Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

Time integration and steady-state continuation for 2d lubrication equations

Lubrication equations allow to describe many structurin processes of thin liquid films. We develop and apply numerical tools suitable for their analysis employing a dynamical systems approach. In particular, we present a time integration algorithm based on exponential propagation and an algorithm for steady-state continuation. In both algorithms a Cayley transform is employed to overcome numerical problems resulting from scale separation in space and time. An adaptive time-step allows to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed on the one hand to analyse different phases of the dewetting of a liquid film on a horizontal homogeneous substrate. On the other hand, we consider the depinning of drops pinned by a wettability defect. Time-stepping and path-following are used in both cases to analyse steady-state solutions and their bifurcations as well as dynamic processes on short and long time-scales. Both examples are treated for two- and three-dimensional physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations, respectively.Comment: 33 pages, 16 figure

Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium

Infiltration of water in dry porous media is subject to a powerful gravity-driven instability. Although the phenomenon of unstable infiltration is well known, its description using continuum mathematical models has posed a significant challenge for several decades. The classical model of water flow in the unsaturated flow, the Richards equation, is unable to reproduce the instability. Here, we present a computational study of a model of unsaturated flow in porous media that extends the Richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively. The extended model is based on a phase-field formulation and is fourth-order in space. The new model poses a set of challenges for numerical discretizations, such as resolution of evolving interfaces, stiffness in space and time, treatment of singularly perturbed equations, and discretization of higher-order spatial partial–differential operators. We develop a numerical algorithm based on Isogeometric Analysis, a generalization of the finite element method that permits the use of globally-smooth basis functions, leading to a simple and efficient discretization of higher-order spatial operators in variational form. We illustrate the accuracy, efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media. We simulate, for the first time, unstable gravity-driven infiltration in three dimensions, and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium. Our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux.United States. Dept. of Energy (Early Career Award Grant DE-SC0003907

On a fractional step-splitting scheme for the Cahn-Hilliard equation

PURPOSE – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. DESIGN / METHODOLOGY / APPROACH – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. FINDINGS – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. ORIGINALITY / VALUE – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.http://www.emeraldinsight.com/loi/echb201

Isogeometric analysis and hierarchical refinement for multi-field contact problems

The present work deals with multi-field contact problems in the context of IGA. In particular, a thermomechanical as well as a fracture mechanical system is considered, where novel formulations are introduced for both. The corresponding discrete contact formulations are based on a variationally consistent mortar approach adapted for NURBS discretized and hierarchical refined surfaces. Finally, the capabilities of the proposed framework are demonstrated within numerous numerical examples

Phase Separation Dynamics in Isotropic Ion-Intercalation Particles

Lithium-ion batteries exhibit complex nonlinear dynamics, resulting from diffusion and phase transformations coupled to ion intercalation reactions. Using the recently developed Cahn-Hilliard reaction (CHR) theory, we investigate a simple mathematical model of ion intercalation in a spherical solid nanoparticle, which predicts transitions from solid-solution radial diffusion to two-phase shrinking-core dynamics. This general approach extends previous Li-ion battery models, which either neglect phase separation or postulate a spherical shrinking-core phase boundary, by predicting phase separation only under appropriate circumstances. The effect of the applied current is captured by generalized Butler-Volmer kinetics, formulated in terms of diffusional chemical potentials, and the model consistently links the evolving concentration profile to the battery voltage. We examine sources of charge/discharge asymmetry, such as asymmetric charge transfer and surface "wetting" by ions within the solid, which can lead to three distinct phase regions. In order to solve the fourth-order nonlinear CHR initial-boundary-value problem, a control-volume discretization is developed in spherical coordinates. The basic physics are illustrated by simulating many representative cases, including a simple model of the popular cathode material, lithium iron phosphate (neglecting crystal anisotropy and coherency strain). Analytical approximations are also derived for the voltage plateau as a function of the applied current

B-Spline meshing for high-order finite element analyses of multi-physics problems

Multi-physics problems often involve differential equations of higher-order, which cannot be solved with standard finiteelement methods. B-splines as finite element basis functions provide the required continuity and smoothness. However, the meshgeneration for arbitrarily shaped domains is non-intuitively and traditional techniques often lead to distorted elements.Here a strategy is presented to design isoparametric B-spline based meshes for curves, surfaces, and volumes. The error of thehomeomorphic transformation into curved boundaries is estimated. For selected two and three-dimensional shapes, the knotvectors and the control points are calculated.Exemplarily, a finite element analysis of a helical structure subjected to a chemo-mechanical deformation with phase decompositionis performed