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 Nicolas Moës 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 Belytschko 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 1999, 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 physics-numerics-geometry 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 Method (FMM). 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.