A coupled finite element-boundary element method for modeling Diffusion equation in 3D multi-modality optical imaging

Subhadra Srinivasan; Hamid R. Ghadyani; Brian W. Pogue; Keith D. Paulsen

doi:10.1364/BOE.1.000398

A coupled finite element-boundary element method for modeling Diffusion equation in 3D multi-modality optical imaging

Subhadra Srinivasan, Hamid R. Ghadyani, Brian W. Pogue, and Keith D. Paulsen

Biomedical Optics Express
Vol. 1,
Issue 2,
pp. 398-413
(2010)
•https://doi.org/10.1364/BOE.1.000398

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Subhadra Srinivasan, Hamid R. Ghadyani, Brian W. Pogue, and Keith D. Paulsen, "A coupled finite element-boundary element method for modeling Diffusion equation in 3D multi-modality optical imaging," Biomed. Opt. Express 1, 398-413 (2010)

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Related Topics
Table of Contents Category
- Image Reconstruction and Inverse Problems
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Image reconstruction techniques (170.3010)
- Light propagation in tissues (170.3660)
- Medical and biological imaging (170.3880)

About this Article
History
- Original Manuscript: June 1, 2010
- Revised Manuscript: July 21, 2010
- Manuscript Accepted: July 23, 2010
- Published: August 2, 2010
Virtual Issues
Biomedical Optics Express Optical Imaging and Spectroscopy (2010)

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Open Access

Abstract

Three dimensional image reconstruction for multi-modality optical spectroscopy systems needs computationally efficient forward solvers with minimum meshing complexity, while allowing the flexibility to apply spatial constraints. Existing models based on the finite element method (FEM) require full 3D volume meshing to incorporate constraints related to anatomical structure via techniques such as regularization. Alternate approaches such as the boundary element method (BEM) require only surface discretization but assume homogeneous or piece-wise constant domains that can be limiting. Here, a coupled finite element-boundary element method (coupled FE-BEM) approach is demonstrated for modeling light diffusion in 3D, which uses surfaces to model exterior tissues with BEM and a small number of volume nodes to model interior tissues with FEM. Such a coupled FE-BEM technique combines strengths of FEM and BEM by assuming homogeneous outer tissue regions and heterogeneous inner tissue regions. Results with FE-BEM show agreement with existing numerical models, having RMS differences of less than 0.5 for the logarithm of intensity and 2.5 degrees for phase of frequency domain boundary data. The coupled FE-BEM approach can model heterogeneity using a fraction of the volume nodes (4-22%) required by conventional FEM techniques. Comparisons of computational times showed that the coupled FE-BEM was faster than stand-alone FEM when the ratio of the number of surface to volume nodes in the mesh (N_s/N_v) was less than 20% and was comparable to stand-alone BEM ( ± 10%).

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References

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2010 (6)

M. Schweiger, O. Dorn, A. Zacharopoulos, I. Nissila, and S. R. Arridge, “3D level set reconstruction of model and experimental data in Diffuse Optical Tomography,” Opt. Express 18(1), 150–164 (2010).
[Crossref] [PubMed]

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[Crossref] [PubMed]

R. B. Schulz, A. Ale, A. Sarantopoulos, M. Freyer, E. Soehngen, M. Zientkowska, and V. Ntziachristos, “Hybrid system for simultaneous fluorescence and x-ray computed tomography,” IEEE Trans. Med. Imaging 29(2), 465–473 (2010).
[Crossref] [PubMed]

D. Hyde, E. L. Miller, D. H. Brooks, and V. Ntziachristos, “Data specific spatially varying regularization for multimodal fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29(2), 365–374 (2010).
[Crossref] [PubMed]

A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. E. Grimson, and W. Wells, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage 49(1), 561–567 (2010).
[Crossref] [PubMed]

H. Ghadyani, J. M. Sullivan, and Z. Wu, “Boundary recovery for delaunay tetrahedral meshes using local topological transformations,” Finite Elem. Anal. Des. 46(1-2), 74–83 (2010).
[Crossref] [PubMed]

2009 (4)

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Philos. Transact. A Math. Phys. Eng. Sci. 367(1900), 3073–3093 (2009).
[Crossref] [PubMed]

