Lisa Lun

An approximate block Newton method for coupled iterations of nonlinear solvers: Theory and conjugate heat transfer applications

A new, approximate block Newton (ABN) method is derived and tested for the coupled solution of nonlinear models, each of which is treated as a modular, black box. Such an approach is motivated by a desire to maintain software flexibility without sacrificing solution efficiency or robustness. Though block Newton methods of similar type have been proposed and studied, we present a unique derivation and use it to sort out some of the more confusing points in the literature. In particular, we show that our ABN 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. These problems are represented by partitioned spatial regions, each modeled by independent heat transfer codes and linked by temperature and flux matching conditions at the boundaries common to the partitions. Whereas a typical block Gauss-Seidel iteration fails about half the time for the model problem, quadratic convergence is achieved by the ABN method under all conditions studied here. Additional performance advantages over existing methods are demonstrated and discussed.

Control of interface shape of cadmium zinc telluride grown via an electrodynamic gradient freeze furnace

We consider the feasibility of closed-loop control to control the interface shape of cadmium zinc telluride (CZT) grown via an electrodynamic gradient freeze (EDG) furnace. A simple proportional control algorithm is applied to a quasi-steady-state model to control the interface shape by adjusting the thermal gradient of the furnace temperatures. Three scenarios are enacted: actuation along the entire external boundary, along the solid portion only, and along the melt portion only. Actuation along the solid only proved to be the most successful of the three scenarios, and produced the unexpected benefit of a favorable convex interface shape.

Large-Scale Numerical Modeling of Melt and Solution Crystal Growth

We present an overview of mathematical models and their large‐scale numerical solution for simulating different phenomena and scales in melt and solution crystal growth. Samples of both classical analyses and state‐of‐the‐art computations are presented. It is argued that the fundamental multi‐scale nature of crystal growth precludes any one approach for modeling, rather successful crystal growth modeling relies on an artful blend of rigor and practicality.

A Schur complement formulation for solving free-boundary, Stefan problems of phase change

A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm-Newton approach, which employs Newtons method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt-solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.

Modeling the growth of CZT by the EDG process

The overall goal of this research is to develop and apply computational modeling to better understand the processes used to grow bulk crystals employed in radiation detectors. Specifically, the work discussed here aims at understanding the growth of cadmium zinc telluride (CZT), a material of long interest to the detector community. We consider the growth of CZT via gradient freeze processes in electrodynamic multizone furnaces and show how crucible mounting and design are predicted to affect conditions for crystal growth. Analysis of these systems will be essential for for significant materials improvement, i.e., growing larger crystals with superior quality and at a lower cost.