Talk:Extended finite element method

Who invented this method?

The article suggests that the method was developed in by Ted Belytschko and collaborators. However, the partition of unity method is described by Babuska and Melenk in a 1997 paper. Is that the origin of XFEM/PUFEM? RuppertsAlgorithm (talk) 13:34, 22 December 2008 (UTC)

You are absolutely right. Before Belytschko and Co. started publishing on this topic, giving it their own name (XFEM), and making it usable especially for the fracture modeling community, Babuska and Melenk had laid the foundations and the theoretical concept with the Partition of Unity Method. This was already published in 1996 (see, J.M. Melenk and I. Babuska. The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Eng., 139:289–314, 1996). Feel free to improve the article. If you ask me, it should be moved from XFEM to PUM anyway. Tomeasy _{T C} 14:25, 22 December 2008 (UTC)

Actually, the partition-of-unity idea is only one part of the X-FEM. The X-FEM was the first to introduce the concept of introducing geometric features, such as cracks or voids, that are not explicitly meshed. This idea is combined with the PUM to form X-FEM. That is why the X-FEM has become so popular. —Preceding unsigned comment added by 134.253.26.6 (talk) 00:14, 14 January 2009 (UTC)

Why extended?

The article doesn't explain in much detail how this method differs from standard finite elements. Is it just standard finite elements with a certain class of discontinuous or non-smooth basis functions, or is there more to it than that? —David Eppstein (talk) 22:54, 6 July 2013 (UTC)

Someone can correct me if this is wrong, but my understanding is that the "standard finite elements" involve constructing spaces that span polynomial spaces so that the Bramble-Hilbert lemma can be applied to guarantee convergence. Extended finite elements involve adding more basis functions not designed to span polynomial spaces (needed for approximation estimates) but for building equation specific information into the solution. If you look at the abstract of the Babuska-Melenk paper, they describe the PUM method as "a new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved". The X-FEM literature is a little more complex, involving building geometric features that don't have to be meshed into the solution (as another contributer mentions above). RuppertsAlgorithm (talk) 00:50, 8 July 2013 (UTC)