impact using small-diameter FBG sensors,” Composites Part A 36, 903–908 (2005).
    [CrossRef]
  6. A. R. Chambers, M. C. Mowlem, and L. Dokos, “Evaluating impact damage in CFRP using fibre optic sensors,” Compos. Sci. Technol. 67, 1235–1242 (2007).
    [CrossRef]
  7. J.. Frieden, J. Cugnosi, J. Botsis, and T. Gmur, “Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors—Part 1: impact and localization,” Compos. Struct. 94, 438–445 (2012).
    [CrossRef]
  8. K. Worden and W. J. Staszewski, “Impact location and quantification on a composite panel using neural networks and a genetic algorithm,” Strain 36, 61–68 (2000).
    [CrossRef]
  9. J. Haywood, P. T. Coverley, W. J. Staszewki, and K. Worden, “An automatic impact monitor for composite panel employing smart sensor technology,” Smart Mater. Struc. 14, 265–271 (2005).
    [CrossRef]
  10. K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides—application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
    [CrossRef]
  11. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
    [CrossRef]
  12. F. Yanhui and S. S. Fernando, “Normalized wavelet packets quantifiers for condition monitoring,” Mech. Syst. Signal Process. 23, 712–723 (2009).
    [CrossRef]
  13. I. W. Selesnick, R. G. Baraniuk, and N. Kingsbury, “The dual-tree complex wavelet transform, wavelet transform,” IEEE Signal Process. Mag. 22, 123–151 (2005).
    [CrossRef]
  14. X. Chen, L. Yong, and R. Harrison, “Type-2 fuzzy logic-based classifier fusion for support vector machines,” Appl. Soft Comput. 8, 1222–1231 (2008).
    [CrossRef]
  15. D. Kim and S. Z. Cho, “Pattern selection for support vector regression based response modeling,” Expert Syst. Appl. 39, 8975–8985 (2012).
    [CrossRef]
  16. J. S. Cheng, D. J. Yu, and Y. Yang, “Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform,” Mech. Syst. Signal Process. 21, 1197–1211 (2007).
    [CrossRef]

2012 (2)

J.. Frieden, J. Cugnosi, J. Botsis, and T. Gmur, “Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors—Part 1: impact and localization,” Compos. Struct. 94, 438–445 (2012).
[CrossRef]

D. Kim and S. Z. Cho, “Pattern selection for support vector regression based response modeling,” Expert Syst. Appl. 39, 8975–8985 (2012).
[CrossRef]

2011 (1)

E. Kirkby, R. D. Oliveria, V. Michaud, and J. A. Manson, “Impact localization with FBG for a self-healing carbon fibre composite structure,” Compos. Struct. 94, 8–14(2011).
[CrossRef]

2009 (1)

F. Yanhui and S. S. Fernando, “Normalized wavelet packets quantifiers for condition monitoring,” Mech. Syst. Signal Process. 23, 712–723 (2009).
[CrossRef]

2008 (1)

X. Chen, L. Yong, and R. Harrison, “Type-2 fuzzy logic-based classifier fusion for support vector machines,” Appl. Soft Comput. 8, 1222–1231 (2008).
[CrossRef]

2007 (2)

J. S. Cheng, D. J. Yu, and Y. Yang, “Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform,” Mech. Syst. Signal Process. 21, 1197–1211 (2007).
[CrossRef]

A. R. Chambers, M. C. Mowlem, and L. Dokos, “Evaluating impact damage in CFRP using fibre optic sensors,” Compos. Sci. Technol. 67, 1235–1242 (2007).
[CrossRef]

2006 (1)

2005 (3)

I. W. Selesnick, R. G. Baraniuk, and N. Kingsbury, “The dual-tree complex wavelet transform, wavelet transform,” IEEE Signal Process. Mag. 22, 123–151 (2005).
[CrossRef]

J. Haywood, P. T. Coverley, W. J. Staszewki, and K. Worden, “An automatic impact monitor for composite panel employing smart sensor technology,” Smart Mater. Struc. 14, 265–271 (2005).
[CrossRef]

S. Takeda, S. Minakuchi, Y. Okabe, and N. Takeda, “Delamination monitoring of laminated composites subjected to low-velocity impact using small-diameter FBG sensors,” Composites Part A 36, 903–908 (2005).
[CrossRef]

2004 (1)

2000 (1)

K. Worden and W. J. Staszewski, “Impact location and quantification on a composite panel using neural networks and a genetic algorithm,” Strain 36, 61–68 (2000).
[CrossRef]

1998 (1)

C. Doyle, “In-situ process and condition monitoring of advanced fibre-reinforced composite materials using optical fibre sensors,” Smart Mater. Struc. 7, 145–158(1998).
[CrossRef]

1997 (1)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

1978 (1)

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides—application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Baraniuk, R. G.

