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Simulation of cracks with XFEMand hanging nodes
Alaskar Alizada, Thomas-Peter Fries
Research group: “Numerical methods for discontinuities“
ECCOMAS XFEM Conference, Aachen, 28-30 September, 2009
Alaskar Alizada Simulation of cracks with XFEM and hanging nodes Slide: 2
Motivation
Refined meshes and hanging nodes
XFEM formulation for hanging nodes
Numerical results
Conclusions & Outlook
Overview
Alaskar Alizada Slide: 3
Motivation
Simulation of cracks with XFEM and hanging nodes
XFEM can capture jumps and kinks within elements byenriching 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
Alaskar Alizada Slide: 4
Motivation
Simulation of cracks with XFEM and hanging nodes
Closed interface Open interface
Refined meshes
Alaskar Alizada Slide: 5Simulation of cracks with XFEM and hanging nodes
Heaviside enrichmentalong the crack path
Branch enrichment functionsat crack-tip
+Mesh refinement
at crack-tip
Heaviside enrichmentalong the crack path
+
Motivation
Alaskar Alizada Slide: 6
Refined meshes and hanging nodes
Simulation of cracks with XFEM and hanging nodes
Allowed element types in mesh.
Only one node in the middle of each edge is allowed.
Alaskar Alizada Slide: 7
Refined meshes and hanging nodes
Simulation of cracks with XFEM 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
Alaskar Alizada Slide: 8
XFEM formulation for hanging nodes
Simulation of cracks with XFEM and hanging nodes
Heaviside enrichment function
where is a level-set function.
- I*
Alaskar Alizada Slide: 9Simulation of cracks with XFEM and hanging nodes
Constrained approximation. Hanging nodes have no DoF.The shape functions at hanging nodes are interpolated.
XFEM formulation for hanging nodes
A
B
CD
E
FEM:
XFEM: and have to be replaced but how ? Not trivial.
Alaskar Alizada Slide: 10Simulation of cracks with XFEM and hanging nodes
XFEM formulation for hanging nodes
= +
Hanging nodes also have DoF. The shape functions athanging nodes - .
If then standard bi-linear shape function are used forregular nodes, then .
Alaskar Alizada Slide: 11Simulation of cracks with XFEM and hanging nodes
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:
Alaskar Alizada Slide: 13Simulation of cracks with XFEM and hanging nodes
XFEM formulation for hanging nodes
A
B
CD
E
E
D
A
C
B
Alaskar Alizada Slide: 14
Numerical results
Simulation of cracks with XFEM and hanging nodes
Edge crack problem
Crack mode I
Alaskar Alizada Slide: 15
Numerical results
Simulation of cracks with XFEM and hanging nodes
refined meshrefinement level = 2
deformed mesh
Alaskar Alizada Slide: 16Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 17Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 18Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 19Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 20Simulation of cracks with XFEM and hanging nodes
Numerical resultsMixed mode test case: SIF
Alaskar Alizada Slide: 21Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 22Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 23
shear edge crack [Belytschko, Black, 1999]
Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 24
shear edge crack [Belytschko, Black, 1999]
Simulation of cracks with XFEM and hanging nodes
Numerical results
refined mesh deformed mesh
Alaskar Alizada Slide: 25Simulation of cracks with XFEM and hanging nodes
Numerical results
Alaskar Alizada Slide: 26
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.
Simulation of cracks with XFEM and hanging nodes
This allows to use the proposed idea even if the analyticalsolution is not known.
Alaskar Alizada Slide: 27
Thank youfor your attention!
www.xfem.rwth-aachen.de
Simulation of cracks with XFEM and hanging nodes
