Abstract

We have developed fluorescence enhanced optical tomography based upon fully adaptive finite element method (FEM) using tetrahedral dual-meshing wherein one of the two meshes discretizes the forward variables and the other discretizes the unknown parameters to be estimated. We used the 8-subtetrahedron subdivision scheme to create the nested dual-mesh in which each are independently refined. However, two tetrahedrons from the two different meshes pose an intersection problem that needs to be resolved in order to find the common regions that the forward variables (the fluorescent diffuse photon fluence fields) and the parameter estimates (the fluorescent absorption coefficients) can be mutually assigned. Using an efficient intersection algorithm in the nested tetrahedral environments previously developed by the authors, we demonstrate fully adaptive tomography using a posteriori error estimates. Performing the iterative reconstructions using the simulated boundary measurement data, we demonstrate that small fluorescent targets embedded in the breast simulating phantom in point illumination/detection geometry can be resolved at reasonable computational cost.

©2007 Optical Society of America

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References

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  1. K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging 14, 504–514 (1995).
    [Crossref] [PubMed]
  2. H. Jiang, K. Paulsen, U. Osterberg, and M. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998)
    [Crossref] [PubMed]
  3. M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. 44, 2703–2721 (1999).
    [Crossref] [PubMed]
  4. H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. 42, 3117–3129 (2003).
    [Crossref] [PubMed]
  5. M. Huang and Q. Zhu, “Dual-mesh tomography reconstruction method with a depth correction that uses a priori ultrasound information,” Appl. Opt. 43, 1654–1662 (2004).
    [Crossref] [PubMed]
  6. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence enhanced optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
    [Crossref] [PubMed]
  7. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14, 6516–6534 (2006).
    [Crossref] [PubMed]
  8. J. H. Lee, A. Joshi, and E. M. Sevick-Muraca, “Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation,” (submitted).
  9. A. Liu and B. Joe, “Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision,” Math. Comput. 65, 1183–1200 (1996).
    [Crossref]
  10. D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error analysis,” Int. J. Numer. Meth. Engng. 19, 1593–1619 (1983).
    [Crossref]
  11. E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.
  12. J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, New York, 1999).
    [Crossref]
  13. M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
    [Crossref]
  14. A. K. Sahu, R. Roy, A. Joshi, and E. M. Sevick-Muraca, “Evaluation of anatomical structure and non-uniform distribution of imaging agent in near-infrared fluorescence-enhanced optical tomography,” Opt. Express 13, 10182–10199 (2005)
    [Crossref] [PubMed]

2006 (1)

2005 (1)

2004 (2)

2003 (2)

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. 42, 3117–3129 (2003).
[Crossref] [PubMed]

1999 (1)

M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[Crossref] [PubMed]

1998 (1)

H. Jiang, K. Paulsen, U. Osterberg, and M. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998)
[Crossref] [PubMed]

1996 (1)

A. Liu and B. Joe, “Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision,” Math. Comput. 65, 1183–1200 (1996).
[Crossref]

1995 (1)

K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging 14, 504–514 (1995).
[Crossref] [PubMed]

1983 (1)

D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error analysis,” Int. J. Numer. Meth. Engng. 19, 1593–1619 (1983).
[Crossref]

Arridge, S. R.

M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[Crossref] [PubMed]

Babuska, I.

D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error analysis,” Int. J. Numer. Meth. Engng. 19, 1593–1619 (1983).
[Crossref]

Bangerth, W.

Brooksby, B.

Dehghani, H.

Epstein, M. J.

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

Fedele, F.

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

Gago, J. P. De S. R.

D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error analysis,” Int. J. Numer. Meth. Engng. 19, 1593–1619 (1983).
[Crossref]

Godavarty, A.

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.

Houston, J. P.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.

Huang, M.

Jiang, H.

H. Jiang, K. Paulsen, U. Osterberg, and M. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998)
[Crossref] [PubMed]

Joe, B.

A. Liu and B. Joe, “Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision,” Math. Comput. 65, 1183–1200 (1996).
[Crossref]

Joshi, A.

Kelly, D. W.

D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error analysis,” Int. J. Numer. Meth. Engng. 19, 1593–1619 (1983).
[Crossref]

Kuwana, E.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.

Laible, J.

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

Lee, J. H.

J. H. Lee, A. Joshi, and E. M. Sevick-Muraca, “Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation,” (submitted).

Liu, A.

A. Liu and B. Joe, “Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision,” Math. Comput. 65, 1183–1200 (1996).
[Crossref]

Meaney, P. M.

K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging 14, 504–514 (1995).
[Crossref] [PubMed]

Moskowitz, M. J.

K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging 14, 504–514 (1995).
[Crossref] [PubMed]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, New York, 1999).
[Crossref]

Osterberg, U.