X. Zhang, C. T. Badea, and G. A. Johnson, “Three-dimensional reconstruction in free-space whole-body fluorescence tomography of mice using optically reconstructed surface and atlas anatomy,” J. Biomed. Opt. 14(6), 064010 (2009).
[Crossref] [PubMed]

C. Li, G. Wang, J. Qi, and S. R. Cherry, “Three-dimensional fluorescence optical tomography in small-animal imaging using simultaneous positron-emission-tomography priors,” Opt. Lett. 34(19), 2933–2935 (2009).
[Crossref] [PubMed]

A. D. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. R. Arridge, “3D shape based reconstruction of experimental data in Diffuse Optical Tomography,” Opt. Express 17(21), 18940–18956 (2009).
[Crossref] [PubMed]

2008 (7)

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35(5), 1682–1697 (2008).
[Crossref] [PubMed]

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27(6), 752–765 (2008).
[Crossref] [PubMed]

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc. Natl. Acad. Sci. U.S.A. 105(49), 19126–19131 (2008).
[Crossref] [PubMed]

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[Crossref] [PubMed]

M. V. Schulmerich, J. H. Cole, K. A. Dooley, M. D. Morris, J. M. Kreider, S. A. Goldstein, S. Srinivasan, and B. W. Pogue, “Noninvasive Raman tomographic imaging of canine bone tissue,” J. Biomed. Opt. 13(2), 020506 (2008).
[Crossref]

C. M. Carpenter, S. Srinivasan, B. W. Pogue, and K. D. Paulsen, “Methodology development for three-dimensional MR-guided near infrared spectroscopy of breast tumors,” Opt. Express 16(22), 17903–17914 (2008).
[Crossref] [PubMed]

Z. Yuan, Q. Zhang, E. S. Sobel, and H. Jiang, “Tomographic x-ray-guided three-dimensional diffuse optical tomography of osteoarthritis in the finger joints,” J. Biomed. Opt. 13(4), 044006 (2008).
[Crossref] [PubMed]

2007 (4)

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. D. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys. 34(11), 4545–4557 (2007).
[Crossref] [PubMed]

C. M. Carpenter, B. W. Pogue, S. Jiang, H. Dehghani, X. Wang, K. D. Paulsen, W. A. Wells, J. Forero, C. Kogel, J. B. Weaver, S. P. Poplack, and P. A. Kaufman, “Image-guided optical spectroscopy provides molecular-specific information in vivo: MRI-guided spectroscopy of breast cancer hemoglobin, water, and scatterer size,” Opt. Lett. 32(8), 933–935 (2007).
[Crossref] [PubMed]

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 104(10), 4014–4019 (2007).
[Crossref] [PubMed]

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express 15(13), 8043–8058 (2007).
[Crossref] [PubMed]

2006 (1)

S. Srinivasan, B. W. Pogue, H. Dehghani, F. Leblond, and X. Intes, “Data subset algorithm for computationally efficient reconstruction of 3-D spectral imaging in diffuse optical tomography,” Opt. Express 14(12), 5394–5410 (2006).
[Crossref] [PubMed]

2005 (2)

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. H. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10(2), 024033–0240339 (2005).
[Crossref] [PubMed]

G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. 50(17), 3941–3956 (2005).
[Crossref] [PubMed]

2004 (1)

B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycle changes,” J. Biomed. Opt. 9(3), 541–552 (2004).
[Crossref] [PubMed]

2001 (1)

C. P. Bradley, G. M. Harris, and A. J. Pullan, “The computational performance of a high-order coupled FEM/BEM procedure in electropotential problems,” IEEE Trans. Biomed. Eng. 48(11), 1238–1250 (2001).
[Crossref] [PubMed]

2000 (1)

G. Fischer, B. Tilg, R. Modre, G. J. Huiskamp, J. Fetzer, W. Rucker, and P. Wach, “A bidomain model based BEM-FEM coupling formulation for anisotropic cardiac tissue,” Ann. Biomed. Eng. 28(10), 1229–1243 (2000).
[Crossref] [PubMed]

1998 (1)

M. Schweiger and S. R. Arridge, “Comparison of two- and three-dimensional reconstruction methods in optical tomography,” Appl. Opt. 37(31), 7419–7428 (1998).
[Crossref] [PubMed]

1996 (1)

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. 35(19), 3447–3458 (1996).
[Crossref]

1993 (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

1992 (1)

