(773d) Efficient and Robust Strategies For Coupling Models Of Melt Crystal Growth Processes

Even at the continuum level, melt crystal growth problem complexity presents a significant technical challenge, requiring the integration of a large-scale furnace model with a strongly-coupled, multi-physics, transport problem of the Stefan, moving-boundary type. Problem nonlinearity can be severe, due to high-temperature, radiative heat transfer and strong, richly structured flows of a transitional nature. Robust computing of steady-state solutions under these conditions can be achieved by Newton-Raphson iteration, but its desirable quadratic convergence property relies on our ability to compute a sufficiently accurate approximation to the inverse of a Jacobian matrix of the global system of equations. It is costly to develop monolithic software that simultaneously represents all chosen phenomena at all scales in a single model of melt crystal growth. From a practical standpoint, problems of this scope favor a partitioned approach, in which a few major subdomains of the problem are tackled independently by existing software best suited to the task. Such methods can be used to link together existing best-in-class tools to tackle complex multi-physics and multi-scale problems, without requiring extraordinary programming effort.

Towards this end, we have developed an approximate block Newton (ABN) method to couple arbitrary, black-box nonlinear solvers . This ABN method preserves the quadratic approximation properties of an exact Newton iteration. The notion of a solver is abstract, encompassing any interpolations or other transformations of data exchanged between solvers. It is shown that the method behaves like a Newton iteration preconditioned by an inexact Newton solver derived from subproblem Jacobians. The method is demonstrated on several conjugate heat transfer problems modeled after melt crystal growth processes. Whereas a typical block Gauss--Seidel iteration fails about half the time for such problems, quadratic convergence is achieved by the ABN method under all conditions studied. Prospects for quantitative process modeling and the ability to represent three-dimensional and transient phenomena in bulk crystal growth are also discussed.

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Supported in part by DOE/NNSA, DE-FG52-08NA28768, the content of which does not necessarily reflect the position or policy of the United States Government, and no official endorsement should be inferred.