I. W. Selesnick, R. G. Baraniuk, and N. Kingsbury, “The dual-tree complex wavelet transform, wavelet transform,” IEEE Signal Process. Mag. 22, 123–151 (2005).
[CrossRef]

Barton, J. S.

Bennion, I.

Botsis, J.

J.. Frieden, J. Cugnosi, J. Botsis, and T. Gmur, “Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors—Part 1: impact and localization,” Compos. Struct. 94, 438–445 (2012).
[CrossRef]

Chambers, A. R.

A. R. Chambers, M. C. Mowlem, and L. Dokos, “Evaluating impact damage in CFRP using fibre optic sensors,” Compos. Sci. Technol. 67, 1235–1242 (2007).
[CrossRef]

Chen, X.

X. Chen, L. Yong, and R. Harrison, “Type-2 fuzzy logic-based classifier fusion for support vector machines,” Appl. Soft Comput. 8, 1222–1231 (2008).
[CrossRef]

Cheng, J. S.

J. S. Cheng, D. J. Yu, and Y. Yang, “Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform,” Mech. Syst. Signal Process. 21, 1197–1211 (2007).
[CrossRef]

Cho, S. Z.

D. Kim and S. Z. Cho, “Pattern selection for support vector regression based response modeling,” Expert Syst. Appl. 39, 8975–8985 (2012).
[CrossRef]

Coverley, P. T.

J. Haywood, P. T. Coverley, W. J. Staszewki, and K. Worden, “An automatic impact monitor for composite panel employing smart sensor technology,” Smart Mater. Struc. 14, 265–271 (2005).
[CrossRef]

Cugnosi, J.

J.. Frieden, J. Cugnosi, J. Botsis, and T. Gmur, “Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors—Part 1: impact and localization,” Compos. Struct. 94, 438–445 (2012).
[CrossRef]

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Dokos, L.

A. R. Chambers, M. C. Mowlem, and L. Dokos, “Evaluating impact damage in CFRP using fibre optic sensors,” Compos. Sci. Technol. 67, 1235–1242 (2007).
[CrossRef]

Doyle, C.

C. Doyle, “In-situ process and condition monitoring of advanced fibre-reinforced composite materials using optical fibre sensors,” Smart Mater. Struc. 7, 145–158(1998).
[CrossRef]

Fender, A.

Fernando, S. S.

F. Yanhui and S. S. Fernando, “Normalized wavelet packets quantifiers for condition monitoring,” Mech. Syst. Signal Process. 23, 712–723 (2009).
[CrossRef]

Fiebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Frieden, J..

J.. Frieden, J. Cugnosi, J. Botsis, and T. Gmur, “Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors—Part 1: impact and localization,” Compos. Struct. 94, 438–445 (2012).
[CrossRef]

Fujii, Y.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides—application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Gmur, T.

J.. Frieden, J. Cugnosi, J. Botsis, and T. Gmur, “Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors—Part 1: impact and localization,” Compos. Struct. 94, 438–445 (2012).
[CrossRef]

Harrison, R.

X. Chen, L. Yong, and R. Harrison, “Type-2 fuzzy logic-based classifier fusion for support vector machines,” Appl. Soft Comput. 8, 1222–1231 (2008).
[CrossRef]

Haywood, J.

J. Haywood, P. T. Coverley, W. J. Staszewki, and K. Worden, “An automatic impact monitor for composite panel employing smart sensor technology,” Smart Mater. Struc. 14, 265–271 (2005).
[CrossRef]

Hill, K. O.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides—application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Jeong, S. H.

Johnson, D. C.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides—application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Jones, B. J. S.

Jones, J. D. C.

Kawasaki, B. S.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides—application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Kim, D.

D. Kim and S. Z. Cho, “Pattern selection for support vector regression based response modeling,” Expert Syst. Appl. 39, 8975–8985 (2012).
[CrossRef]

Kingsbury, N.

I. W. Selesnick, R. G. Baraniuk, and N. Kingsbury, “The dual-tree complex wavelet transform, wavelet transform,” IEEE Signal Process. Mag. 22, 123–151 (2005).
[CrossRef]

Kirkby, E.

E. Kirkby, R. D. Oliveria, V. Michaud, and J. A. Manson, “Impact localization with FBG for a self-healing carbon fibre composite structure,” Compos. Struct. 94, 8–14(2011).
[CrossRef]

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Lee, B. H.

Lee, S. K.

MacPherson, W. N.

Maier, R. R. J.

Manson, J. A.

