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An Immersed Finite Element Method and its Application to Multiphase Problems
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

Multiphase flows are frequently encountered in many important physical and industrial applications. These flows are usually characterized by very complicated structure that involves free moving surfaces inside the fluid domain and discontinuous or even singular material properties of the flow. The application range for the multiphase flow phenomena is extremely wide, ranging from processing industry to environmental problems, from biological applications to food industry and so on. Unfortunately, due to the inherent complexity of these problems, their solution proved to be a considerable challenge. Thus, in the many applications, the predictive capability and physical understanding must rely heavily on numerical models.

In this thesis we develop and analyze a finite element based method for the solution of multiphase problems. This thesis consists of four papers. In paper 1 we develop our finite element based method for the elliptic interface problems. The interface jump conditions that are present due to the discontinuity of the coefficients and presence of the singular forces are derived. Using these jump conditions, we enrich the finite element spaces in order to account for the irregularities in the flow. The resulting method was applied to the interface Stokes problem, modeling a thin elastic rubber band immersed in the homogeneous fluid. In order to apply the introduced method, the interface Stokes problem was rewritten as a sequence of three Poisson problems, one for the pressure and two for the velocity components. Paper 2 is an extension of the ideas used in paper 1. Namely, third order Hermitian polynomials are used as basis functions, their modification according to the interface jump conditions is presented and analyzed, both theoretically and numerically. The rigorous error analysis of the introduced method for two-dimensional elliptic problems is presented in paper 3. The results imply that our method is second order accurate in the L2 norm. Finally, paper 4 concerns with the extension of our method to a coupled interface Stokes problem, that contains both singular forces and discontinuities in the material properties. An application to the Rayleigh-Taylor instability problem is presented.

Place, publisher, year, edition, pages
Stockholm: KTH , 2007. , vii, 25 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2007:15
Keyword [en]
fluid and plasma physics, applied mechanics, computer science
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4514ISBN: 978-91-7178-775-0 (print)OAI: oai:DiVA.org:kth-4514DiVA: diva2:12630
Public defence
2007-11-12, Sal Fe, KTH, Lindstedtsvägen 26, Stockholm, 13:00
Opponent
Supervisors
Note
QC 20100806Available from: 2007-10-23 Created: 2007-10-23 Last updated: 2012-03-22Bibliographically approved
List of papers
1. A Fixed Grid Finite Element Method for Elliptic Interface Problems
Open this publication in new window or tab >>A Fixed Grid Finite Element Method for Elliptic Interface Problems
2006 (English)In: Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, 2006, 1-23 p.Conference paper, Published paper (Refereed)
Abstract [en]

A finite element method for two-dimensional elliptic interface problem is presented. Due to the presence of these interfaces the problem will contain discontinuities in the coefficients and singular source terms that are represented by delta functions along the interface. As a result, the solution to the interface problem and its deratives may have jump discontinuities. The new method is specifically designed to handle this feature of the solution in the context of non-interface fitted grids.

The main idea is to modify the standard basis function in the vicinity of the interface such that the jump conditions are well approximated. The resulting finite element space is, in general, non-conforming. The interface itself is represented by a set of Lagrangian markers together with a parametric description connecting them. To illustrate the abilities of the method, numerical tests are presented. For all the considered test problems, the new method has been shown to have super-linear or second order of convergence. Our approach is also compared with the standard finite element method. Finally, the method has been successfully applied to the Stokes interface problems.

