Efficient Solvers for the Phase-Field Crystal Equation: Development and Analysis of a Block-Preconditioner

A preconditioner to improve the convergence properties of Krylov subspace solvers is derived and analyzed in this work. This method is adapted to linear systems arising from a finite-element discretization of a phase-field crystal equation

A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation

We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn-Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn-Hilliard equaion. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text overlap with arXiv:1709.0400

Cracking and damage from crystallization in pores: Coupled chemo-hydro-mechanics and phase-field modeling

Cracking and damage from crystallization of minerals in pores center on a wide range of problems, from weathering and deterioration of structures to storage of CO2 via in situ carbonation. Here we develop a theoretical and computational framework for modeling these crystallization-induced deformation and fracture in fluid-infiltrated porous materials. Conservation laws are formulated for coupled chemo-hydro-mechanical processes in a multiphase material composed of the solid matrix, liquid solution, gas, and crystals. We then derive an expression for the effective stress tensor that is energy-conjugate to the strain rate of a porous material containing crystals growing in pores. This form of effective stress incorporates the excess pore pressure exerted by crystal growth – the crystallization pressure – which has been recognized as the direct cause of deformation and fracture during crystallization in pores. Continuum thermodynamics is further exploited to formalize a constitutive framework for porous media subject to crystal growth. The chemo-hydro-mechanical model is then coupled with a phase-field approach to fracture which enables simulation of complex fractures without explicitly tracking their geometry. For robust and efficient solution of the initial–boundary value problem at hand, we utilize a combination of finite element and finite volume methods and devise a block-partitioned preconditioning strategy. Through numerical examples we demonstrate the capability of the proposed modeling frameworkfor simulating complex interactions among unsaturated flow, crystallization kinetics, and cracking in the solid matrix

An efficient numerical framework for the amplitude expansion of the phase-field crystal model

The study of polycrystalline materials requires theoretical and computational techniques enabling multiscale investigations. The amplitude expansion of the phase field crystal model (APFC) allows for describing crystal lattice properties on diffusive timescales by focusing on continuous fields varying on length scales larger than the atomic spacing. Thus, it allows for the simulation of large systems still retaining details of the crystal lattice. Fostered by the applications of this approach, we present here an efficient numerical framework to solve its equations. In particular, we consider a real space approach exploiting the finite element method. An optimized preconditioner is developed in order to improve the convergence of the linear solver. Moreover, a mesh adaptivity criterion based on the local rotation of the polycrystal is used. This results in an unprecedented capability of simulating large, three-dimensional systems including the dynamical description of the microstructures in polycrystalline materials together with their dislocation networks.Comment: 12 pages, 7 figure

Quantum Chemistry for Solvated Molecules on Graphical Processing Units Using Polarizable Continuum Models

The conductor-like polarization model (C-PCM) with switching/Gaussian smooth discretization is a widely used implicit solvation model in chemical simulations. However, its application in quantum mechanical calculations of large-scale biomolecular systems can be limited by computational expense of both the gas phase electronic structure and the solvation interaction. We have previously used graphical processing units (GPUs) to accelerate the first of these steps. Here, we extend the use of GPUs to accelerate electronic structure calculations including C-PCM solvation. Implementation on the GPU leads to significant acceleration of the generation of the required integrals for C-PCM. We further propose two strategies to improve the solution of the required linear equations: a dynamic convergence threshold and a randomized block-Jacobi preconditioner. These strategies are not specific to GPUs and are expected to be beneficial for both CPU and GPU implementations. We benchmark the performance of the new implementation using over 20 small proteins in solvent environment. Using a single GPU, our method evaluates the C-PCM related integrals and their derivatives more than 10× faster than that with a conventional CPU-based implementation. Our improvements to the linear solver provide a further 3× acceleration. The overall calculations including C-PCM solvation require, typically, 20–40% more effort than that for their gas phase counterparts for a moderate basis set and molecule surface discretization level. The relative cost of the C-PCM solvation correction decreases as the basis sets and/or cavity radii increase. Therefore, description of solvation with this model should be routine. We also discuss applications to the study of the conformational landscape of an amyloid fibril.United States. Office of Naval Research (N00014-14-1-0590

Modelling the optics of high resolution liquid crystal devices by the finite differences in the frequency domain method

A procedure combining accurate liquid crystal and electromagnetic modelling is developed for the analysis of wave propagation through liquid crystal devices. This is required to study the optics of high resolution liquid crystal cells or cells containing very small features, where diffraction effects occur. It is also necessary for the study of optical waveguiding devices using liquid crystal as variable permittivity substrates. An accurate finite element modelling program is used to find the permittivity tensor distribution, which is then used to find the response of the device to an excitation electromagnetic field by means of a finite difference in the frequency domain (FDFD) approach