Preconditioning nonlocal multi-phase flow

We propose an efficient solver for saddle point problems arising from finite element approximations of nonlocal multi-phase Allen-Cahn variational inequalities. The solver is seen to behave mesh independently and to have only a very mild dependence on the number of phase field variables. In addition we prove convergence, in three GMRES iterations, of the approximation of the two phase problem, regardless of mesh size or interfacial width. Numerical results are presented that illustrate the competitiveness of this approach

An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator

Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements

We consider the e�cient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an e�ective Schur complement approximation. Numerical results illustrate the competitiveness of this approach

Multiscale thermo-hydro-mechanical-chemical coupling effects for fluid-infiltrating crystalline solids and geomaterials: theory, implementation, and validation

Extreme climate change and demanding energy resources have led to new geotechnical engineering challenges critical for sustainable development and resilient infrastructure of our society. Applications such as geological disposal of nuclear waste and carbon dioxide, artificial ground freezing, and hydraulic fractures all require an in-depth understanding of the thermo-hydro-mechanical coupling mechanisms of geomaterials subjected to various environmental impact. This dissertation presents a multiphysical computational framework dedicated to address the issues related to those unconventional applications. Our objective is not only incorporating multiphysical coupling effects at the constitutive laws, but also taking into account the nonlocal effects originated from the flow of pore-fluid, thermal convection and diffusion among solid and fluid constituents, and crystallization and recrystallization of crystals in the pore space across length scales. By considering these coupling mechanisms, we introduce a single unified model capable of predicting complex thermo-hydro-mechanical responses of geological and porous media across wide spectra of temperature, confining pressure and loading rate. This modeling framework applies to two applications, i.e., the freezing and thawing of frozen soil and the modeling of anisotropic crystal plasticity/fracture response of rock salt. Highlights of the key ingredients of the models cover the stabilization procedure used for the multi-field finite element, the return mapping algorithm for crystal plasticity, the micromorphic regularization of the Modified Cam-Clay model, and the strategy for enhancing computational efficiency of solvers, such as pre-conditioner, adaptive meshing, and internal variable mapping. By introducing the multiphysical coupling mechanisms explicitly, our computational geomechanics model is able to deliver more accurate and consistent results without introducing a significant amount of additional material parameters. In a parallel effort, we analyze the impact of thermo-hydro-mechanical (THM) coupling effects on the dynamic wave propagation and strain localization in a fully saturated softening porous medium. The investigation starts with deriving the characteristic polynomial corresponding to the governing equations of the THM system. The theoretical analysis based on the Abel–Ruffini theorem reveals that the roots of the characteristic polynomial for the THM problem cannot be expressed algebraically. Our analysis concludes that the rate-dependence introduced by multiphysical coupling may not regularize the THM governing equations when softening occurs