DEFORMATIONS IN SOLID MECHANICS
Students involved in this project will conduct research in solid mechanics (linear elasticity). Students will use the optical device ARAMIS, a non-contact optical 3D deformation measuring system, to analyze, calculate and document deformations. With ARAMIS, students can determine the coordinates, displacements and strains on the surface of objects under measuring. Students will compare the experimental results with theoretical and numerical outcomes. The most traditional way of describing the material behavior is by using partial differential equations (PDE). Students will study the governing system of PDE in linear isotropic elasticity, also known as the Navier equations, and learn how to solve them analytically (with the help of Maple) and numerically (with the help of MATLAB).
OPTIMALITY MODELING IN MATHEMATICAL BIOLOGY OF PLANTS
Optimality modeling is an important approach in modern biology used to explain the structure and function of living organisms. Researchers assume that natural selection results in organisms with traits that are optimal according to criteria related to evolutionary advantages, which leads to the idea of optimality modeling. In this project, we will build and test optimality models of water transport in plants. Plant transport of water from roots to leaves is strongly linked to to photosynthesis, the process plants use to assimilate carbon dioxide from the atmosphere and synthesize organic compounds to sustain their growth and reproduce. Thus, the amount of carbon captured from the atmosphere or the amount of water pulled from the soil, are natural optimality criteria for understanding why plants are built and behave the way they do.
How do plants pull water? It is not by the capillary effect alone if that's what you thought. Plants create huge negative pressure gradients by evaporating water through their leaves. They create huge negative pressures at the leaf level through some intricate biophysical mechanisms. Optimization of water transport is subject to costs of maintaining the network of conduits, and constraints, like water availability and atmospheric humidity. Constructing the optimality models will be a collaboration between student groups working on mathematical and experimental tasks. The experimental group will develop techniques for measuring pressures, flows and hydraulic characteristics in various parts of the plants. The mathematical group will set up constrained optimization models and make predictions about plants' behavior using analytic and computational methods. Specific optimality criteria and cost expressions will be adjusted to match the experimental results. Tested models will be used to predict the desired behavior of plants in novel environmental conditions, such as droughts.
Mathematically, the models raise interesting optimal control problems involving differential and difference equations with additional memory effects. It turns out that hydraulic characteristics of plants are not merely functions of current environmental parameters, but "remember" some of their history, like past droughts. Taking these effects into account leads to challenging questions, both experimental and mathematical, at the cutting edge of current research in mathematical biology.