K. D. Paulsen and W. Liu, “Memory and operations count scaling of coupled finite element and boundary element systems of equations,” Int. J. Numer. Methods Eng. 33(6), 1289–1303 (1992).
[Crossref]

1991 (1)

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue I. models of radiation transport and their application,” Lasers Med. Sci. 6(2), 155–168 (1991).
[Crossref]

1989 (1)

P. Vaupel, F. Kallinowski, and P. Okunieff, “Blood flow, oxygen and nutrient supply, and metabolic microenvironment of human tumors: a review,” Cancer Res. 49(23), 6449–6465 (1989).
[PubMed]

Aikawa, E.

Ale, A.

Arridge, S. R.

M. Schweiger and S. R. Arridge, “Comparison of two- and three-dimensional reconstruction methods in optical tomography,” Appl. Opt. 37(31), 7419–7428 (1998).
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Badea, C. T.

Boas, D. A.

Boverman, G.

Bradley, C. P.

Brooks, D. H.

Brukilacchio, T. J.

Butler, J.

Carpenter, C.

Carpenter, C. M.

Cerussi, A.

Chaves, T.

Cherry, S. R.

Chorlton, M.

Cole, J. H.

Custo, A.

Dan, I.

Davis, S. C.

de Kleine, R. H.

Dehghani, H.

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Dooley, K. A.

Dorn, O.

Durkin, A.

Fang, Q.

Fetzer, J.

Fischer, G.

Fischl, B.

Forero, J.

Freyer, M.

Ghadyani, H.

Gibbs-Strauss, S. L.

Gibson, A.

Goldstein, S. A.

Grimson, W. E.

Harris, G. M.

Hillman, E.

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Hoopes, P. J.

Hsiang, D.

Huiskamp, G. J.

Hyde, D.

Intes, X.

Isaacson, D.

Jiang, H.

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. 35(19), 3447–3458 (1996).
[Crossref]

Jiang, S.

Johnson, G. A.

Kallinowski, F.

P. Vaupel, F. Kallinowski, and P. Okunieff, “Blood flow, oxygen and nutrient supply, and metabolic microenvironment of human tumors: a review,” Cancer Res. 49(23), 6449–6465 (1989).
[PubMed]

Kaufman, P. A.

Kirsch, D. G.

Kogel, C.

Kolehmainen, V.

Kopans, D. B.

Kreider, J. M.

Leblond, F.

Li, A.

Li, C.

Liu, W.

Lynch, D. R.

Mehta, R.

Mesquita, R.

Miller, E. L.

Modre, R.

Moore, R. H.

Morris, M. D.

Niedre, M. J.

Nissila, I.

Ntziachristos, V.

O’Hara, J. A.

Okunieff, P.

P. Vaupel, F. Kallinowski, and P. Okunieff, “Blood flow, oxygen and nutrient supply, and metabolic microenvironment of human tumors: a review,” Cancer Res. 49(23), 6449–6465 (1989).
[PubMed]

Patterson, M. S.

Paulsen, K. D.

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. 35(19), 3447–3458 (1996).
[Crossref]

Payne, H. L.

Pittet, M. J.

R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature 452(7187), 580–589 (2008).
[Crossref] [PubMed]

Pogue, B. W.

Poplack, S. P.

Pullan, A. J.

Qi, J.

Rafferty, E.

Rucker, W.

Samkoe, K. S.

Sarantopoulos, A.

Schulmerich, M. V.

Schulz, R. B.

Schweiger, M.

M. Schweiger and S. R. Arridge, “Comparison of two- and three-dimensional reconstruction methods in optical tomography,” Appl. Opt. 37(31), 7419–7428 (1998).
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20(2), 299–309 (1993).
[Crossref] [PubMed]

Shah, N.

Sobel, E. S.

Soehngen, E.

Soho, S.

Song, X.

Srinivasan, S.

Stott, J. J.

Sullivan, J. M.

Tilg, B.

Tosteson, T. D.

Tromberg, B. J.

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Fig. 1

A schematic showing steps from medical image data to obtaining a volumetric mesh for computation with examples from breast data. These steps have to be routinely performed before image reconstruction can be done for 3D multi-modality optical imaging. Methods for image segmentation vary between applications; here thresholding and region-growing techniques were applied for breast tissue. Surface rendering is automatically generated by many open source softwares, but getting a reliable volume mesh can be time-consuming and more difficult to automate.

