Two phase flows simulation with XFEM and hanging nodes · Alaskar Alizada Slide: 3 Motivation XFEM can capture jumps and kinks within elements by enriching the approximation space

Alaskar Alizada, Thomas-Peter Fries

Research group: “Numerical methods for discontinuities“

3rd GACM, Hannover, 21-23 September, 2009

Two phase flows simulationwith XFEM and hanging nodes

Alaskar Alizada Slide: 2

Motivation

Refined meshes and hanging nodes

XFEM formulation for hanging nodes

Numerical results

Conclusions & Outlook

Overview

XFEM and hanging nodes


Motivation XFEM can capture jumps and kinks within elements by

enriching the approximation space. Therefore, in general, no mesh manipulation is needed. However, in addition to jumps and kinks, high gradients at

interface can appear.

Therefore, mesh refinement and XFEM can be useful.

path



Motivation

Closed interface Open interface

Refined meshes



Heaviside enrichmentalong the crack path

Branch enrichment functionsat crack-tip

+Mesh refinement

at crack-tip

Heaviside enrichmentalong the crack path

+

Motivation



Refined meshes and hanging nodes Allowed element types in mesh.

Only one node in the middle of each edge is allowed.



Refined meshes and hanging nodes

... it ensures the sparseness of the global systemmatrix (small bandwidth).

This requirement is desired, because...

Note: Also elements next to the cut elements can beaffected by the refinement algorithm.

... it allows no jumps of the element sizes in the mesh.

hh/8




Heaviside enrichment function

where is a level-set function.

- I*



Constrained approximation. Hanging nodes have no DoF.The shape functions at hanging nodes are interpolated.


A

B

CD

E

FEM:

XFEM: and have to be replaced but how ? Not trivial.




= +

Hanging nodes also have DoF. The shape functions athanging nodes - .

If then standard bi-linear shape function are used forregular nodes, then .



XFEM formulation for hanging nodes Shape functions on regular nodes should be changed for

the partition of unity property:

where are regular nodes, - hanging nodes, - new shape functions for regular nodes, - shape functions for hanging nodes.

Now hanging nodes have DoFs and the partition of unityproperty is fulfilled.


Alaskar Alizada Slide: 12Simulation of cracks with XFEM and hanging nodes

XFEM formulation for hanging nodes Proof the partition of unity property:



A

B

CD

E

E

D

A

C

B



Numerical results

Edge crack problem

Crack mode I



Numerical resultsrefined mesh

refinement level = 2deformed mesh



Numerical results



Numerical results



Numerical results



Numerical results



Numerical resultsMixed mode test case: SIF



Numerical results



Numerical results



shear edge crack [Belytschko, Black, 1999]

Numerical results



shear edge crack [Belytschko, Black, 1999]

Numerical results

refined mesh deformed mesh



Numerical results



Conclusions & Outlook Using XFEM and refinement in the vicinity of discontinuities

shows very good approximation results.

Special shape functions with partition of unity propertyintroduced consider hanging nodes as standard DoF.

The application of the proposed idea for other materialmodels, where analytical solution is not known, will be donein the future.

This allows to use the proposed idea even if the analyticalsolution is not known.



Thank youfor your attention!

www.xfem.rwth-aachen.de


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Two phase flows simulation with XFEM and hanging nodes · Alaskar Alizada Slide: 3 Motivation XFEM can capture jumps and kinks within elements by enriching the approximation space