Communications in Applied Mathematics and Computational Science

The extended finite element method for boundary layer problems in biofilm growth

Bryan Smith, Benjamin Vaughan, and David Chopp

Full-text: Open access

Abstract

In this paper, we use the eXtended Finite Element Method, with customized enrichment functions determined by asymptotic analysis, to study boundary layer behavior in elliptic equations with discontinuous coefficients. In particular, we look at equations where the coefficients are discontinuous across a boundary internal to the domain. We also show how to implement this method for Dirichlet conditions at an interface. The method requires neither the mesh to conform to the internal boundary, nor the mesh to have additional refinement near the interface, making this an ideal method for moving interface type problems. We then apply this method to equations for linearized biofilm growth to study the effects of biofilm geometry on the availability of substrate and the effect of tip-splitting in biofilm growth.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 2, Number 1 (2007), 35-56.

Dates
Received: 8 July 2006
Accepted: 1 April 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1513798529

Digital Object Identifier
doi:10.2140/camcos.2007.2.35

Mathematical Reviews number (MathSciNet)
MR2327082

Zentralblatt MATH identifier
1131.65092

Subjects
Primary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 92B05: General biology and biomathematics

Keywords
X-FEM extended finite element method level set method elliptic equations Helmholtz equation biofilms

Citation

Smith, Bryan; Vaughan, Benjamin; Chopp, David. The extended finite element method for boundary layer problems in biofilm growth. Commun. Appl. Math. Comput. Sci. 2 (2007), no. 1, 35--56. doi:10.2140/camcos.2007.2.35. https://projecteuclid.org/euclid.camcos/1513798529


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References

  • W. G. Characklis and K. C. Marshall, Biofilms, Wiley, 1990.
  • J. Chessa and T. Belytschko, An extended finite element method for two-phase fluids, Transactions of the ASME 70 (2003), 10–17.
  • J. Chessa et al., The extended finite element method (xfem) for solidification problems., International Journal of Numerical Methods in Engineering 53 (2002), 1959–1977.
  • D. L. Chopp, Simulation of biofilms using the level set method, Parametric and geometric deformable models: an application in biomaterials and medical imagery, Springer, 2006.
  • D. L. Chopp, M. J. Kirisits, B. Moran, and M. R. Parsek, The dependence of quorum sensing on the depth of a growing biofilm, Bulletin of Mathematical Biology 65 (2003), 1053–1079.
  • J. Dockery and I. Klapper, Finger formation in biofilm layers, SIAM J. Appl. Math. 62 (2001), no. 3, 853–869.
  • J. E. Dolbow, N. Moës, and T. Belytschko, Discontinuous enrichment in finite elements with a partition of unity method, Finite Elements in Analysis and Design 36 (2000), 235–260.
  • ––––, An extended finite element method for modeling crack growth with frictional contact, Computational Methods in Applied Mechanics and Engineering 190 (2001), 6825–6846.
  • F. Gibou, R. Fedkiw, L. Cheng, and M. Kang, A second-order-accurate symmetric discretization of the poisson equation on irregular domains, Journal of Computational Physics 176 (2002), 205–227.
  • H. Ji, D. Chopp, and J. E. Dolbow, A hybrid extended finite element/level set method for modeling phase transformation, International Journal of Numerical Methods in Engineering 54 (2002), 1209–1233.
  • H. Johansen and P. Colella, A cartesian grid embedded boundary method for poisson's equation on irregular domains, Journal of Computational Physics 147 (1998), 60–85.
  • B. L. V. Jr., B. G. Smith, and D. L. Chopp, A comparison of the extended finite element method for elliptic equations with discontinuous coefficients and singular sources, Communications in Applied Mathematics and Computational Science 1 (2006), no. 1, 207–228.
  • J. S. Langer, Instabilities and pattern formation in crystal growth, Review of Modern Physics 52 (1980), no. 1, 1–28.
  • R. J. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal Numerical Analysis 31 (1994), 1019–1044.
  • J. M. Melenk and I. Babuška, The partition of unity finite element method: basic theory and applications, Computational Methods in Applied Mechanics and Engineering 139 (1996), 289–314.
  • N. Moës, J. Dolbow, and T. Belytschko, A finite element method for crack growth without remeshing, International Journal of Numerical Methods in Engineering 46 (1999), 131–150.
  • S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms base on hamilton-jacobi formulations, Journal of Computational Physics 79 (1988), 12–49.
  • M. Stolarska, D. L. Chopp, N. Moës, and T. Belytschko, Modelling crack growth by level sets in the extended finite element method, International Journal of Numerical Methods in Engineering 51 (2001), 943–960.
  • N. Sukumar, D. Chopp, N. Moës, and T. Belytschko, Modeling holes and inclusions by level sets in the extended finite element method, International Journal of Numerical Methods in Engineering 48 (2000), 1549–1570.
  • N. Sukumar, D. L. Chopp, and B. Moran, Extended finite element method and fast marching method for three-dimensional fatigue crack propagation, Engineering Fracture Mechanics 70 (2003), 29–48.
  • G. J. Wagner, N. Moës, W. K. Liu, and T. Belytschko, The extended finite element method for rigid particles in stokes flow, International Journal of Numerical Methods in Engineering 51 (2001), 293–313.