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Delta Function Approximations in Level Set Methods by Distance Function Extension
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.ORCID iD: 0000-0002-4911-467X
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2010 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, no 6, 2199-2219 p.Article in journal (Refereed) Published
Abstract [en]

 In [A.-K. Tornberg, B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys. 200 (2004) 462-488], it was shown for simple examples that the then most common way to regularize delta functions in connection to level set methods produces inconsistent approximations with errors that are not reduced with grid refinement. Since then, several clever approximations have been derived to overcome this problem. However, the great appeal of the old method was its simplicity. In this paper it is shown that the old method - a one-dimensional delta function approximation extended to higher dimensions by a distance function - can be made accurate with a different class of one-dimensional delta function approximations. The prize to pay is a wider support of the resulting delta function approximations.

Place, publisher, year, edition, pages
2010. Vol. 229, no 6, 2199-2219 p.
Keyword [en]
Level set method, Delta function, Consistent approximations, Discretization, Distance function
National Category
Computer and Information Science
Identifiers
URN: urn:nbn:se:kth:diva-10506DOI: 10.1016/j.jcp.2009.11.030ISI: 000275092000015Scopus ID: 2-s2.0-73649126152OAI: oai:DiVA.org:kth-10506DiVA: diva2:218471
Funder
Knut and Alice Wallenberg Foundation
Note
QC 20100907. Uppdaterat från Manuskript till Artikel (20100907)Available from: 2009-05-19 Created: 2009-05-19 Last updated: 2011-05-03Bibliographically approved
In thesis
1. Numerical Modeling of Fluid Interface Phenomena
Open this publication in new window or tab >>Numerical Modeling of Fluid Interface Phenomena
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Place, publisher, year, edition, pages
Stockholm: KTH, 2009. ix, 41 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2009:7
Identifiers
urn:nbn:se:kth:diva-10507 (URN)978-91-7415-344-6 (ISBN)
Presentation
2009-05-10, D42, Lindstedsvägen 5, plan 4, Kungliga Tekniska högskolan, 10:00 (English)
Opponent
Supervisors
Available from: 2009-05-26 Created: 2009-05-19 Last updated: 2010-11-03Bibliographically approved
2. Numerical Methods for Fluid Interface Problems
Open this publication in new window or tab >>Numerical Methods for Fluid Interface Problems
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns numerical techniques for two phase flowsimulations; the two phases are immiscible and incompressible fluids. Strategies for accurate simulations are suggested. In particular, accurate approximations of the weakly discontinuousvelocity field, the discontinuous pressure, and the surface tension force and a new model for simulations of contact line dynamics are proposed.

In two phase flow problems discontinuities arise in the pressure and the gradient of the velocity field due to surface tension forces and differences in the fluids' viscosity. In this thesis, a new finite element method which allows for discontinuities along an interface that can be arbitrarily located with respect to the mesh is presented. Using standard linear finite elements, the method is for an elliptic PDE proven to have optimal convergence order and a system matrix with condition number bounded independently of the position of the interface.The new finite element method is extended to the incompressible Stokes equations for two fluid systemsand enables accurate approximations of the weakly discontinuous velocity field and the discontinuous pressure.

An alternative way to handle discontinuities is regularization. In this thesis, consistent regularizations of Dirac delta functions with support on interfaces are proposed. These regularized delta functions make it easy to approximate surface tension forces in level set methods.

A new model for simulating contact line dynamics is also proposed. Capillary dominated flows are considered and it is assumed that contact line movement is driven by the deviation of the contact angle from its static value. This idea is used together with the conservative level set method. The need for fluid slip at the boundary is eliminated by providing a diffusive mechanism for contact line movement. Numerical experiments in two space dimensions show that the method is able to qualitatively correctly capture contact line dynamics.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011
Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:07
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-33111 (URN)978-91-7415-969-1 (ISBN)
Public defence
2011-05-20, Sal D3, Lindstedtsvägen 5, KTH, Stockholm, 14:21 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note
QC 20110503Available from: 2011-05-03 Created: 2011-04-28 Last updated: 2012-05-24Bibliographically approved

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