Computational methods for materials science, high frequency wave propagation, and quantum mechanics

该项目涉及建模、仿真和高效计算技术上重要和智力上有趣的问题。在材料科学领域，研究了两种新的非晶格方法，用于计算晶格结构可能发生变化的晶体材料的演化。例如，当缺陷形成很重要时，这是需要的。该项目还将解决非均匀材料中高频波传播的有效计算。这是计算科学和应用数学中的一个经典问题，因为它对地球物理和光学问题很有用，而且它是一个数学上丰富的问题，需要深入了解波动方程的行为。PI还计划开发用于研究分子量子动力学的数值方法。量子力学在理解化学反应中的重要性怎么强调都不为过，这项研究提供了为各种光诱导化学反应提供详细见解和预测能力的潜力。该项目还包括研究生的培训。第一个项目涉及晶体材料演化的非晶格计算模型的开发。在一种情况下，PI计划制定一种非晶格动力学蒙特卡罗方法来模拟外延生长，在另一种情况下，提出了相场晶体方法的替代方案。在这两种情况下，系统的潜在能量都是合适的分子间势。此外，在这两种情况下，计算的时间尺度都要比分子动力学大几个数量级。另一个项目涉及波动方程高频解的计算。目标是利用高频解的渐近结构来开发有效的数值方法，这些方法不会由于使用依赖几何光学或高斯光束的方法所产生的渐近近似而产生误差。PI还计划开发数值方法来研究分子的量子动力学，当圆锥相交发挥重要作用时。一个完整的量子力学处理将意味着在20到30个维度上解决薛定谔方程，即使是小分子。我们的计划是通过使用表面跳跃方法作为能量景观的探针来识别重要模式。然后，只需使用这一较小的模式集，就可以精确地获得完整的量子处理。

This project involves modeling, simulation, and efficient computation of technologically important and intellectually interesting problems. In the area of material science the research investigates two new off-lattice methods for the computational modeling of the evolution of crystalline materials in which the lattice structure may change. This is needed when defect formation is important, for example. The project will also address the efficient computation of high frequency wave propagation in nonhomogeneous materials. This classic problem in computational science and applied mathematics has attracted much attention because it is useful for problems in geophysics and optics and it is a mathematically rich problem requiring insight into the behavior of wave equation. The PI also plans to develop numerical methods for studying quantum dynamics of molecules. The importance of quantum mechanics in understanding chemical reactions cannot be overstated, and this research offers the potential of providing detailed insight and predictive power for a variety of photo-induced chemical reactions. The project also involves training of graduate students.The first project involves developement of off-lattice computation models for the evolution of crystalline materials. In one case, the PI plans to formulate an off-lattice kinetic Monte Carlo method for the simulation of epitaxial growth and in other an alternative to the phase field crystal method is proposed. In both cases the underlying energy of the system is a suitable intermolecular potential. In addition, in both cases one will be computing on time scales orders of magnitude larger than molecular dynamics. Another project involves the computation of high frequency solutions of the wave equation. The goal is to exploit the asymptotic structure of high frequency solutions to develop efficient numerical methods that do not have the errors that result from asymptotic approximations that occur when one uses methods that rely on geometrical optics or Gaussian beams. The PI also plans to develop numerical methods for the studying quantum dynamics of molecules when conical intersections play an important role. A full quantum mechanical treatment would mean solving the Schroedinger equation in 20 to 30 dimensions even for small molecules. Our plan is to identify the important modes by using surface hopping methods as a probe of the energy landscape. A full quantum treatment can then be accurately obtained just using this smaller set of modes.