The investigator and his colleagues study the interaction between the defect structures in materials (such as phase and grain boundaries) and the background heterogeneous medium. The heterogeneity can come from intrinsic atomic structures, externally imposed forcing, or even numerical approximations. The emphasis is on understanding the structure and propagation of interfaces. Due to the highly nonlinear interaction between the materials interface and background inhomogeneity, many new features arise, most notably boundaries between patterns instead of pure phases, and pinning and de-pinning transitions. In terms of mathematical disciplines, the project is related to the concept of homogenization of singularly perturbed problems in which multiple scales coexist. To tackle nonlinear problems as in the current setting, new techniques and interpretations beyond traditional approaches are required. The project investigates the structure and propagation of boundaries between patterns, and the connection between pulsating waves and singular perturbations. Both the discrete and continuum regimes are analyzed.
The project is highly interdisciplinary and is well-motivated by problems coming from applied science, in particular materials science. It brings together rigorous applied analysis, numerical simulations and modeling in a variety of physical phenomena. It emphasizes the connections between multiple scales, as practical materials often are composed of structures that have greatly different length scales. Results can lead to better understanding of how applied forcing and changes of external environments can affect the behavior of materials responses. This in turn can lead to the manufacturing of stronger and better materials.
This proposal analyzes mathematical models for the structure and dynamics of fronts and interfaces arised from materials science. The goal is to understand the behaviors of the solution as a function of external driving forces and the underlying background heterogeneous environment. These external factors can be related to impurities, defects, lattice effects, and even numerical simulations. Some of the key findings include (i) the characterization of singular structures of the solutions as the external parameters changes, in particular when they pass through some critical values, (ii) how the solution behaves with respect to the underlying geometry, (iii) the incorporation of stochastic effects. The outcomes are very useful in predicting materials response toward the changes of outside environment and hence can lead to better control and manufacture of the materials. The projects involved are highly interdiscinplinary. They incoporate modeling, analysis and simulations. Mathematics and other disciplines are brought together. They also provide suitable platform for the training of human resources. Under the duration of this grant, the PI has graduated three PhD students and supervised two post-doctoral scholars. The projects provide them opportunities to learn versatile techniques, such as asymptotic analysis, singular perturbations, stochastic analysis, which can be used in a variety of other problems. Some of these junior people have started their own career in research in applied science. The PI has taught various courses tailored to a wide range of students with interests in applied sciences. These include partial differential equations, probability, asymptotic and nonlinear analysis. Some of these can lead to the creation of new courses in curriculum so that more students will benefit in the future. The PI has also (co-)organized two conferences that brought together people with mathematical interests and background so as to engage in lively discussion and the creation of new research directions.
