Forward and inverse simulations of heat transfer and grain growth in thin films

Abstract

In the first simulation module, a finite element model of the micro-heater system is used to define a discrete set of non-linear equations used as a basis for the inverse problem solution. A direct minimization method with Tikhonov regularization is used to iteratively find the local optimal solution. A uniform and a linear temperature distribution could be attained in the central region above the micro-heater array, but the temperatures near the edges of the domain could not be controlled due to heat loss at the edges. Thus, to control the temperature field over the full width of the domain, the heater array must extend beyond the domain of interest. The regularization method allows for a smoother solution compared with solution without regularization.

Thermal processing is used by metals manufacturers to generate desired microstructural features and thus, properties. However, the processing conditions are typically specified based on experience and trial and error, leading to a longer development cycle and resulting process parameters that may not be optimized. In this work we solve the inverse problem for current input to generate desired grain size distribution in a copper thin film during thermal processing using a micro-heater array. The entire problem is solved based on successful development of two simulation modules: inverse finite element heat transfer simulation and inverse Monte Carlo grain growth simulation.

In the second simulation module, we develop a Monte Carlo (MC) algorithm to model material microstructure evolution in the presence of a non-uniform temperature field that may vary with time and space. We propose a novel parallel MC algorithm for accurately scaling to physical time and space. We first scale the MC model to physical observation by fitting experimental data. Based on the scaling relationship, we derive a grid site selection probability (SSP) function to consider the effect of a spatially varying temperature field. The SSP function is based on the differential MC step, which allows it to naturally consider time varying temperature fields too. We verify the model and compare the predictions to other existing formulations in two-dimensional cases with only spatially varying fields, where the predicted microstructure evolution in regions of constant field are expected to be the same as for the isothermal case. We also test the model in a more realistic three-dimensional case with temperature field varying in both space and time. We believe the newly proposed approach is promising for modeling material physics problems that involves time-dependent local gradients. Lastly, the MC method is adjusted to consider anisotropic grain boundary energy.

A complete inverse problem is formulated by coupling the inverse finite element heat transfer simulation module with an inverse MC grain growth simulation module. Given the target grain size distribution in the film, the historical current in the heater lines of the micro-heater array is solved by driving the two modules. Specifically, the grain size distribution is measured in the MC simulation and it is used to update target temperature distribution for the inverse heat transfer simulation to solve for, and the corresponding current input is updated. This inverse problem is solved and scaled on the RPI super computer IBM Blue Gene Q. Performance of parallelization is investigated and discussed.