H. Jiang, K. Paulsen, U. Osterberg, and M. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998)
[Crossref] [PubMed]

Patterson, M.

H. Jiang, K. Paulsen, U. Osterberg, and M. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998)
[Crossref] [PubMed]

Paulsen, K.

H. Jiang, K. Paulsen, U. Osterberg, and M. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998)
[Crossref] [PubMed]

Paulsen, K. D.

H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. 42, 3117–3129 (2003).
[Crossref] [PubMed]

K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging 14, 504–514 (1995).
[Crossref] [PubMed]

Pogue, B. W.

Roy, R.

A. K. Sahu, R. Roy, A. Joshi, and E. M. Sevick-Muraca, “Evaluation of anatomical structure and non-uniform distribution of imaging agent in near-infrared fluorescence-enhanced optical tomography,” Opt. Express 13, 10182–10199 (2005)
[Crossref] [PubMed]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.

Sahu, A. K.

Schweiger, M.

M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[Crossref] [PubMed]

Sevick-Muraca, E. M.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14, 6516–6534 (2006).
[Crossref] [PubMed]

A. K. Sahu, R. Roy, A. Joshi, and E. M. Sevick-Muraca, “Evaluation of anatomical structure and non-uniform distribution of imaging agent in near-infrared fluorescence-enhanced optical tomography,” Opt. Express 13, 10182–10199 (2005)
[Crossref] [PubMed]

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence enhanced optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.

J. H. Lee, A. Joshi, and E. M. Sevick-Muraca, “Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation,” (submitted).

Shudong, J.

Sullivan, Jr., J. M.

K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging 14, 504–514 (1995).
[Crossref] [PubMed]

Thompson, A. B.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, New York, 1999).
[Crossref]

Zhang, C.

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

Zhu, Q.

Zienkiewicz, O. C.

D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error analysis,” Int. J. Numer. Meth. Engng. 19, 1593–1619 (1983).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Med. Imaging (2)

K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging 14, 504–514 (1995).
[Crossref] [PubMed]

M. J. Epstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging,  22, 1215–1223 (2003).
[Crossref]

Int. J. Numer. Meth. Engng. (1)

D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska, “A posteriori error analysis and adaptive processes in the finite element method: Part I-Error analysis,” Int. J. Numer. Meth. Engng. 19, 1593–1619 (1983).
[Crossref]

Math. Comput. (1)

A. Liu and B. Joe, “Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision,” Math. Comput. 65, 1183–1200 (1996).
[Crossref]

Med. Phys. (1)

H. Jiang, K. Paulsen, U. Osterberg, and M. Patterson, “Frequency-domain near-infrared photo diffusion imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25, 183–193 (1998)
[Crossref] [PubMed]

Opt. Express (3)

Phys. Med. Biol. (1)

M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[Crossref] [PubMed]

Other (3)

J. H. Lee, A. Joshi, and E. M. Sevick-Muraca, “Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation,” (submitted).

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook (CRC Press, 2003), Chap. 33.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, New York, 1999).
[Crossref]

Supplementary Material (20)

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» Media 17: AVI (89 KB)     
» Media 18: AVI (572 KB)     
» Media 19: AVI (422 KB)     
» Media 20: AVI (1091 KB)     

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Figures (11)

Fig. 1.
Fig. 1. 8-subtetrahedron subdivision scheme for nested tetrahedral meshing [9]. The operations shown in (a) through (d) are termed SUB8, SUB2, SUB4a , and SUB4b , respectively. The subtetrahedrons resulting from the operations (a) to (d) are labeled as S 8, S 2, S 4a , and S 4b , respectively.

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Fig. 2.
Fig. 2. Nontrivial intersections resolved by FINT [8]. The figures show the disjoint tetrahedron pieces obtained by FINT for the intersection between the type (a) S 2-, (b) S 4a -, and (c) S 4b -tetrahedron and the tetrahedrons given from the other mesh embedded therein.

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Fig. 3.
Fig. 3. Flow diagram for adaptive reconstruction scheme. GN means Gauss-Newtonm, D is the radius of the trust-region and κ is the measure given by Eq. (44).

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Fig. 4.
Fig. 4. Source/detector configurations for the breast simulating phantom where (a) shows the 27 source locations and (b) shows 128 detector locations.

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Fig. 5.
Fig. 5. Configurations for spherical fluorescent targets in 5 mm diameter where (a) shows the location of single target and (b) shows the locations of two targets. The diameter is d=5 mm and g is the gap between the two targets.

