Embedded Boundary Methods


The long-term goal of this project is to develop automatic algorithms and software to simulate fluid flow in complex geometries. Our approach is based on the use of Cartesian meshes with embedded boundaries. Most of the domain is covered by completely regular Cartesian cells, where fast and efficient finite volume schemes have been developed. Most of the difficulty both mathematically and in implementation comes from the cut cells which intersect the boundary of the solid objects.

Grid generation is extremely fast, robust, and automatic. Our method takes as input a watertight triangulation of the surface, and returns the Cartesian mesh, where the cut cells are identified along with their volume, face centroids, and the cut triangles inside of each cut cell. The tools we've developed for this can generate a mesh with millions of cells in minutes on a desktop workstation. Some examples are shown here; see the Cart3D web site for more.


fuel rod mesh - face on

 fuel rod mesh
oneraMg solution

 oneraM6 mesh


The ongoing challenge is to find stable and accurate difference schemes where the Cartesian cells intersect the body. For steady state problems, second order finite volume schemes have been developed.  Since for time-dependent problems, explicit schemes usually have a CFL condition that depends on the cell volume, new approaches are needed to develop a scheme that maintains stability in the presence of arbitrarily small  cut cell sizes.  We are developing h-box methods -- difference schemes with an enlarged stencil in rotated coordinates -- to tackle this problem.


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