The main research emphasis of our laboratory is on the development of mathematically rigorous techniques for the computational design and control of materials processes including deformation, solidification and crystal growth processes. Our interests lie in understanding and controlling the effects of microstructure evolution in material properties.

Many of the materials problems of interest share a common mathematical structure that includes the control of stochastic partial differential equation or discrete system based multiscale and multi-physics processes. Understanding uncertainty propagation across length scales is a key ingredient of this research. Stochastic spectral methods, wavelet-based approximations, Bayesian inference and information theory are increasingly used to develop a unified framework for modeling materials across length scales. We are also very active in interfacing robust control of continuum systems with information technologies including machine learning techniques in order to develop real time feedback mechanisms for the control of complex materials processes in the presence of uncertainty.

Material Process Design and Control Problems of Current Interest

The main material process design problems of current interest are summarized below:

• Quantitative representation of the microstructure of polycrystal materials and knowledge discovery by statistical exploration of structure/property/process relations. Abstract representation of microstructural elements is fundamental for optimizing microstructure-sensitive material properties.
• Materials-by-design: First-principle calculations, molecular dynamics, mesoscopic models, computational thermodynamics, phase field & level set simulation methods and virtual interrogation of microstructures.
• Development of gradient-based optimization algorithms for the design of multi-stage metal forming processes as applied to the manufacturing of aircraft components. Emphasis is given to the control of microstructure-sensitive properties.
• Control of the effect of various thermo-physical processes on the obtained crystal structure in the crystal growth of metallic, semiconductor and organic materials. Current emphasis is given in addressing melt flow control via optimal furnace design and using non-uniform externally applied magnetic fields.
• Control of solidification on substrates and optimal design of mold surface topographies in directional (chill) and continuous casting processes that lead to desired shell surface topologies and near-surface microstructures.
• Development of information technologies for the real time control of solidification processes.

A sample of recent computational methods developed by the MPDC group include those listed below.

• Bayesian computation for inverse problems in transport processes.
• Stochastic variational multiscale methods for transport in highly heterogeneous media -- spectral stochastic and stochastic support methods.
• Stochastic modeling in the deformation of heterogeneous materials.
• Statistical learning (machine learning) techniques for exploring material databases.
• Model reduction techniques applied to continuum system design.
• Continuum sensitivity methods for the computational design of deformation processes.
• Level set methods for modeling dendritic solidification in the presence of melt flow.
• Functional optimization and adjoint techniques for the control of complex flow and thermal transport processes.