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Fig. 6.
Fig. 6. Initial tetrahedral dual-meshes: (a) and (b) are used for obtaining simulated boundary measurement data, while (c) and (d) are used for iterative reconstruction problem. In the figures, (a) and (c) are the initial meshes discretizing fluence fields, while (b) and (d) are the initial meshes discretizing the parameter μaxf map. The separately discretized fluence fields and parameter μaxf map are coupled by FINT.

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Fig. 7.
Fig. 7. (1.39 MB) Movies of the adaptive refinements of the dual-mesh and iterative reconstructions for the single target. The movies show evolutions in the forward mesh in (a) [Media 1], the inverse mesh in (b) [Media 2], two isosurfaces of 80% and 50% of maximum of the reconstructed μaxf value in (c) and (e) [Media 3] [Media 4], and target values above 0.1% of the maximum of the reconstructed μaxf value for y=0 slice in (g) [Media 5]. Figures of ideal targets are also shown in (d), (f), and (h) for comparision.

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Fig. 8.
Fig. 8. (2.64 MB) Movies of the adaptive refinements of the dual-mesh and iterative reconstructions for the two targets with 1 cm gap. The movies show evolutions in the forward mesh in (a) [Media 6], the inverse mesh in (b) [Media 7], two isosurfaces of 80% and 50% of maximum of the reconstructed μaxf value in (c) and (e) [Media 8] [Media 9], and target values above 0.1% of the maximum of the reconstructed μaxf value for y=0 slice in (g) [Media 10]. Figures of ideal targets are also shown in (d), (f), and (h) for comparision.

Download Full Size | PPT Slide | PDF

Fig. 9.
Fig. 9. (3.38 MB) Movies of the adaptive refinements of the dual-mesh and iterative reconstructions for the two targets with 5 mm gap. The movies show evolutions in the forward mesh in (a) [Media 11], the inverse mesh in (b) [Media 12], two isosurfaces of 80% and 50% of maximum of the reconstructed μaxf value in (c) and (e) [Media 13] [Media 14], and target values above 0.1% of the maximum of the reconstructed μaxf value for y=0 slice in (g) [Media 15]. Figures of ideal targets are also shown in (d), (f), and (h) for comparision.

Download Full Size | PPT Slide | PDF

Fig. 10.
Fig. 10. (2.41 MB) Movies of the adaptive refinements of the dual-mesh and iterative reconstructions for the two targets with 2.5 mm gap. The movies show evolutions in the forward mesh in (a) [Media 16], the inverse mesh in (b) [Media 17], two isosurfaces of 80% and 50% of maximum of the reconstructed μaxf value in (c) and (e) [Media 18] [Media 19], and target values above 0.1% of the maximum of the reconstructed μaxf value for y=0 slice in (g) [Media 20]. Figures of ideal targets are also shown in (d), (f), and (h) for comparision.

Download Full Size | PPT Slide | PDF

Fig. 11.
Fig. 11. Changes in the number of nodes in the forward/inverse meshes, total accumulated computing time, and maximum value of the reconstructed μaxf distribution map, according to the number of iterations in reconstruction

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Tables (1)

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Table. 1. Locations of two spherical targets. All units are in cm.

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Equations (46)

