Physically-based Tempered Fractional-order Operators for Efficient Multiscale Simulations
Abstract
For multiscale problems involving complex material structure or interface geometries, it is widely expected that fractional derivatives should be related to fractal system properties (e.g., roughness, tortuosity, relaxation time distribution). However, the theoretical basis for this behavior has not been established, as the only definitive link between fractals and fractional-order differential equations (FDEs) is that the sample paths generated by the continuous stochastic processes have fractal dimensions that are functionally related to the fractional exponents. Further, the infinite non-locality implied by FDEs is problematic for many applications, particularly for multiscale calculations on heterogeneous domains. Such strong non-locality presents distinct challenges for efficient computation in fluid flow and FSI problems. The objectives of this project are: 1: Development of mathematics for upscaled FDEs to provide a formal basis for use of fractional-order operators for upscaling material deformation, momentum fluxes, and mass fluxes in complex materials and interface geometries that cannot be fully resolved over all relevant spatial and temporal scales. 2: Synthesis of existing experimental results for a) flow and diffusion in materials with complex structure, and b) turbulent flows over rough walls, porous interfaces, and canopies. 3: Development of physically and mathematically justified multiscale models by unification of rigorously upscaled equations (Thrust 1) to inform formulation of kernels and closure schemes in multiscale computational models with parameterization of these functions and testing of the resulting upscaled simulations using existing datasets (Thrust 2). Synthesis of findings from experiments and numerical simulations will allow us to evaluate the generality of numerical closure schemes, kernels, and shape functions for a wide range of problems exhibiting similar mechanisms of non-locality, and seek robust formulations of these functions based on material properties and system geometries.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 14, 2019
- Source ID
- W911NF1510569
Entities
People
- Aaron I Packman
Organizations
- Army Contracting Command
- Northwestern University
- United States Army