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Fig. 2

Schematic of a two-layered region in 2D having homogeneous distribution of optical properties in region I and heterogeneous distribution in region II.

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Fig. 3

Surface renderings of the six test cases used in this study are shown, with two-three regions created. Clockwise from top left, the six test cases show cases (1) the outer breast contour and tumor created from clinical MRI (2) outer breast and simulated spherical inclusion (3) Outer breast, sphere and tumor (4) Outer breast, larger sphere and tumor (5) Outer breast and two spherical inclusions and (6) Outer breast, fibroglandular and tumor tissues.

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Fig. 4

Logarithm of photon fluence obtained using coupled FE-BEM for a single source in test cases 1, and 6. Left: Results from test case 1 showing outer boundary; Middle: inner tumor boundary by making outer surface transparent; Right: Results from test case 6 for inner tissues.

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Fig. 5

Comparison of (a) logarithm of intensity and (b) phase at the detector locations on the boundary ( = 240 measurement points) obtained from BEM, FEM and coupled FE-BEM for test case 1.

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Fig. 6

2-D cross-sections along the center of the interior spherical inclusion in test case 2 for $μ_{a}$ (left column) and logarithm of fluence (right column). The background was always homogeneous. Top row shows cross-section of sphere for a homogeneous domain (1:1 contrast between sphere and background), Middle row shows 2:1 contrast between sphere and background and bottom row shows a spatially varying distribution in the sphere (2:1 varying). As expected the fluence decreases with increasing heterogeneity.

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Fig. 7

Ratio of computational time of coupled FE-BEM to stand-alone FEM for the six test cases, plotted as a function of % surface to volume nodes (top) from the respective meshes (N_s/N) where Ns is the number of boundary nodes in the coupled mesh and N is the number of nodes in the FEM mesh and % surface area to volume ratio (bottom) of the total tissue domain.

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Fig. 8

Ratio of computational time of coupled FE-BEM model to BEM for the six test cases, plotted as a function of % surface to volume nodes (top) of the interior tissue (N_b/N_v) where N_b is the number of nodes on boundary of interior tissue and N_v is the number of volume nodes of interior tissue, and % surface area to volume ratio (ISA/IV) (bottom) of the interior tissue domain.

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Tables (1)

Table 1 Mesh sizes for the different test cases used in the simulations. The first two columns of mesh sizes correspond to the coupled FE-BEM and the last two columns correspond to mesh sizes for BEM and FEM

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Equations (26)

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- \nabla . D (r) \nabla Φ (r, ω) + (μ_{a} (r) + \frac{i ω}{c}) Φ (r, ω) = q (r, ω)

D (r) = \frac{1}{(3 (μ_{a} (r) + μ_{s}^{'} (r)))}

Φ (r, ω) + \frac{D}{α} {\frac{\partial Φ}{\partial n} |}_{d Ω} = 0

〈 - \nabla D \nabla Φ, ϕ_{j} 〉 + 〈 (μ_{a} + \frac{i ω}{c}) Φ, ϕ_{j} 〉 = 〈 q, ϕ_{j} 〉

〈 D \nabla Φ, \nabla ϕ_{j} 〉 - \oint D \frac{\partial Φ}{\partial n} ϕ_{j} d s + 〈 (μ_{a} + \frac{i ω}{c}) Φ, ϕ_{j} 〉 = 0

(〈 D_{i} \nabla ϕ_{i}, \nabla ϕ_{j} 〉 + 〈 {(μ_{a} + \frac{i ω}{c})}_{i} ϕ_{i}, ϕ_{j} 〉) \sum_{i = 1}^{N_{v}} Φ_{i} = (\oint ϕ_{i} ϕ_{j} d s) \sum_{i = 1}^{N_{v}} D_{i} \frac{\partial Φ_{i}}{\partial n}

