Homogenization of time-fractional diffusion equations with periodic coefficients

We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data $a(x)\in L^{2}(D)$ in a bounded domain $D\subset \mathbb{R}^d$ with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient $\kappa^{\epsilon}(x)$ is smooth and periodic with the period $\epsilon>0$ being sufficiently small. We derive that its first order approximation has a convergence rate of $\mathcal{O}(\epsilon^{1/2})$ when the dimension $d\leq 2$ and $\mathcal{O}(\epsilon^{1/6})$ when $d=3$. Several numerical tests are presented to show the performance of the first order approximation

Model Reduction Techniques and Parareal Algorithm for Multiscale problems

A broad range of scientific and engineering problems involve multiple scales. For example, composite material properties and subsurface properties can vary over many length scales. Direct numerical methods of multiscale problems is often difficult due to the fact that a very fine mesh of the domain is required to reflect the heterogeneous coefficients. From a computational point of view, the major challenge to solve these problems is the size of the computation, even with the aid of supercomputers. On the other hand, from an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple scale systems, such as the effective conductivity, permeability, elastic moduli and eddy diffusivity. Therefore, it is desirable to develop fast and effective numerical methods that capture the small scale effect on the large scales, but do not require resolving all the small features. There has been extensive research effort devoted to developing computational methods for multiscale problems. Among the most popular and developed techniques are homogenization method, multiscale finite element methods and parareal algorithm. The goal of homogenization methods and multiscale finite element methods is to construct numerical solvers on the coarse grid. Their resulting linear systems are typically much smaller than using fine grid. Parareal algorithm facilitates speeding up the numerical solver to time dependent equations on the condition of sufficient processors. Typically, parareal algorithm could result in less wall-clock time than sequentially computing. In this dissertation, we will design and apply model reduction techniques to time-fractional diffusion equations, parabolic equations and stokes equations in heterogeneous media. Homogenization approach is studied for the time-fractional diffusion equation. We discuss constraint energy minimizing generalized multiscale finite element method for the incompressible Stokes flow problem in a perforated domain. In this dissertation, we present two methodologies for parabolic problems with heterogeneous coefficients: a novel approach coupling multiscale methods with parareal algorithm and an efficient numerical solver coupling space-time finite element method and Non-local multi-continua technique. The former aims for time-independent permeability field and the latter for time-dependent permeability field

Superionic iron oxide-hydroxide in Earth’s deep mantle

H2O ice becomes a superionic phase under the high pressure and temperature conditions of deep planetary interiors of ice planets such as Neptune and Uranus, which affects interior structures and generates magnetic fields. The solid Earth, however, contains only hydrous minerals with negligible amount of ice. Here we combine high pressure and temperature electrical conductivity experiments, Raman spectroscopy, and first-principles simulations, to investigate the state of hydrogen in the pyrite type FeO2Hx (x ≤ 1) which is a potential H-bearing phase near the core-mantle boundary. We find that when the pressure increases beyond 73 GPa at room temperature, symmetric hydroxyl bonds are softened and the H+ (or proton) become diffusive within the vicinity of its crystallographic site. Increasing temperature under pressure, the diffusivity of hydrogen is extended beyond individual unit cell to cover the entire solid, and the electrical conductivity soars, indicating a transition to the superionic state which is characterized by freely-moving proton and solid FeO2 lattice. The highly diffusive hydrogen provides fresh transport mechanisms for charge and mass, which dictate the geophysical behaviors of electrical conductivity and magnetism, as well as geochemical processes of redox, hydrogen circulation, and hydrogen isotopic mixing in Earth’s deep mantle

Ultrasonic vibration assisted grinding of CFRP composites: Effect of fiber orientation and vibration velocity on grinding forces and surface quality

Ultrasonic vibration assisted grinding (UVAG) of carbon fiber reinforced plastic (CFRP) composites was carried out using a monolayer brazed diamond tool. Effects of the fiber orientation and vibration velocity on grinding forces and surface quality were analyzed. The maximum vibration velocity was varied by adjusting the grinding tool length. The results reveals that the grinding force is decreased with the increase of vibration velocity. The maximal grinding force is obtained with the fiber orientation at 135°, while the minimum grinding force is produced when the fiber orientation is 45°. Surface damages, such as groove scratch, fiber breakage are reduced with the increase of vibration velocity. Additionally, the fiber orientation plays a critical role in the surface morphology. Keywords: Vibration velocity, Fiber orientation, Grinding force, Surface quality, Carbon fiber reinforced plastic composites, Ultrasonic vibration assisted grindin