Final Report-Scalable Nonlinear Solvers for Fully Implicit Coupled Nuclear Fual Modeling.

摘要

The focus of the project is on the development and customization of some highly scalable domain decomposition based preconditioning techniques for the numerical solution of nonlinear, coupled systems of partial differential equations (PDEs) arising from nuclear fuel simulations. These high-order PDEs represent multiple interacting physical fields (for example, heat conduction, oxygen transport, solid deformation), each is modeled by a certain type of Cahn-Hilliard and/or Allen-Cahn equations. Most existing approaches involve a careful splitting of the fields and the use of held-by-held iterations to obtain a solution of the coupled problem. Such approaches have many advantages such as ease of implementation since only single held solvers are needed, but also exhibit disadvantages. For example, certain nonlinear interactions between the fields may not be fully captured, and for unsteady problems, stable time integration schemes are difficult to design. In addition, when implemented on large scale parallel computers, the sequential nature of the held-by-held iterations substantially reduces the parallel efficiency. To overcome the disadvantages, fully coupled approaches have been investigated in order to obtain full physics simulations. The time integration of the Cahn-Hilliard equation is nontrivial. The nonlinear fourth- order term imposes severe time-step size restrictions for explicit methods. Under the overly simplihed assumption that the mobility is a constant, one can use a semi-implicit method that treats the fourth-order term implicitly, while the nonlinear second-order term is treated explicitly. This technique allows a somewhat larger time step than explicit methods while avoiding the use of nonlinear solvers. However, in this project we are interested in the more realistic variable mobility case for which a fully implicit method was developed.