Two-dimensional wave propagation in layered periodic media

We study two-dimensional wave propagation in materials whose properties vary periodically in one direction only. High order homogenization is carried out to derive a dispersive effective medium approximation. One-dimensional materials with constant impedance exhibit no effective dispersion. We show that a new kind of effective dispersion may arise in two dimensions, even in materials with constant impedance. This dispersion is a macroscopic effect of microscopic diffraction caused by spatial variation in the sound speed. We analyze this dispersive effect by using high-order homogenization to derive an anisotropic, dispersive effective medium. We generalize to two dimensions a homogenization approach that has been used previously for one-dimensional problems. Pseudospectral solutions of the effective medium equations agree to high accuracy with finite volume direct numerical simulations of the variable-coefficient equations

A monolithic conservative level set method with built-in redistancing

We introduce a new level set method for representing evolving interfaces. In the case of divergence-free velocity fields, the new method satisfies a conservation principle. Conservation is important for many applications such as modeling two-phase incompressible flow. In the present implementation, the conserved quantity is defined as the integral of a smoothed characteristic function. The new approach embeds level sets into a volume of fluid formulation. The evolution of an approximate signed distance function is governed by a conservation law for its (smoothed) sign. The non-linear level set transport equation is regularized by adding a flux correction term that assures a non-singular Jacobian and penalizes deviations from a distance function. The result is a locally conservative level set method with built-in elliptic redistancing. The continuous model is monolithic in the sense that the level set transport model, the volume of fluid law of mass conservation, and the minimization problem that preserves the approximate distance function property are incorporated into a single equation. There is no need for any extra stabilization, artificial compression, flux limiting, redistancing, mass correction, and other numerical fixes which are commonly used in level set or volume of fluid methods. In addition, there is just one free parameter that controls the strength of regularization and penalization in the model. The accuracy and conservation properties of the monolithic finite element / level set method are illustrated by the results of numerical studies for passive advection of free interfaces

High-Order Maximum Principle Preserving (MPP) Techniques for Solving Conservation Laws with Applications on Multiphase Flow

We develop numerical methods to solve the linear scalar conservation law fulfilling the maximum principle. To do this we use continuous and discontinuous Galerkin finite elements and achieve the preservation on the maximum principle via the Flux Corrected Transport (FCT) method. We use high-order polynomial spaces with Bernstein basis functions and obtain the optimal convergence rates with spaces of up to third order for smooth solutions that are monotone. This methodology produces good quality results for spaces up to (around) third order. However, when higher-order spaces are used non-physical oscillations are introduced, which is true nevertheless the methods are maximum principle preserving. These oscillations can be highly reduced by defining tighter bounds. Using discontinuous Galerkin finite elements we present a new FCT-like methodology based on single cell flux corrections. This method combines a mass conservative low-order Maximum Principle Preserving (MPP) solution with a non-mass conservative high-order MPP solution. The process is designed to recover mass conservation locally (with respect to degrees of freedom). Using this scheme we obtain the optimal convergence rates with spaces of up to third order for smooth solutions that are monotone. The method is designed to overcome problems when high-order spaces are used and, under this context, we obtained better results than with the standard FCT method. We present two methods to transport a smoothed Heaviside level set function using a one-stage reinitialization based on artificial compression. The first method allows arbitrarily large compression which might lead to non-physical behavior. To overcome this difficulty the second method self-balances the artificial dissipation and compression. Finally, we use the level set solver with a Navier-Stokes solver to simulate incompressible two-phase flow

Título: Sentencia 1662 11 07

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