Numerical Analysis of Quasicontinuum Methods

许多对科学和技术非常重要的物理过程需要数值方法，这些方法将小区域中单个原子的相互作用的建模与其余区域中较大原子集之间的相互作用（连续体模型）的建模结合起来。一个重要的例子是裂纹扩展，其中实际的预测模型需要精确地描述裂纹尖端附近小区域内原子的详细相互作用，以及在更大的周围材料中连续体建模的效率。虽然通过计算机模拟材料中所有原子的相互作用可以达到最高的精度，但材料中原子的大量使得这种方法即使对最快的计算机也是不可行的。准连续介质方法通过将裂纹尖端附近的原子模型与周围材料中的连续介质模型耦合在一起，可以在不牺牲可靠预测所需的精度的情况下实现这种模拟。连续体模型通过将选定区域中的大量原子替换为具有代表性的原子，大大减少了计算量，从而达到了所需的精度。拟连续体方法的分析和自适应算法的发展将保证其可靠性和提高其效率。理论和严格的数值实验将发展，以确定最准确和有效的原子-连续体耦合。准连续体方法的发展有可能促进新材料的设计，这些材料能够更好地抵抗失效，并具有对科学和技术重要的其他特性。

Many physical processes of great importance to science and technology require numerical methods that couple modeling of the interactions of individual atoms in small regions with modeling of the interactions among larger sets of atoms (continuum models) in the remaining regions. An important example is crack growth, where practical predictive models require the accuracy of the detailed interaction of the atoms in a small region near the crack tip, along with the efficiency of continuum modeling in the larger surrounding material.Although the greatest accuracy could be achieved by a computer simulation of the interactions of all of the atoms in the material, the large number of atoms in the material makes this approach infeasible for even the fastest computers. The quasicontinuum method can make possible such simulations without sacrificing the accuracy needed for reliable prediction by coupling an atomistic model in the neighborhood of the crack tip with a continuum model in the surrounding material. The continuum model achieves the desired accuracy with a major reduction in computational work by replacing the large numbers of atoms in selected regions with representative atoms.The principal investigator proposes to develop analysis and adaptive algorithms for quasicontinuum methods that will ensure their reliability and improve their efficiency. Theory and rigorous numerical experiments will be developed to determine the most accurate and efficient atomistic-continuum coupling. The development of the quasicontinuum method has the potential to facilitate the design of new materials better able to resist failure and having other properties important for science and technology.