PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Becky Halpenny.

Title:"Mathematical and Numerical Analysis of Nonlocal Models"
Seminar:Ph.D. Oral Comprehensive Examination
Speaker:Xiaochuan Tian, Adviser: Qiang Du, Penn State
Abstract:
The peridynamic (PD) model is an integral-type nonlocal model of materials which provides an alternative set-up to classical continuum mechanics based on partial di erential equations (PDEs). In this talk, numerical methods for nonlocal di usion that can be applied to various singular kernels will be developed and basic properties of these discrete schemes such as discrete maximum principles will be discussed. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. The latter reveals that while some methods tend to converge to the intended solutions, other methods lead to the possibility of convergence to unintended local limits. Thus we reveal the risks of discretizing nonlocal models when the modeling parameter is tied to the discretization parameter. Motivated by this phenomenon, we develop an abstract mathematical framework to help us nd asymptotically compatible schemes as robust discretization of nonlocal models that can be applied to a general state-based peridynamic system under minimum regularity assumptions. The framework may be used to guide computational studies of nonlocal problems; in particular, we develop a non-conforming method for nonlocal problems and its convergence can be proved within the framework. In order to apply the framework to the convergence of non-conforming methods for nonlocal problems, a more general compactness result of limiting behaviors of nonlocal energy norms is needed, which is an extension of a compactness result of Bourgain, Brezis and Mironescu.

Room Reservation Information

Room Number:103 Osmond Laboratory
Date:11 / 12 / 2013
Time:09:30am - 11:15am