A comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries

作者：Tauriello Gerardo; Koumoutsakos Petros^*

来源：Journal of Computational Physics, 2015, 283: 388-407.

DOI：10.1016/j.jcp.2014.11.033

• Fictitious domain methods
• Phase field methods
• Volume penalization
• Level set methods
• Parabolic problems

We present a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries embedded using simple Cartesian meshes. The two methods have been widely employed to solve partial differential equations in complex and moving geometries for applications ranging from solid and fluid mechanics to biology and geophysics. Their popularity is largely due to their discretization on Cartesian meshes thus avoiding the need to create body-fitted grids. At the same time, there are questions regarding their accuracy and it appears that the use of each one is confined by disciplinary boundaries. Here, we compare penalization and phase field methods to handle problems with Neumann and Robin boundary conditions. We discuss extensions for Dirichlet boundary conditions and in turn compare with methods that have been explicitly designed to handle Dirichlet boundary conditions. The accuracy of all methods is analyzed using one and two dimensional benchmark problems such as the flow induced by an oscillating wall and by a cylinder performing rotary oscillations. This comparative study provides information to decide which methods to consider for a given application and their incorporation in broader computational frameworks. We demonstrate that phase field methods are more accurate than penalization methods on problems with Neumann boundary conditions and we present an error analysis explaining this result.

• 出版日期2015-2-15