E. Kirkby, R. D. Oliveria, V. Michaud, and J. A. Manson, “Impact localization with FBG for a self-healing carbon fibre composite structure,” Compos. Struct. 94, 8–14(2011).
[CrossRef]

McCulloch, S.

Michaud, V.

E. Kirkby, R. D. Oliveria, V. Michaud, and J. A. Manson, “Impact localization with FBG for a self-healing carbon fibre composite structure,” Compos. Struct. 94, 8–14(2011).
[CrossRef]

Minakuchi, S.

S. Takeda, S. Minakuchi, Y. Okabe, and N. Takeda, “Delamination monitoring of laminated composites subjected to low-velocity impact using small-diameter FBG sensors,” Composites Part A 36, 903–908 (2005).
[CrossRef]

Moore, A. J.

Mowlem, M. C.

A. R. Chambers, M. C. Mowlem, and L. Dokos, “Evaluating impact damage in CFRP using fibre optic sensors,” Compos. Sci. Technol. 67, 1235–1242 (2007).
[CrossRef]

Okabe, Y.

S. Takeda, S. Minakuchi, Y. Okabe, and N. Takeda, “Delamination monitoring of laminated composites subjected to low-velocity impact using small-diameter FBG sensors,” Composites Part A 36, 903–908 (2005).
[CrossRef]

Oliveria, R. D.

E. Kirkby, R. D. Oliveria, V. Michaud, and J. A. Manson, “Impact localization with FBG for a self-healing carbon fibre composite structure,” Compos. Struct. 94, 8–14(2011).
[CrossRef]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Rigg, E. J.

Selesnick, I. W.

I. W. Selesnick, R. G. Baraniuk, and N. Kingsbury, “The dual-tree complex wavelet transform, wavelet transform,” IEEE Signal Process. Mag. 22, 123–151 (2005).
[CrossRef]

Song, I. C.

Staszewki, W. J.

J. Haywood, P. T. Coverley, W. J. Staszewki, and K. Worden, “An automatic impact monitor for composite panel employing smart sensor technology,” Smart Mater. Struc. 14, 265–271 (2005).
[CrossRef]

Staszewski, W. J.

K. Worden and W. J. Staszewski, “Impact location and quantification on a composite panel using neural networks and a genetic algorithm,” Strain 36, 61–68 (2000).
[CrossRef]

Takeda, N.

S. Takeda, S. Minakuchi, Y. Okabe, and N. Takeda, “Delamination monitoring of laminated composites subjected to low-velocity impact using small-diameter FBG sensors,” Composites Part A 36, 903–908 (2005).
[CrossRef]

Takeda, S.

S. Takeda, S. Minakuchi, Y. Okabe, and N. Takeda, “Delamination monitoring of laminated composites subjected to low-velocity impact using small-diameter FBG sensors,” Composites Part A 36, 903–908 (2005).
[CrossRef]

Worden, K.

J. Haywood, P. T. Coverley, W. J. Staszewki, and K. Worden, “An automatic impact monitor for composite panel employing smart sensor technology,” Smart Mater. Struc. 14, 265–271 (2005).
[CrossRef]

K. Worden and W. J. Staszewski, “Impact location and quantification on a composite panel using neural networks and a genetic algorithm,” Strain 36, 61–68 (2000).
[CrossRef]

Yang, Y.

J. S. Cheng, D. J. Yu, and Y. Yang, “Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform,” Mech. Syst. Signal Process. 21, 1197–1211 (2007).
[CrossRef]

Yanhui, F.

F. Yanhui and S. S. Fernando, “Normalized wavelet packets quantifiers for condition monitoring,” Mech. Syst. Signal Process. 23, 712–723 (2009).
[CrossRef]

Yong, L.

X. Chen, L. Yong, and R. Harrison, “Type-2 fuzzy logic-based classifier fusion for support vector machines,” Appl. Soft Comput. 8, 1222–1231 (2008).
[CrossRef]

Yu, D. J.

J. S. Cheng, D. J. Yu, and Y. Yang, “Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform,” Mech. Syst. Signal Process. 21, 1197–1211 (2007).
[CrossRef]

Zhang, L.

