Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T20:06:04.677Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fast Accurate Boundary Integral Method for Potentials on Closely Packed Cells

Published online by Cambridge University Press:  03 June 2015

Wenjun Ying  and
J. Thomas Beale
Wenjun Ying*
Affiliation:
Shanghai Jiao Tong University, Department of Mathematics and Institute of Natural Sciences, Minhang, Shanghai 200240, P.R. China
J. Thomas Beale*
Affiliation:
Duke University, Department of Mathematics, Durham, NC 27708-0320, USA
*
Email:[email protected]
*Corresponding author.Email:[email protected]
Get access

Abstract

Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present a boundary integral method in two dimensions which is specially designed to maintain second order accuracy even if boundaries are arbitrarily close. The method uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast multipole method for efficient summation. We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium. We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples are presented. We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.

Keywords

35J0565N0665N38Laplace equationboundary integral methodfast multipole method
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 4 , October 2013 , pp. 1073 - 1093
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ashihara, T., Yao, T., Namba, T., Ito, M., Ikeda, T., Kawase, A., Toda, S., Suzuki, T., Inagaki, M., Sugimachi, M., Kinoshita, M., Nakazawa, K., Electroporation in a model of cardiac defibrilla-tion, J. Cardiovasc. Electrophysiol. 12 (12) (2001) 1393–1403.Google Scholar
[2]Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, UK, 1997.Google Scholar
[3]Barnes, J., Hut, P., A hierarchical O(NlogN) force calculation algorithm, Nature 324 (2) (1986) 446–449.Google Scholar
[4]Beale, J. T., A grid-based boundary integral method for elliptic problems in three-dimensions, SIAM J. Numer. Anal. 42 (2004) 599–620.Google Scholar
[5]Beale, J. T., Lai, M. C., A method for computing nearly singular integrals, SIAM J. Numer. Anal. 38 (6) (2001) 1902–1925.Google Scholar
[6]Beale, J. T., Layton, A. T., On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. Appl. Math. Comput. Sci. 1 (2006) 91–119.CrossRefGoogle Scholar
[7]Cipra, B. A., The best of the 20th century: Editors name top 10 algorithms, SIAM News 33 (4) (2000) 2.Google Scholar
[8]Colton, D., Kress, R., Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.Google Scholar
[9]Greengard, L., Rokhlin, V., A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987) 325–348.Google Scholar
[10]Hamilton, J. T., Majda, G., On the rokhlin-greengard method with vortex blobs for problems posed in all space or periodic in one direction, J. Comput. Phys. 121 (1995) 29–50.Google Scholar
[11]Hsiao, G. C., Wendland, W. L., Boundary Integral Equations, Springer, 2008.CrossRefGoogle Scholar
[12]Krassowska, W., Neu, J. C., Response of a single cell to an external electric field, Biophysical Journal 66 (1994) 1768–1776.Google Scholar
[13]Mayo, A., Fast high order accurate solution of Laplace’s equation on irregular regions, SIAM J. Sci. Statist. Comput. 6 (1985) 144–157.Google Scholar
[14]Mir, L. M., Bureau, M. F., Gehl, J., Rangara, R., Rouy, D., Caillaud, J. M., Delaere, P., Branellec, D., Schwartz, B., Scherman, D., High-efficiency gene transfer into skeletal muscle mediated by electric pulses, Proc. Natl. Acad. Sci. USA 96 (8) (1999) 4262–4267.Google Scholar
[15]Neumann, E., Kakorin, S., Toensing, K., Fundamentals of electroporative delivery of drugs and genes, Bioelectrochem. Bioenerg. 48 (1) (1999) 3–16.Google Scholar
[16]Pozrikidis, C., Interfacial dynamics for stokes flow, J. Comput. Phys. 169 (2001) 250–301.Google Scholar
[17]Saad, Y., Iterative methods for sparse linear systems, PWS Publishing Company, Boston, 1996.Google Scholar
[18]Saad, Y., Schultz, M. H., GMRES: A generalized minimal residual method for solving non-symmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856–869.Google Scholar
[19]Sauter, S. A., Schwab, C., Boundary Element Methods, Springer Series in Computational Mathematics 39, 2011.Google Scholar
[20]Sersa, G., Cufer, T., Cemazar, M., Rebersek, M., Zvonimir, R., Electro-chemotherapy with bleomycin in the treatment of hypernephroma metastasis: case report and literature review, Tumori 86 (2) (2000) 163–165.Google Scholar
[21]van der Vorst, H. A., BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution nonsymmetric linear systems, SIAM J. Sci. Stat. 13 (1992) 631–644.Google Scholar
[22]Veerapaneni, S. K., Gueyffier, D., Zorin, D., Biros, G., A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d, J. Comput. Phys. 228 (2009) 2334–2353.CrossRefGoogle Scholar
[23]Veerapaneni, S. K., Rahimian, A., Biros, G., Zorin, D., A fast algorithm for simulating vesicle flows in three dimensions, J. Comput. Phys. 230 (2011) 5610–5634.Google Scholar
[24]Ying, L., Biros, G., Zorin, D., A high-order 3d boundary integral equation solver for elliptic PDEs in smooth domains, Journal of Computational Physics 219 (1) (2006) 247–275.Google Scholar
[25]Ying, W.-J., Henriquez, C. S., Hybrid finite element method for describing the electrical response of biological cells to applied fields, IEEE Trans. Biomed. Engrg. 54 (4) (2007) 611–620.CrossRefGoogle Scholar
[26]Zinchenko, A.Z., Davis, R. H., An efficient algorithm for hydrodynamical interaction of many deformable drops, J. Comput. Phys. 157 (2000) 539–587.Google Scholar
[27]Zinchenko, A. Z., Rother, M. A., Davis, R. H., A novel boundary-integral algorithm for viscous interaction of deformable drops, Phys. Fluids 9 (1997) 1493–1511.Google Scholar