Physically-based tempered fractional-order operators for efficient multiscale simulations

In the last two decades, a variety of deterministic and stochastic partial differential equations (PDEs) have been generalized using fractional-order derivative operators to simulate “non-local” phenomena – processes that violate typical continuum assumptions because of strong heterogeneity, fracticality, or correlations. Fractional-order derivatives provide convenient non-local operators that alleviate the scale-dependence of conventional effective continuum parameters by providing an appropriate description of the ensemble effects of large motions and waiting times relative to the domain size and simulation time Improved theory and numerical methods are needed to support development of efficient multi-scale models with physically justified sub-gridscale closure schemes. The goal of this proposal is to unify physics-based mathematical formulations of fractional-order dynamics with high-performance computational methods to enable multiscale simulation of fluid flow, scalar transport, and fluid-solid interactions with complex material structure and interface geometries. This work will advance the fundamental basis for application of FDEs and the state-of-the-art for high-performance simulation of a wide range of complex multiscale, multiphysics problems. We propose to develop new theoretical and numerical methods for critical problems in fluid and solid mechanics involving complex material structure and interface geometries. We also propose to link fractional-order operators across complex interfaces to solve FSI problems that involve coupling between fluid flow, material structure, and material fluxes or deformations. We will develop unstructured-mesh and mesh-free approaches for this class of problems, as they are well suited to handling the complex geometries that yield anomalous flow and transport. We will determine strategies for efficiently implementing fractional-order operators in highly parallelized computational frameworks and on unstructured meshes, as well as Lagrangian nonlocal particle methods for multiphase material systems. To improve model fidelity, it is essential to establish the physical basis of fractional-order operators for important classes of problems. We will focus initially on systems that are clearly understood to yield non-locality in continuum models, particularly problems that involve multi-scale heterogeneity (e.g., fractal material structure and interface geometries) that are known to generate non-locality, yet are extremely difficult to resolve explicitly in numerical models. We will explore upscaled forms of the Voigt, Navier-Stokes, and Advection-Diffusion equations that include fractional-order operators that formally incorporate underlying system complexity. This will provide the essential basis for development of physically-justified closure schemes based on scaling properties of component materials and interfaces. Further, the associated information on scaling cut-offs will provide a basis for development of tempered solutions that constrain the degree of non-locality in multi-scale computational models. We will conduct experiments to elucidate how complex material structure and interface geometries produce non-locality, establish the physical basis for parameterizing fractional-order operators, and provide test data for multi-scale computational models. This proposed work builds on our current conceptual and theoretical understanding of key processes that produce non-locality (Packman, Magin, Bolster, Schumer), our ability to derive upscaled FDEs reflecting these processes (Magin, Schum