Numerical computation of cavitation in nonlinear elasticity

This talk addresses some analytical aspects of the numerical computation of cavitation. The aim is to compute the singular minimizers of the model of Sivaloganathan and Spector (2000), in which cavities are allowed only at a finite number of prescribed flaw points. A regularization of the model was provided by Sivaloganathan, Spector and Tilakraj (2006), for the problem of a single cavity. Some obstacles must be overcome in the case of multiple cavities, related to the invertibility of deformations, and to the non-interpenetration of matter.

Further difficulties were encountered in computations carried out with X. Xu, namely, the inefficiency of the finite element method for this problem, and the mesh-degeneracy at the cavity surfaces. This motivated the use of the non- conformingfiniteelementmethod,andthe additionofasuitablestabilization term. These issues will be described, and some striking new observations on cavitation will be presented.