Novel Numerical Method for Resolving Micro-Structure Based on Supermesh for Multi-Material Systems

A new numerical method is developed for the solution of the diffusion problem in a system of several materials. In such a system, the diffusion coefficients are piecewise continuous and jumps in their values can occur across the complex-shaped interfaces between contiguous materials. The boundary conditions along the complex-shaped interfaces can either be a jump condition boundary condition, a Neumann boundary condition, or a Dirichlet boundary condition. The moment-of-fluid (MOF) procedure is employed to reconstruct the interfaces. This procedure enables accurate reconstruction of any number of material interfaces in a computational cell. Furthermore, MOF is a volume preserving reconstruction, as well as capable of capturing thin filamentary regions without the necessity of adaptive mesh refinement. The proposed method is tested on multi-material diffusion problems which demonstrates its potential to enable numerical simulation of complex flows of technological importance relevant to predicting the heat transfer rate in materials and manufacturing processes. Results using the new method are reported on problems in complex (filamentary) domains, and it is found that the method is very efficient at approximating the temperature, the temperature gradient, and the interfacial heat flux, as compared to traditional approaches. \par The method will also be applied for computing solutions to the Stefan problem involving complex deforming geometries. The interface propagation equation is resolved by using the unsplit cell-integrated semi-Lagrangian method. The level set method is also coupled during this process in order to assist in the initialization of the (transient) provisional velocity field. Our method is validated on both canonical and challenging benchmark tests. Algorithm convergence results based on grid refinement are reported. It is found that the new method approximates solutions to the Stefan problem efficiently, compared to traditional approaches, due to the localized finite volume approximation stencil derived from the underlying supermesh. The new kind of supermesh approach provides a general framework for solving many complex deforming boundary problems containing inherent deforming micro-structural components.