# Three-dimensional h-adaptive XFEM for Two-phase Incompressible Flow

## Outline

Two-fluid flows are of high relevance in many engineering applications, for example in the context of wave, bubble, and drop mechanics. A major challenge for two-fluid flow modelling is due to the existence of the moving interface across which the velocity and pressure fields can be discontinuous. In addition, there is a need to accurately capture the interface position as it evolves with time. If a fixed mesh is used, it is conceivable that the interface will cut through the elements during the course of its evolution. However, it is well-known that the polynomial function spaces of the standard FEM are not able to capture discontinuities within the elements. The aim of this project is to develop a robust two-fluid flow solver (in 2-D and 3-D) based on the XFEM and the level-set method. The XFEM enriches the pressure space so that jumps in the pressure field can be reproduced within elements without the need for remeshing. The level-set method employs an implicit description of the moving interface and allows for complicated topological changes, such as the merging or the pinching apart of bubbles. In addition, h-adaptivity is realized in this work in the form of mesh refinement via hanging nodes in the vicinity of the interface. This ensures a better resolution of the interface position and also the ability to capture steep gradients (which are known to exist in the velocity field in the vicinity of the interface for large viscosity ratios between the two fluids). The accuracy of the flow solver will be demonstrated through several two- and three-dimensional test cases.

## Timeline

Start of candidature :**01 Oct 2007**

Thesis submitted :

**16 Nov 2010**

Thesis defended :

**22 Mar 2011**

## Research Metrics

Published articles =**3**

Total citations =

**66**

Mean citations per publication =

**22**

## Publications

**Journals**

K.W. Cheng, T.P. Fries: Higher-Order XFEM for Curved Strong and Weak Discontinuities,

*Internat. J. Numer. Methods Engrg.*, 82(5), 564 - 590, 2010. (citation count)

T.P. Fries, A. Byfut, A. Alizada, K.W. Cheng, A. Schröder: Hanging Nodes and XFEM,

*Internat. J. Numer. Methods Engrg.*, DOI: 10.1002/nme.3024.

K.W. Cheng, T.P. Fries: Three-Dimensional

*h*-Adaptive XFEM for Two-Phase Incompressible Flow,

*Comp. Mthds, Appl. Mech. Engrg.*, accepted.

**Conference Proceedings/Presentations**

K.W. Cheng, T.P. Fries: Higher-Order Extended Finite Element Method for Arbitrary Weak

Discontinuities, 10th US National Congress on Computational Mechanics, Columbus, Ohio, USA, July 16-19, 2009

K.W. Cheng, T.P. Fries: A Systematic Study of Different XFEM-formulations with respect to

Higher-order Accuracy for Arbitrary Curved Discontinuities, International Conference on Extended Finite Element Methods – Recent Developments and Applications, Aachen, Germany, Sept. 28-30, 2009

K.W. Cheng, T.P. Fries: A comparison of various enrichment functions for weak discontinuities with respect to higher-order XFEM approximations, 3rd GACM Colloquium on Computational Mechanics in Hannover, Germany, Sep. 2009.

K.W. Cheng, T.P. Fries: h-adaptive XFEM for two-phase incompressible flow on a dynamic quadtree mesh, 4th European Conference on Computational Mechanics, Paris, France, May 2010.

T.P. Fries, K.W. Cheng:

**(keynote lecture)**Three-Dimensional h-Adaptive XFEM for Two-Phase Incompressible Flow, International Conference on Extended Finite Element Methods - Partition of Unity Enrichment: Recent Developments and Applications, Cardiff, United Kingdom, June 29 – July 1, 2011.

**Dissertation and Technical Reports**

K.W. Cheng, T.P. Fries: A Literature Review of the Extended Finite Element Method with Emphasis on Higher Order Approximations, AICES Technical Report, AICES-2008-8, RWTH Aachen University, Aachen, Germany, 2008.

K.W. Cheng: h- and p-XFEM with application to two-phase incompressible flow, Ph.D. Thesis, RWTH Aachen University, Aachen, Germany, 2011.

K.W. Cheng: An introduction to classical and statistical thermodynamics, a talk given in partial fulfillment of the requirements for the degree of Doctor of Engineering (Dr.-Ing.), Mechanical Engineering, RWTH Aachen University, Aachen, Germany, 2011.

## Gallery

1)** Triangulation-free higher-order accurate Gaussian quadrature.**

Distribution of integration points on a cut quadratic element: solid line represents the interpolated interface; * denotes the integration points and • denotes the nodal points. The number of integration points is large for the purpose of illustration.