[A] (Φ) = [B] (D \frac{\partial Φ}{\partial n})

\begin{array}{l} A_{k l} = 〈 D \nabla ϕ_{k} \nabla ϕ_{l} 〉 + 〈 (μ_{a} + \frac{i ω}{c}) ϕ_{k} ϕ_{l} 〉 \\ B_{k l} = \oint ϕ_{k} ϕ_{l} \end{array}

\begin{array}{l} (\begin{matrix} A_{b b} & A_{b i} \\ A_{i b} & A_{i i} \end{matrix}) {\begin{matrix} Φ_{b} \\ Φ_{i} \end{matrix}} = (\begin{matrix} B_{b b} & 0 \\ 0 & 0 \end{matrix}) {\begin{matrix} D {\frac{\partial Φ}{\partial n} |}_{Γ}_{b} \\ 0 \end{matrix}} \\ \Rightarrow {\begin{matrix} Φ_{b} \\ Φ_{i} \end{matrix}} = (\begin{matrix} A I_{b b} & A I_{b i} \\ A I_{i b} & A I_{i i} \end{matrix}) (\begin{matrix} B_{b b} & 0 \\ 0 & 0 \end{matrix}) {\begin{matrix} D {\frac{\partial Φ}{\partial n} |}_{b} \\ 0 \end{matrix}} \end{array}

Φ_{b} = [A I_{b}_{b}] B_{b b} (D {\frac{\partial Φ}{\partial n} |}_{b})

\nabla . D_{l} \nabla Φ - k_{l}^{2} Φ = - q_{0} (r, ω)

(μ_{a} (r) + \frac{i ω}{c}) = k_{l}^{2}

D_{l} \nabla^{2} G (r, r_{i}) - k_{l}^{2} G (r, r_{i}) = - δ (r - r_{i})

G_{i} (r, r i) = \frac{\exp (\frac{- k_{l} | r - r_{i} |}{\sqrt{D_{l}}})}{4 π D_{l} | r - r_{i} |}, 3 - D

c_{i} Φ_{i} + \oint D_{l} \frac{\partial G_{i}}{\partial n} Φ - \oint D_{l} \frac{\partial Φ}{\partial n} G_{i} = 〈 q_{0}, G_{i} 〉

c_{i} Φ_{i} + \oint D_{l} \frac{\partial G_{i}}{\partial n} Φ d s - \oint D_{l} \frac{\partial Φ}{\partial n} G_{i} d s = 〈 q_{0}, G_{i} 〉

[\tilde{A}] {Φ_{i}} - [\tilde{B}] {D_{l} \frac{\partial Φ}{\partial n}} = {{\tilde{Q}}_{i}}

\begin{array}{l} {\tilde{A}}_{i, j} = c_{i} δ_{i j} + \oint D_{l} \frac{\partial G_{i}}{\partial n} ψ_{j} d s \\ {\tilde{B}}_{i, j} = \oint G_{i} ψ_{j} d s \\ {\tilde{Q}}_{i} = 〈 q_{0}, G_{i} 〉 \end{array}

(\begin{array}{l} \begin{matrix} {\tilde{A}}_{a a} + α {\tilde{B}}_{a a} & {\tilde{A}}_{a b} & - {\tilde{B}}_{a b} \end{matrix} \\ \begin{matrix} {\tilde{A}}_{b a} + α {\tilde{B}}_{b a} & {\tilde{A}}_{b b} & - {\tilde{B}}_{b b} \end{matrix} \end{array}) {\begin{cases} Φ_{a} \\ Φ_{b} \\ {D \frac{\partial Φ}{\partial n} |}_{b} \end{cases}} = {\begin{cases} {\tilde{Q}}_{a} \\ {\tilde{Q}}_{b} \end{cases}}

\begin{array}{l} Φ_{b} (FEM) = Φ_{b} (BEM) \\ {D \frac{\partial Φ}{\partial n} |}_{b} (FEM) = - {D \frac{\partial Φ}{\partial n} |}_{b} (BEM) \end{array}

