Extended Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries

In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original differential equations such that the equations are solved in the region where a domain parameter takes a specified value while boundary conditions are imposed on the region where the value of the domain parameter varies smoothly across a short distance. The mathematical derivations are straightforward and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples herein: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both surface diffusion and reaction; (3) the mechanical equilibrium equation; and (4) the equation for phase transformation with the presence of additional boundaries. The solutions for several of these cases are validated against corresponding analytical and semi-analytical solutions. The potential of the approach is demonstrated with five applications: surface-reaction-diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces, and a self-propelled droplet.Comment: This document is the revised version of arXiv:0912.1288v

The dynamics of interfaces during coarsening in solid–liquid systems

The isothermal coarsening of dendritic Al–Cu microstructures is examined using time-resolved, in situ synchrotron-based X-ray tomographic microscopy. By obtaining the data at the coarsening temperature (in situ) with speeds that are on the order of or faster than the ongoing evolution of the microstructure (time-resolved, 4-D), we examine the dynamic morphological evolution of the solid–liquid interfaces in two solid volume fractions; as such, the relationship between the velocity of the evolving interface and characteristics that define the morphology of the interface is determined. We find that, while there is a correlation between velocity and mean curvature of the interface, there is a significant dispersion in the velocities for a given value of mean curvature. In addition, the Gaussian curvature plays a role in determining the interface velocity even though it has no effect on the chemical potential at the interface. We find that there are many interface patches of various morphologies and size scales that are not evolving during the coarsening process. At higher solid volume fractions and longer coarsening times, more of the structure is active in the coarsening process, and there is an increase in localized diffusional interactions