Cracks and crack propagation with XFEM and hanging nodes in 2D€¦ · Alaskar Alizada Slide: 3 Motivation XFEM can capture jumps and kinks within elements by enriching the approximation

Cracks and crack propagation with XFEM and hanging nodes in 2D

Alaskar Alizada, Thomas-Peter Fries

Research group: “Numerical methods for discontinuities“

ECCM 2010, Paris, 16-21 May, 2010

Alaskar Alizada Cracks and crack propagation with XFEM and hanging nodes in 2D Slide: 2

  Motivation

  XFEM formulation for hanging nodes

  Shape functions at hanging nodes

  Special treatment of crack-tip element

  Numerical results

  Conclusions & Outlook

Overview

Alaskar Alizada Slide: 3

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.



Motivation


Goals of the refinement:

1  Capture high gradients near the interface

Avoid model-dependent enrichment functions (branch enrichments)


Motivation


2  Resolve the mesh for accurate results

refined mesh coarse mesh

Goals of the refinement:


XFEM formulation for hanging nodes

- I*


Heaviside enrichment function

where is a level-set function.


Shape functions at hanging nodes


  Hanging nodes have no DoFs = „Constrained approximation“:

a.  Lagrange multipliers

b.  Connectivity matrix

c.  Special conforming shape functions at the regular nodes. Hanging nodes have no shape functions can not be enriched


= +

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

  Hanging nodes have DoFs.

a.  Special conforming shape functions are used [Gupta, 1978]




  Shape functions on regular nodes should be changed for the partition of unity property:

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

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



Alaskar Alizada Slide: 10 Cracks and crack propagation with XFEM and hanging nodes in 2D

  Fries, Byfut, Alizada, Cheng, Schröder: Hanging nodes and XFEM, IJNME, submitted.

Shape functions at hanging nodes   Conforming set of shape functions defined at hanging

nodes with PU property.


Special treatment of crack-tip element



Special treatment of crack-tip element

[Zi, Belytschko, 2003]

cracked

uncracked



Numerical results Angled inclined crack problem

[Liu, Xiao, Karihaloo, 2004]



Numerical results

refined mesh, 30°



Numerical results



Numerical results



Numerical results Double cantilever beam [Belytschko, Black, 1999]



Numerical results



Numerical results


Single edge notched beam [Areias, Belytschko, 2005]


Numerical results



Numerical results



Numerical results



Conclusions   Using XFEM and refinement for stationary cracks as well as for

crack propagation problems show very good approximation results.

  Special conforming shape functions with partition of unity property were introduced considering hanging nodes as standard DoF.

  The application of the proposed idea for other material models, such as cohesive crack model, will be done

(WCCM 2010).


  Special procedure for the crack-tip element is applied.

Outlook


Thank you for your attention!

www.xfem.rwth-aachen.de


Documents

Cracks and crack propagation with XFEM and hanging nodes in 2D€¦ · Alaskar Alizada Slide: 3 Motivation XFEM can capture jumps and kinks within elements by enriching the approximation