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

min p I ( p ) = 1 2 ∑ i = 1 N S ∑ j = 1 N D [ P j T Φ m i ( p ) − y j i ] 2
p k ≥ 0 , k = 1,2 , ⋯ , N μ
{ − ∇ ∙ [ D x ( r ) ∇ Φ x r ω ] + k x Φ x r ω = 0 − ∇ ∙ [ D m ( r ) ∇ Φ m r ω ] + k m Φ m r ω = β x m Φ x r ω
{ 2 D x n ∙ ∇ Φ x + γ Φ x = Q ( r ) 2 D m n ∙ ∇ Φ m + γ Φ m = 0
u α = { φ i ( r α ) , φ j ( r α ) , φ k ( r α ) , φ l ( r α ) } ,
v α = { ψ i ' ( r α ) , ψ j ' ( r α ) , ψ k ' ( r α ) , ψ l ' ( r α ) } ,
π α ( r β ) = δ αβ
∫ Δ = π p a π q b π r c π s d d v = 6 V Δ a ! b ! c ! d ! ( a + b + c + d + 3 ) ! ,
U n α = φ n ( r α ) , V n α = ψ n ( r α ) .
{ φ i ( r ) , ψ i ( r ) } = ∑ α { U i α , V i α } π α ( r )
{ D x , m ( r ) , μ a x f ( r ) } = ∑ i { D x , m i , μ a x f i } ψ i ( r )
Φ x , m ( r ) = ∑ i Φ x , m i φ i ( r )
[ K x O − B xm K m ] [ Φ x Φ m ] = [ Q O ]
K x , m ij = ∑ k D x , m k Z k , i j + ∑ k μ a xf , a m f k Θ k , i j + k x , m 0 E i j + γ 2 Γ i j ,
B x m i j = χ ∑ k μ a x f k Θ k , i j ,
Q i = 1 2 ∫ ∂ Ω φ i ( r ) Q ( r ) d a ,
Z k , i j = ∫ Ω ψ k ( r ) ∇ φ i ( r ) . ∇ φ j ( r ) d v ,
Θ k , i j = ∫ Ω ψ k ( r ) φ i ( r ) φ j ( r ) d v ,
E i j = ∫ Ω φ i ( r ) φ j ( r ) d v ,
Γ i j = ∫ ∂ Ω φ i ( r ) φ j ( r ) da ,
k x , m 0 = i ω c + μ a x i , a m i , χ = q ( 1 − i ω τ ) .
{ Z , Θ , E } = ∑ Δ { Z Δ , Θ Δ , E Δ }
Z k , i j Δ = ∑ α β γ V k γ U i α U j β ∫ Δ π γ ∇ π α ∙ ∇ π β d v
Θ k , i j Δ = ∑ α β γ V k γ U i α U j β ∫ Δ π γ π α π β d v ,
E i j Δ = ∑ α β U i α U j β ∫ Δ π α π β d v ,
K x , m i j = ∑ Δ ( ∑ k D x , m k Z k , i j Δ + ∑ k μ a x f , a m f k Θ k , i j Δ ) + k x , m 0 ∑ Δ ( ∑ k E i j Δ ) + γ 2 Γ i j ,
B x m i j = χ ∑ Δ ( ∑ k μ a x f k Θ k , ij Δ )
D x , m ( r ) ≈ 1 4 ∑ i D x , m i , μ a x f ( r ) ≈ 1 4 ∑ i μ a x f i .
Z i j Δ = ∑ α β U i α U j β ∫ Δ ∇ π α . ∇ π β d v ,
K x , m i j = 1 4 ∑ Δ ( ∑ k D x , m k ) Z i j Δ + 1 4 ∑ Δ ( ∑ k μ a x f , a m f k ) E i j Δ + k x , m 0 ∑ Δ ( ∑ k E i j Δ ) + γ 2 Γ ij ,
B x m i j = 1 4 χ ∑ Δ ( ∑ k μ a x f k ) E i j Δ ,
[ K x O − B x m K m ] [ ∂ Φ x i ∂ p k ∂ Φ x i ∂ p k ] = − [ ∂ K x ∂ p k O − ∂ B x m ∂ p k ∂ K m ∂ p k ] [ Φ x i Φ m i ]
( [ K x − B x m O K m ] [ Ψ x Ψ m ] ) T [ ∂ Φ x i ∂ p k ∂ Φ x i ∂ p k ] = − [ Ψ x T Ψ m T ] [ ∂ K x ∂ p k O − ∂ B x m ∂ p k ∂ K m ∂ p k ] [ Φ k i Φ m i ]
J k , ij = P j T ( ∂ Φ m i ∂ p k ) .
[ K x − B x m O K m ] [ Ψ x Ψ m ] = [ O P j ] ,
J k , i j = − Ψ x j T ( ∂ K x ∂ p k ) Φ x i + Ψ m j T ( ∂ B x m ∂ p k ) Φ x i − Ψ m j T ( ∂ K m ∂ p k ) Φ m i ,
P j k = φ k ( r j )
J k , i j ≅ − Ψ x j T ( ∂ K x ∂ p k ) Φ x i + Ψ m j T ( ∂ B x m ∂ p k ) Φ x i .
J = ∑ Δ J Δ
J k , i j Δ = 1 4 ( ∑ γ ∂ D x γ ∂ p k δ k γ ) ∑ α β ( Ψ x j ) α Z α β Δ ( Φ x i ) β − 1 4 ( ∑ γ δ k γ ) ∑ α β ( Ψ x j ) α E α β Δ ( Φ x i ) β + 1 4 χ ( ∑ γ δ k γ ) ∑ α β ( Ψ m j ) α E α β Δ ( Φ x i ) β .
ε T [ f ] = h ∮ ∂ T ∣ ∂ f ∂ n ∣ ± 2 d a , ,
e Φ T = ∑ s = 1 N S ( w x , s ε T [ Φ x s ] + w m , s ε T [ Φ m s ] ) + ∑ d = 1 N D ( w x a , d ε T [ Ψ x d ] + w m a , d ε T [ Ψ x d ] )
e μ T = ε T [ μ a x f ] .
e Φ , μ T > η Φ , μ max T { e Φ , μ T } , 0 < η Φ , μ < 1 .
κ = ∣ [ p c + p d ] 2 − p m ∣ p max − p m i n
κ > θ , 0 < θ < 1 2 ,

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