Gunduz Caginalp - US grants

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences. It is for the purchase of special purpose equipment dedicated to the support of research in the mathematical sciences. In general, this equipment is required by several research projects, and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers, and with substantial cost-sharing from the institution submitting the proposal. This is an instance of university, industrial, and government cooperation in the support of basic research in the mathematical sciences. The equipment in this project will be used to support research on Coupled Oscillators and Neural Networks; Travelling Waves in Combustion Theory; On the Relation between Painleve Transcendents and Inverse Scattering; Nonlinear Boundary Value Problems in Ordinary Differential Equations; and On Systems of Differential Equations which Describe Phase Boundaries.

This research will continue an investigation of the phase field approach to solidification. This approach has resulted in a uniform way of understanding phase boundaries. The difference between a thin and sharp interface will be considered, especially the effect of the difference on instability. Anisotropy will be studied using various analytical and numerical methods. Also considered will be the interface between a solid and liquid when the interface is in contact with an external boundary. The related problem for alloys will be investigated.

With this award the principal investigator will continue his analytical studies of free boundary problems using asymptotic methods and he will initiate attempts to construct efficient numerical schemes for the resolution of interfacial phenomena. Overall he plans to develop a unified theory of free boundary problems that arise in phase transitions based upon the Phase Field Model that he himself has helped to popularize. The principal investigator will consider progressively realistic models of alloys and contact with external containers. Many technologically important natural phenomena involve the behavior of different media along the interfaces or surfaces of contact between them. With this award the principal investigator will continue his important work on developing approximate solutions of models of phase transitions using asymptotic methods and he will begin developing numerical schemes in order to resolve increasingly complicated interfacial phenomena. This combined asymptotic-numerical approach should pay big dividends toward understanding the physics of transition layers and free boundary problems.

This research concerns the modeling and analysis of various free- boundary problems related to phase transition. In particular, the general approach of phase field model is applied to several basic problems: alloys, triple points, and crystal growth. The goal is to gain insight into the freezing and melting processes in alloys, and the mechanisms of crystal growth. The results of this research will contribute to the objectives of the advanced materials processing program.

9703530 Caginalp This proposal is being jointly funded by Division of Mathematical Sciences (DMS) and Division of Materials Research (DMR). This proposal involves research on two topics: (a) the study of interfaces, and (b) the use of renormalization and scaling methods to study systems of parabolic differential equations. The study of interfaces proceeds along the lines of previous work that utilizes the phase field model. The study of alloys is particularly important in terms of the ''freezing in'' of solute into the solid. This is an area in which mathematics can make a substantial contribution since the models necessarily involve differential equations that are degenerate, as the diffusivity is close to zero in the solid phase. In the second part of the study, the PI will utilize renormalization and scaling methods to obtaing results on systems of parabolic differential equations in which an asymptotic self-similarity can be expected. The work involves an extension of the PI's work that used these methods to compute anomalous exponents in nonlinear diffusion equations. The objective is to extend these results to systems of parabolic differential equations, thereby adding a powerful tool to the methodology of these systems. Finally, the study will apply these methods to reaction-diffusion systems such as the phase field equations in order to extract key features of global behavior. This study will provide a formalism for studying complicated problems involving materials science. In particular, interface problems arise in many industrial applications and pose important challenges in terms of theory and large scale computation. The development of a consistent set of equations and methodology for high speed computation is valuable as a starting point for many industrial applications such as casting of alloys. The study will also utilize renormalization techniques that have been so successful in understanding subtle behavior in thermodynamics. With the development of these techniques in this dynamical context, an extremely complicated engineering problem can potentially be understood in terms of manageable parts with particular characteristic behavior.