Keyword
elliptic equations; finite element method; interface; singular source term; discontinous coefficients; immersed interface method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-7561 (URN)
Conference
the International Conference on Numerical Analysis and Applied Mathematics, ICNAAM, Greece, 2006
Note
QC 20100806Available from: 2007-10-23 Created: 2007-10-23 Last updated: 2010-08-06Bibliographically approved
2. Highly accurate finite element method for one-dimensional elliptic interface problems
Open this publication in new window or tab >>Highly accurate finite element method for one-dimensional elliptic interface problems
2009 (English)In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 59, no 1, 119-134 p.Article in journal (Refereed) Published
Abstract [en]

A high order finite element method for one-dimensional elliptic interface problems is presented. Due to presence of these interfaces the problem will contain discontinuities in the coefficients and singularities in the right hand side that are represented by delta functional with the support on the interfaces. As a result, the solution to the interface problem and its derivatives may have jump discontinuities. The proposed method is specifically designed to handle this features of the solution using non-body fitted grids, i.e. the grids are not aligned with the interfaces.The finite element method will be based on third order Hermitian interpolation. The main idea is to modify the basis functions in the vicinity of the interface such that the jump conditions are well approximated. A rigorous error analysis shows that the presented finite element method is fourth order accurate in L-2 norm. The numerical results agree well with the theoretical analysis. The basic idea can easily be generalized to other finite element ansatz functions.

Keyword
Discontinuous coefficients; Elliptic interface problem; Finite element method; Immersed interface method; Singular source; Electric network analysis; Error analysis; Probability density function; Problem solving; Basic ideas; Basis functions; Discontinuous coefficients; Elliptic interface problem; Elliptic interface problems; Finite elements; Fourth orders; Hermitian; High order finite element methods; Immersed interface method; Interface problems; Jump conditions; Jump discontinuities; Numerical results; Right hands; Singular source; Theoretical analyses; Third orders
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-7562 (URN)10.1016/j.apnum.2007.12.003 (DOI)000261485800007 ()2-s2.0-55149091553 (Scopus ID)
Note
QC 20100806. Uppdaterad från Submitted till Published 20100806.Available from: 2007-10-23 Created: 2007-10-23 Last updated: 2017-12-14Bibliographically approved
3. An immersed finite element method and its convergence for elliptic interface problems with discontinuous coefficients and singular sources
Open this publication in new window or tab >>An immersed finite element method and its convergence for elliptic interface problems with discontinuous coefficients and singular sources
2007 (English)Report (Other academic)
Abstract [en]

This paper is concerned with the analysis of an immersed ginite element method for two dimensional elliptic interface problems. The main idea of the method is to use specifically designed macro elements in the vicinity of the interface, such that the jump conditions are well approximated. In general, the resulting immersed finite element space is non-conforming. It is shown that the presented method is second order accurate in L² norm. The provided numerical results agree with the theoretical estimates.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. 23 p.
Series
TRITA-CSC, ISSN 0348-2952 ; 2007:03
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-7563 (URN)KTH/NA-07/03-SE (ISRN)
Note
QC 20100806Available from: 2007-10-23 Created: 2007-10-23 Last updated: 2010-08-06Bibliographically approved
4. A non-conforming finite element method for interface Stokes problems and its application to two-phase Rayleigh-Taylor instability with solid obstacles
Open this publication in new window or tab >>A non-conforming finite element method for interface Stokes problems and its application to two-phase Rayleigh-Taylor instability with solid obstacles
2007 (English)Report (Other academic)
Abstract [en]

In this paper we establish an immersed finite element method for the solution of interface Stokes problems. The main idea of the method is to use a fixed, uniform mesh everywhere over the computational domain except the vicinity of the interface, where specifically designed macro elements are employed, such that the jump conditions are well approximated. In general, the resulting immersed finite element space is non-conforming. The interface itself is represented with the help of Lagrangian markers. The capabillity of the method is illustrated in the case of a Rayleigh-Taylor two-phase flow instability problem with solid obstacles governed by the Stokes equations.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. 28 p.
Series
TRITA-CSC, ISSN 0348-2952 ; 2007:4
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-7564 (URN)KTH/NA-07/04-SE (ISRN)
Note
QC 20100806Available from: 2007-10-23 Created: 2007-10-23 Last updated: 2010-08-06Bibliographically approved

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Citation style
  • apa
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Output format
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