Summer Program June 25-29, 2012: Nonlocal Continuum Models for Diffusion, Mechanics, and Other Applications

NEW DATE:  June 25-29, 2012

This program was held at the Radisson RTP  in Research Triangle Park, NC. The location is in close proximity to SAMSI.

Research foci

Nonlocal discrete models are common in many applications. For example, molecular dynamics and other particle methods having interactions that extend beyond nearest neighbors are in common use. Less popular are nonlocal continuum models which are very much dominated in their usage by local continuum models and especially differential equation models.

However, in recent years, there has been burgeoning interest in mathematical, scientific, and engineering circles in nonlocal continuum models, especially for solid mechanics, diffusion, and wave propagation. This interest is motivated by the desire to model singular or anomalous behavior such as cracks and fracture in solids and, more generally, by the need to develop multiscale models, that is, models that are valid and tractable over a wide range of temporal and spatial scales. As a result, there are by now researchers all over the world investigating diverse aspects of nonlocal continuum models.

Participants in the workshop discussed modeling, mathematical, statistical, computational, and applications issues such as

  • kernel choices
  • connections between nonlocal continuum models and discrete models such as Molecular Dynamics
  • well posedness of the equations
  • limiting behaviors of solutions
  • Finite element and other discretization methods
  • efficient solution methods for discretized systems
  • uncertainty quantification
  • applications including but not limited to mechanics, image processing, graphs, diffusion, and wave propagation

Uncertainty quantification (UQ) through nonlocal models is especially intriguing because of the very different nature of local and nonlocal models with respect to such effects as dispersion, smoothing or the lack thereof.

Description of activities

The goal of this summer workshop was to bring together such researchers from the mathematical, statistical, computational, scientific, and engineering communities to first talk about their interests and research and then, by having groups of participants work together to establish lasting, synergistic connections that lead to new research. We were also interested in having participants who may not be involved in research into nonlocal continuum models, but that have expertise in areas that will influence future research into those models.