Reminiscences . . . .
Exploring the X-FEM
My fortuitous involvement and subsequent interest in the eXtended Finite Element
Method (X-FEM) began
in early 1999 when I joined
Northwestern as a post-doc. It was an opportune time for
had just conceived the idea of using the H(x) step function
to model the crack discontinuity. This has proven to be an elegant
solution and an
improvement over Thomas Black's pioneering thesis work
in 1998 (see
and Black (1999)), where the near-tip asymptotic fields are used within a
partition-of-unity method to model the entire crack.
Nicolas and John Dolbow
carried out its implementation for 2-d cracks.
Later on, Christophe Daux (a student from
Cachan) spent the summer and
extended the method for voids, and branched and intersecting cracks.
I completed the 3-dimensional implementation for cracks in the
summer of 1999,
and the results were indeed very promising.
A search for an apt title during one of the many discussions on
the technique brought us to a converged solution: X-FEM.
John completed a few more fracture applications (Mindlin-Reissner plates,
cracks under compression) which clearly demonstrated the versatility
of the technique in 2-d fracture; in Fall 1999, he headed to
Duke University. In Fall of
Nicolas and I spent
a few months involved in rather
mindless conversations where we did more
talking than coding! After a lot of talk, we
converged to a design that kept
the key components of
distinct with interactions only where required.
This fit our needs and goals so that more mechanics-applications
and numerical techniques (such as level sets) could be readily interfaced
within the framework (C++ code).
Concurrently, we also got involved with
David Chopp in
Applied Mathematics for it
appeared that the X-FEM interface (aha!) to level set
techniques was the next logical step; David had earlier
explored with Brian Moran and
Zulfiqar Ali the use
of level sets for finite element crack modeling, but finite element
themselves weren't a natural tool for the link-up. With
the arrival of the X-FEM, the timing seemed apt for the linkage. David
had already conceived the level set representation for two-dimensional cracks,
and Magdalena Stolarska was implementing his idea at that time.
Magda and Nicolas pursued its
interface to the X-FEM for 2-D crack growth simulations;
I explored the modeling of material interfaces with David and also
worked with him on planar crack propagation in 3-D using the Fast Marching
Completed the material interface study in early 2000 and the X-FEM/FMM work
in the summer of 2000, before heading to Princeton for a research associate
position in July 2000, enroute to UC Davis in January 2001 where I
am currently located.