\begin{array}{l} (\begin{array}{l} \begin{matrix} {\tilde{A}}_{a a} + α {\tilde{B}}_{a a} & {\tilde{A}}_{a b} & - {\tilde{B}}_{a b} \end{matrix} \\ \begin{matrix} {\tilde{A}}_{b a} + α {\tilde{B}}_{b a} & {\tilde{A}}_{b b} & - {\tilde{B}}_{b b} \end{matrix} \end{array}) {\begin{cases} Φ_{a} \\ - A I_{b b} B_{b b} {D \frac{\partial Φ}{\partial n} |}_{b} \\ {D \frac{\partial Φ}{\partial n} |}_{b} \end{cases}} = {\begin{cases} {\tilde{Q}}_{a} \\ {\tilde{Q}}_{b} \end{cases}} \\ \Rightarrow (\begin{matrix} {\tilde{A}}_{a a} + α {\tilde{B}}_{a a} & - {\tilde{A}}_{a b} A I_{b b} B_{b b} - {\tilde{B}}_{a b} \\ {\tilde{A}}_{b a} + α {\tilde{B}}_{b a} & - {\tilde{A}}_{b b} A I_{b b} B_{b b} - {\tilde{B}}_{b b} \end{matrix}) {\begin{cases} Φ_{a} \\ {D \frac{\partial Φ}{\partial n} |}_{b} \end{cases}} = {\begin{cases} {\tilde{Q}}_{a} \\ {\tilde{Q}}_{b} \end{cases}} \end{array}

(\begin{matrix} {\tilde{A}}_{a a} & {\tilde{A}}_{a b} \\ {\tilde{A}}_{b a} & {\tilde{A}}_{b b} \end{matrix}) {\begin{cases} Φ_{a} \\ Φ_{b} \end{cases}} - (\begin{matrix} {\tilde{B}}_{a a} & {\tilde{B}}_{a b} \\ {\tilde{B}}_{b a} & {\tilde{B}}_{b b} \end{matrix}) {\begin{cases} D_{I} {\frac{\partial Φ}{\partial n} |}_{a} \\ D_{I} {\frac{\partial Φ}{\partial n} |}_{a}_{b} \end{cases}} = {\begin{cases} {\tilde{Q}}_{a} \\ {\tilde{Q}}_{b} \end{cases}}

(\begin{array}{l} \begin{matrix} {\tilde{A}}_{a a} & {\tilde{A}}_{a b} & - {\tilde{B}}_{a a} & - {\tilde{B}}_{a b} \end{matrix} \\ \begin{matrix} {\tilde{A}}_{b a} & {\tilde{A}}_{b b} & - {\tilde{B}}_{b a} & - {\tilde{B}}_{b b} \end{matrix} \end{array}) {\begin{matrix} Φ_{a} \\ Φ_{b} \\ D_{I} {\frac{\partial Φ}{\partial n} |}_{a} \\ D_{I} {\frac{\partial Φ}{\partial n} |}_{b} \end{matrix}} = {\begin{cases} {\tilde{Q}}_{a} \\ {\tilde{Q}}_{b} \end{cases}}

\begin{array}{l} (\begin{array}{l} \begin{matrix} {\tilde{A}}_{a a} & {\tilde{A}}_{a b} & - {\tilde{B}}_{a a} & - {\tilde{B}}_{a b} \end{matrix} \\ \begin{matrix} {\tilde{A}}_{b a} & {\tilde{A}}_{b b} & - {\tilde{B}}_{b a} & - {\tilde{B}}_{b b} \end{matrix} \end{array}) {\begin{matrix} Φ_{a} \\ Φ_{b} \\ - α Φ_{a} \\ D_{I} {\frac{\partial Φ}{\partial n} |}_{b} \end{matrix}} = {\begin{cases} {\tilde{Q}}_{a} \\ {\tilde{Q}}_{b} \end{cases}} \\ \Rightarrow (\begin{array}{l} \begin{matrix} {\tilde{A}}_{a a} + α {\tilde{B}}_{a a} & {\tilde{A}}_{a b} & - {\tilde{B}}_{a b} \end{matrix} \\ \begin{matrix} {\tilde{A}}_{b a} + α {\tilde{B}}_{b a} & {\tilde{A}}_{b b} & - {\tilde{B}}_{b b} \end{matrix} \end{array}) {\begin{matrix} Φ_{a} \\ Φ_{b} \\ D_{I} {\frac{\partial Φ}{\partial n} |}_{b} \end{matrix}} = {\begin{cases} {\tilde{Q}}_{a} \\ {\tilde{Q}}_{b} \end{cases}} \end{array}

