Volume 14, Issue 5
An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures

Jian Du, Robert D. Guy, Aaron L. Fogelson, Grady B. Wright & James P. Keener

Commun. Comput. Phys., 14 (2013), pp. 1322-1346.

Published online: 2013-11

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  • Abstract

Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist ateachpoint of space andtheir relativeabundanceis describedby their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction of one phase goes to zero in part of the computational domain. A small and constant network volume fraction is temporarily added throughout the domain in setting up the discrete linear equations and the same set of equation is solved everywhere. These equations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficient and produces an accurate solution of these equations for the full range of relevant regularization parameter values.


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@Article{CiCP-14-1322, author = {Jian Du, Robert D. Guy, Aaron L. Fogelson, Grady B. Wright and James P. Keener}, title = {An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures}, journal = {Communications in Computational Physics}, year = {2013}, volume = {14}, number = {5}, pages = {1322--1346}, abstract = {

Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist ateachpoint of space andtheir relativeabundanceis describedby their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction of one phase goes to zero in part of the computational domain. A small and constant network volume fraction is temporarily added throughout the domain in setting up the discrete linear equations and the same set of equation is solved everywhere. These equations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficient and produces an accurate solution of these equations for the full range of relevant regularization parameter values.


}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.180512.210313a}, url = {http://global-sci.org/intro/article_detail/cicp/7204.html} }
TY - JOUR T1 - An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures AU - Jian Du, Robert D. Guy, Aaron L. Fogelson, Grady B. Wright & James P. Keener JO - Communications in Computational Physics VL - 5 SP - 1322 EP - 1346 PY - 2013 DA - 2013/11 SN - 14 DO - http://dor.org/10.4208/cicp.180512.210313a UR - https://global-sci.org/intro/cicp/7204.html KW - AB -

Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist ateachpoint of space andtheir relativeabundanceis describedby their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction of one phase goes to zero in part of the computational domain. A small and constant network volume fraction is temporarily added throughout the domain in setting up the discrete linear equations and the same set of equation is solved everywhere. These equations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficient and produces an accurate solution of these equations for the full range of relevant regularization parameter values.


Jian Du, Robert D. Guy, Aaron L. Fogelson, Grady B. Wright & James P. Keener. (1970). An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures. Communications in Computational Physics. 14 (5). 1322-1346. doi:10.4208/cicp.180512.210313a
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