Past Event: Oden Institute Seminar

Spatially compatible meshfree discretization

Nat Trask, Sandia National Laboratories

Abstract

While meshfree methods have long promised a natural means of handling problems with large deformation with little numerical dissipation, they have struggled to obtain properties that are often taken for granted in mesh-based methods. From the perspective of compatible discretization, the lack of a mesh means that there is no chain complex upon which to develop a discrete exterior calculus. For this reason, particle methods that are simultaneously able to achieve high-order accuracy and discrete conservation principles have remained elusive. In this talk, we will present recent work from the Compadre (compatible particle discretization) project where we seek to develop meshfree discretizations that achieve these properties. In the first part of the talk, we present a computationally efficient meshfree Gauss divergence theorem which assigns virtual notions of volume and area to particles. With a consistent summation-by-parts theorem in hand, we then develop a meshfree analogue to the finite volume method and demonstrate its robustness when considering Darcy flows with jumps in material properties. In the second part of the talk, we present a meshfree approach to remedy well-known issues with numerical discretizations of peridynamics. While the non-local continuum theory of peridynamics provides an attractive framework for studying fracture with reduced regularity restrictions, particle discretizations of peridynamics fail to obtain a notion of asymptotic compatibility in which the discrete non-local solution recovers the exact local solution as the non-local interaction is reduced. We present a new optimization-based strong form method constructed to enforce reproduction of a given class of nonlocal operators, for which we prove asymptotic compatibility and demonstrate its implementation in a standard engineering workflow. Bio Nat Trask is a senior member of technical staff at Sandia National Laboratories in the Center for Computation and Visualization, where he previously held the National Science Foundation MSPRF postdoctoral position working with Pavel Bochev. He completed his PhD in the division of applied mathematics at Brown University in 2015, where he worked with Martin Maxey and George Karniadakis. Prior to that, he obtained a masters and bachelors degree from the department of mechanical engineering at the University of Massachusetts Amherst, where he studied fuel injection and combustion.