Theory and Implementation of Local Boundary Conditions in Nonlocal Problems


Abstract: We present novel nonlocal governing operators with local boundary conditions used in nonlocal wave propagation. The construction of the operators is inspired by Peridynamics. The operators agree with the original Peridynamic operator in the bulk of the domain and simultaneously enforce local boundary conditions.  Our construction is systematic and easy to follow. The novel operators will help apply Peridynamics to problems that require local boundary conditions such as contact, impact, shear, and traction.

We have proved that the nonlocal diffusion operator is a function of the classical (local) operator. This observation opened a gateway to incorporate local boundary conditions to nonlocal problems on bounded domains. The main tool we use to define the novel governing operators is functional calculus, in which we replace the classical governing operator by a suitable function of it.  We reveal a close connection between the classical and nonlocal wave equations. Namely, the combination of the function piece (even and odd parts) and the extension type used in d'Alembert's formula is identical to that in the construction of our nonlocal operators. We present exact solutions and utilize the resulting error to verify numerical experiments. We explain the critical role of the Hilbert-Schmidt property in enforcing local boundary conditionsrigorously. We also present numerical experiments of impact problems that correspond to inhomogeneous local boundary conditions in the form of displacement and strain fields.

Biography: Burak Aksoylu is an ORAU Senior Fellow at CCDC Army Research Laboratory at Aberdeen Proving Ground, MD. He is also an adjunct faculty at Wayne State University in the Department of Mathematics. He has been a faculty member at TOBB University Economics and Technology, Ankara, Turkey in the Department of Mathematics and Louisiana State University in the Department of Mathematics and in the Center for Computation and Technology. His research has been funded by National Science Foundation, European Commission Marie Curie Program, and Research Council of Turkey. His first postdoc was at California Institute of Technology in the Department of Computing + Mathematical Sciences in the area of computer graphics with a focus on digital geometry processing. His second postdoc was at the University of Texas at Austin in the Institute for Computational Engineering and Sciences in the area of geosciences with a focus on robust preconditioning. He received his PhD at University of California, San Diego from the Department of Mathematics.

In the earlier part of his career, he has worked in the area of preconditioning and linear solvers for numerical solution of PDEs. His research concentrates on the design, analysis, and implementation of rigorous numerical methods. More recently, he has been studying problems with nonlocal governing operators, e.g., nonlocal wave equations. In particular, he has been working in the area of Peridynamics, a nonlocal formulation of continuum mechanics, used mostly for the simulation of crack propagation in materials. Although Peridynamics is an applied field, analytical aspects of his research are strong. For instance, he regularly employs operator theory tools in nonlocal problems and enjoys combining the powers of abstract theory with numerical computing.

Website: http://math.wayne.edu/~burak/