Researchers in Duke's Computational Mechanics Laboratory are actively developing new methodologies and algorithms to robustly capture the physics of evolving interfaces. Our work is distinguished by the use of the finite element method for this class of problems. Typically, the FEM is only used to capture evolving interfaces when the mesh is continuously adapted (re-meshed) to "fit" the interface. This can be incredibly costly and usually requires the use of many heuristics, particularly when topology changes are frequent.
We take a very different approach to this class of problems. We use fixed (or nearly fixed) meshes, and allow the interface to evolve through the mesh. In order to properly capture the physics near the interface, we enhance the basis using the eXtended Finite Element Method developed by Professor Dolbow and colleagues. We then couple this enhanced basis to the level-set method to capture both the geometry and physics of the interface with evolving functions. We strive for the greatest degree of generality and do not limit ourselves to structured Cartesian meshes (as many level-set researchers do). The challenges include properly coupling the two methods, robustly enforcing interfacial conditions and constraints, and addressing discrete conservation concerns.
Our research in this area has been and continues to be sponsored by Sandia National Laboratories, the National Science Foundation, and the Department of Energy through the MICS program.
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References
- Hautefeuille M, Annavarapu C, Dolbow JE. Robust imposition of Dirichlet boundary conditions on embedded surfaces, International Journal for Numerical Methods in Engineering 90 (1): 40-64, 2012
- Annavarapu C, Hautefeuille M, Dolbow JE. A robust Nitsche’s formulation for interface problems, Computer Methods in Applied Mechanics and Engineering Volumes 225–228: 44-54, 2012
- Mourad HM, Dolbow J, and Harari I. A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces, International Journal for Numerical Methods in Engineering, 2006
- Mourad HA, Dolbow J, Garikipati K. An assumed-gradient finite element method for the level set equation, International Journal for Numerical Methods in Engineering 64 (8): 1009-1032, 2005
- Ji H, Dolbow JE. On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method , International Journal for Numerical Methods in Engineering 61 (14): 2508-2535, 2004
- Ji H, Chopp D, Dolbow JE. A hybrid extended finite element/level set method for modeling phase transformations, International Journal for Numerical Methods in Engineering 54 (8): 1209-1233, 2002