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Enriched DOFs in XFEM‍

Dear all,
 
Any one has idea on if the enriched DOFs should be
constrained in XFEM?  Numerous numerical examples from literature have
shown that the enriched DOFs needn't be constrained.
 
But what if
only one element is used in computation? I tried 1-D truss element  and
2-D quadrilateral element with heaviside enrichment function. For the
1-D truss element, the size of the element stiffness matrix is 4*4 and
the rank of the stiffness matrix is 3 which indicates only one DOF is
required to be constrained just like the standard FEM case. I also find that the constrained DOF must be
the standard DOF not the enriched DOF, otherwise the stiffness matrix
would still be singular. For the 2-D element, the size of the element
stiffness matrix is 16*16 and the rank of it is 10, which means 6 DOFs
has to be constrained.The 6 constrained DOFs include both standard and
enriched DOFs.This is different from the standard FEM where only 3 DOFs needs to be constrained.
 
Anyone can explain this?
 
Thanks
 
regards
Xiaodong

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