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On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method

Publication ,  Journal Article
Ji, H; Dolbow, JE
Published in: International Journal for Numerical Methods in Engineering
December 14, 2004

We consider a problem stemming from recent models of phase transitions in stimulus-responsive hydrogels, wherein a sharp interface separates swelled and collapsed phases. Extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly 'fit' the interface geometry are emphasized. Attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux ar the interface. We show that as the reciprocal interfacial mobility vanishes, it plays the role of a penalty parameter enforcing a pure Dirichlet constraint, eventually triggering oscillations in the interfacial velocity. We also examine alternative formulations employing a Lagrange multiplier to enforce this constraint. It is shown that the most convenient choice of basis for the Lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms, suggesting a lack of stability in the discrete formulation. Under such conditions, neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the Lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux. Fortunately, the adaptation and use of local, domain-integral methodologies considerably improves the flux evaluations. Several example problems are presented to compare and contrast the various techniques and methods. Copyright © 2004 John Wiley & Sons, Ltd.

Duke Scholars

Published In

International Journal for Numerical Methods in Engineering

DOI

ISSN

0029-5981

Publication Date

December 14, 2004

Volume

61

Issue

14

Start / End Page

2508 / 2535

Related Subject Headings

  • Applied Mathematics
  • 40 Engineering
  • 09 Engineering
 

Citation

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Ji, H., & Dolbow, J. E. (2004). On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. International Journal for Numerical Methods in Engineering, 61(14), 2508–2535. https://doi.org/10.1002/nme.1167
Ji, H., and J. E. Dolbow. “On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method.” International Journal for Numerical Methods in Engineering 61, no. 14 (December 14, 2004): 2508–35. https://doi.org/10.1002/nme.1167.
Ji H, Dolbow JE. On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. International Journal for Numerical Methods in Engineering. 2004 Dec 14;61(14):2508–35.
Ji, H., and J. E. Dolbow. “On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method.” International Journal for Numerical Methods in Engineering, vol. 61, no. 14, Dec. 2004, pp. 2508–35. Scopus, doi:10.1002/nme.1167.
Ji H, Dolbow JE. On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. International Journal for Numerical Methods in Engineering. 2004 Dec 14;61(14):2508–2535.

Published In

International Journal for Numerical Methods in Engineering

DOI

ISSN

0029-5981

Publication Date

December 14, 2004

Volume

61

Issue

14

Start / End Page

2508 / 2535

Related Subject Headings

  • Applied Mathematics
  • 40 Engineering
  • 09 Engineering