(\begin{matrix} {\tilde{A}}_{a a I} + α {\tilde{B}}_{a a I} & {\tilde{A}}_{a b I} & - {\tilde{B}}_{a b I} \\ \begin{array}{l} {\tilde{A}}_{b a} + α {\tilde{B}}_{b a I} \end{array} & \begin{array}{l} {\tilde{A}}_{b b I} \\ {\tilde{A}}_{b b I I} \\ {\tilde{A}}_{c b I I} \end{array} & \begin{array}{l} - {\tilde{B}}_{b b I} \\ {\tilde{B}}_{b b I I} \\ {\tilde{B}}_{c b I I} \end{array} & \begin{array}{l} {\tilde{A}}_{b c I I} \\ {\tilde{A}}_{c c I I} \end{array} & \begin{array}{l} - {\tilde{B}}_{b c I I} \\ - {\tilde{B}}_{c c I I} \end{array} \end{matrix}) {\begin{matrix} Φ_{a I} \\ Φ_{b I} \\ \begin{array}{l} D_{I} {\frac{\partial Φ}{\partial n} |}_{b I} \\ \begin{matrix} Φ_{c I I} \\ D_{I I} {\frac{\partial Φ}{\partial n} |}_{c I I} \end{matrix} \end{array} \end{matrix}} = {\begin{cases} {\tilde{Q}}_{a I} \\ {\tilde{Q}}_{b I} \\ 0 \\ 0 \end{cases}}

\begin{array}{l} (\begin{matrix} {\tilde{A}}_{a a I} + α {\tilde{B}}_{a a I} & {\tilde{A}}_{a b I} & - {\tilde{B}}_{a b I} \\ \begin{array}{l} {\tilde{A}}_{b a} + α {\tilde{B}}_{b a I} \end{array} & \begin{array}{l} {\tilde{A}}_{b b I} \\ {\tilde{A}}_{b b I I} \\ {\tilde{A}}_{c b I I} \end{array} & \begin{array}{l} - {\tilde{B}}_{b b I} \\ {\tilde{B}}_{b b I I} \\ {\tilde{B}}_{c b I I} \end{array} & \begin{array}{l} {\tilde{A}}_{b c I I} \\ {\tilde{A}}_{c c I I} \end{array} & \begin{array}{l} - {\tilde{B}}_{b c I I} \\ - {\tilde{B}}_{c c I I} \end{array} \end{matrix}) {\begin{matrix} Φ_{a I} \\ Φ_{b I} \\ \begin{array}{l} D_{I} {\frac{\partial Φ}{\partial n} |}_{b I} \\ \begin{matrix} - A I_{c c} B_{c c} D_{I I} {\frac{\partial Φ}{\partial n} |}_{c I I} \\ D_{I I} {\frac{\partial Φ}{\partial n} |}_{c I I} \end{matrix} \end{array} \end{matrix}} = {\begin{cases} {\tilde{Q}}_{a I} \\ {\tilde{Q}}_{b I} \\ 0 \\ 0 \end{cases}} \\ \Rightarrow (\begin{matrix} {\tilde{A}}_{a a I} + α {\tilde{B}}_{a a I} & {\tilde{A}}_{a b I} & - {\tilde{B}}_{a b I} \\ {\tilde{A}}_{b a} + α {\tilde{B}}_{b a I} & {\tilde{A}}_{b b I} & - {\tilde{B}}_{b b I} \\ {\tilde{A}}_{b b I I} & {\tilde{B}}_{b b I I} & - {\tilde{A}}_{b c I I} A I_{c c} B_{c c} - {\tilde{B}}_{b c I I} \\ {\tilde{A}}_{c b I I} & {\tilde{B}}_{c b I I} & - {\tilde{A}}_{c c I I} A I_{c c} B_{c c} - {\tilde{B}}_{c c I I} \end{matrix}) {\begin{matrix} Φ_{a I} \\ Φ_{b I} \\ D_{I} {\frac{\partial Φ}{\partial n} |}_{b I} \\ D_{I I} {\frac{\partial Φ}{\partial n} |}_{c I I} \end{matrix}} = {\begin{cases} {\tilde{Q}}_{a I} \\ {\tilde{Q}}_{b I} \\ 0 \\ 0 \end{cases}} \end{array}

Test Case #	#Surface Nodes	# Volume Nodes	# Nodes BEM Mesh	# Nodes FEM Mesh
1	6471	798	6471	70423
2	6869	2171	6869	61468
3	7346	798	7346	65949
4	8297	798	8297	50203
5	8695	2171	8695	51041
6	11415	798	11415	50243