Time integration scheme for Dynamic Crack Propagation in XFEM?

Hello,

I am trying to implement the time integration scheme proposed by Combescure et. al [1] to simulate a dynamic crack propagation problem in XFEM (2D edge crack under impact loading). But I have an interesting question to ask.

In their time integration scheme [1], to take care of the changing discretization, at each time integration step (or crack propagation step) the authors introduce new enrichment degrees of freedom (while keeping the old enrichment degrees of freddom as well). My question is, if a crack tip propagates let's say 1000 times (i.e. 1000 crack propagation steps) within an element, then what will be the correct displacement approximation for that element after 1000 time steps?

1. Uh=∑Niui + ∑(HNj)bj  + ∑(FNk)ck

Where, i=nodes with conventional degrees of freedom ui

            j=nodes with heaviside degrees of freedom bj

            k= nodes with crack-tip degrees of freedom ck (ck corresponds to crack-tip degrees of freedom after 1000th time step).

2. Uh=∑Niui + ∑(HNj)bj  + {∑(FNk1)ck1 + ∑(FNk2)ck2 + ∑(FNk3)ck3 + .....................+ ∑(FNk1000)ck1000}

where, ck1 =crack tip degrees of freedom after 1st time step/crack propagation step

           ck2 =crack tip degrees of freedom after 2nd time step/crack propagation step

...............................

...............................

            ck1000 =crack tip degrees of freedom after 1000th time step/crack propagation step

From time integration scheme proposed by Combescure et. al [1], it looks like they approximate with with equation (2) mentioned above! While, looking at the paper by Moes et.al [2], it looks like their approximation will be similar to equation (1)! 

Which approximation is correct? Or if they are equivalent? (if yes, how?)

I will really appreciate any insight/suggestions! Thank you all.

Ali Bhuiyan

References:

[1]J. Rethore, A Gravouil, A. Combescure, An energy-conserving scheme for dynamic crack growth using the extended finite element method, International Journal for Numerical Methods in Engineering, 2005, 63:631-659

[2]N. Moes, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 1999, 46, 131-150