Zhao, D.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides—application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Appl. Soft Comput. (1)

X. Chen, L. Yong, and R. Harrison, “Type-2 fuzzy logic-based classifier fusion for support vector machines,” Appl. Soft Comput. 8, 1222–1231 (2008).
[CrossRef]

Compos. Sci. Technol. (1)

A. R. Chambers, M. C. Mowlem, and L. Dokos, “Evaluating impact damage in CFRP using fibre optic sensors,” Compos. Sci. Technol. 67, 1235–1242 (2007).
[CrossRef]

Compos. Struct. (2)

J.. Frieden, J. Cugnosi, J. Botsis, and T. Gmur, “Low energy impact damage monitoring of composites using dynamic strain signals from FBG sensors—Part 1: impact and localization,” Compos. Struct. 94, 438–445 (2012).
[CrossRef]

E. Kirkby, R. D. Oliveria, V. Michaud, and J. A. Manson, “Impact localization with FBG for a self-healing carbon fibre composite structure,” Compos. Struct. 94, 8–14(2011).
[CrossRef]

Composites Part A (1)

S. Takeda, S. Minakuchi, Y. Okabe, and N. Takeda, “Delamination monitoring of laminated composites subjected to low-velocity impact using small-diameter FBG sensors,” Composites Part A 36, 903–908 (2005).
[CrossRef]

Expert Syst. Appl. (1)

D. Kim and S. Z. Cho, “Pattern selection for support vector regression based response modeling,” Expert Syst. Appl. 39, 8975–8985 (2012).
[CrossRef]

IEEE Signal Process. Mag. (1)

I. W. Selesnick, R. G. Baraniuk, and N. Kingsbury, “The dual-tree complex wavelet transform, wavelet transform,” IEEE Signal Process. Mag. 22, 123–151 (2005).
[CrossRef]

J. Lightwave Technol. (1)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Fiebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463(1997).
[CrossRef]

Mech. Syst. Signal Process. (2)

F. Yanhui and S. S. Fernando, “Normalized wavelet packets quantifiers for condition monitoring,” Mech. Syst. Signal Process. 23, 712–723 (2009).
[CrossRef]

J. S. Cheng, D. J. Yu, and Y. Yang, “Application of support vector regression machines to the processing of end effects of Hilbert-Huang transform,” Mech. Syst. Signal Process. 21, 1197–1211 (2007).
[CrossRef]

Smart Mater. Struc. (2)

J. Haywood, P. T. Coverley, W. J. Staszewki, and K. Worden, “An automatic impact monitor for composite panel employing smart sensor technology,” Smart Mater. Struc. 14, 265–271 (2005).
[CrossRef]

C. Doyle, “In-situ process and condition monitoring of advanced fibre-reinforced composite materials using optical fibre sensors,” Smart Mater. Struc. 7, 145–158(1998).
[CrossRef]

Strain (1)

K. Worden and W. J. Staszewski, “Impact location and quantification on a composite panel using neural networks and a genetic algorithm,” Strain 36, 61–68 (2000).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Arrangement of FBG on the CFRP laminate.

Fig. 2.
Fig. 2.

Optical fiber sensing system for monitoring LVI.

Fig. 3.
Fig. 3.

Relative positions of impact location and FBG sensors.

Fig. 4.
Fig. 4.

FBG signals and corresponding frequency spectrums.

Fig. 5.
Fig. 5.

Wavelet packet energy spectrums of LVI response signals.

Fig. 6.
Fig. 6.

Wavelet packet energy spectrums of LVI response signals monitored by eight FBG sensors.

Fig. 7.
Fig. 7.

Contour map of E5,6 values monitored by FBG 1.1.

Fig. 8.
Fig. 8.

Contour map of E5,6 values monitored by FBG 2.1.

Fig. 9.
Fig. 9.

Comparisons of real LVI points and predicted LVI points.

Tables (1)

Tables Icon

Table 1. Center Wavelength and Locations of FBG Sensors

Equations (14)

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

λB=2nΛ,
ΔλBλB={1(n22)[P12ν(P11+P12)]}ε+[α+(dn/dT)n]ΔT,
ΔλBλB=Peε,
E5,j=|S5,j|2dt.
f(x)=ωΦ(x)+bΦ.
Remp=i=1le(f(xi)yi),
e(x,y,f)=|yf(x)|ε={0|f(x)y|ε|f(x)y|ε|f(x)y|>ε.
R(w)=Remp+12w2=i=1le(f(xi)yi)+12w2.
i=1l(aiai*)=0ai0,ai*Cl,i=1,2,,l,
Q(ai,ai*)=12i=1lj=1l(ai*ai)(aj*aj)k(xiyj)+εi=1l(ai*+ai)i=1lyi(ai*ai).
f(x)=i=1l(aiai*)k(xi,xj)+b,
P=[EFBG1.2(6)EFBG1.2(6)EFBG2.1(6)EFBG2.2(6)EFBG3.1(6)EFBG3.2(6)EFBG4.1(6)EFBG4.2(6)],
T=[XY].
E(x,y)=i=1p(xix¯i)2+(yiy¯i)2i=1px¯i2+y¯i2,

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