The project involves the modeling, analysis, algorithmic invention, and numerical analysis of novel models for three physical systems, namely diffusion, solid mechanics, and plasmas. In the first system, the PI considers a nonlocal, integral model for non-Fickian (or anomalous) diffusion. The model has as special cases fractional Laplacian and fractional derivative models but generalizes these models in several ways such as allowing for spatial heterogeneity and anisotropy, less smooth solution behavior, and a simple means for treating problems posed on bounded domains. The project addresses several issues that are crucial to the efficient, robust, and accurate approximation of the nonlocal diffusion model. In particular, the effects of truncating kernels and data so that computations can be done on finite domains are analyzed, combined finite element/domain and data truncation error estimates are obtained, and a goal oriented, adjoint equation-based grid adaptation methodology is developed, analyzed, and tested. In the second system, the PI considers the nonlocal, spatial derivative free peridynamics model for solid mechanics. The model allows for discontinuous solutions so it is especially well suited for modeling defects. As such, its efficacy has been demonstrated in several sophisticated applications, including the fracture and failure of composites, crack instability, fracture of polycrystals, and nanofiber networks. The project involves the development of a computational methodology that takes full advantage of the multiscale properties inherent in peridynamics and which results from the phenomenological horizon parameter that limits the extent of interactions. In the third system, both the Vlasov-Poisson and cold-ion systems for modeling the expansion of ions from high to low-density regions are considered. The project includes the development of a new cold-ion model that remains valid beyond the time of singularity creation, showing that solutions of the new model remain the limit (as the ion temperature tends to zero) of solutions of the Vlasov-Poisson system, studying analytically and computationally the transition that occurs as the density ratio decreases from solutions having no singularities to ones that do, treating problems with multiple species of ions, and developing efficient numerical methods for two and three-dimensional settings.
The project addresses fundamental issues that arise in the mathematical and computational treatment of three physical systems that are of huge importance in a wide variety of applications. These include but are not limited to the nucleation and propagation of defects, (cracks, delaminations, etc.) in solid bodies (airplane wings, nuclear reactor containment vessels, etc.); anomalous diffusive behavior observed in subsurface water, oil, and gas flows, in animal foraging behaviors, in polymeric flows, in exotic materials, etc.; and in ionized flows in lasers, space propulsion systems, supernovae, etc. Obtaining useful information about such complex systems requires massive computational efforts which can be greatly aided by improvements in mathematical models, qualitative theoretical information about solutions of those models, and, most of all, by better, more efficient and more accurate computational algorithms. Because of the wide-ranging settings that the project impacts, the results obtained will be of great interest to scientists, engineers, and policy makers involved in, among other things, the theoretical study of physical phenomena, in the design and manufacture of new devices, and in the assessment of risks and design of remediations.
