Mathematical Sciences: Computational Problems in Fluids and Materials

9404310洛文布大规模计算和计算数学现在是研究那些具有随机性和强非线性特征的物理过程的主要工具。这些过程包括扩散主导系统中复杂图案的形成和增长问题，以及海洋自由表面上的波浪相互作用。与之密切相关的是奇异或近乎奇异行为的问题，例如水流的拓扑类型发生变化，或水波破裂。这项建议的主要目标是通过开发和应用最先进的数值方法进行大规模计算，并通过分析和建模了解重要的组成过程，来研究流体和材料中的此类问题。所有这些问题，虽然出现在完全不同的物理环境中，但都强烈地受到表面张力或能量的影响，并且往往具有相同的数学结构。在这项建议中，我们将数学和数值分析、建模和大规模科学计算结合在一起，研究流体力学和材料科学中的某些基本问题。一个这样的问题是理解扩散主导系统中的图案形成。这对金属合金的生产具有重要意义。在生产过程中观察到的合金内部形成的复杂图案严重影响了所得到的合金的整体性能，如它们的强度和韧性。其他问题涉及流体流动中自由表面的运动。两个重要的例子是海洋表面上波浪的相互作用，以及经历拓扑型变化的流动的动力学，例如通过两种流体的混合产生气泡的特征。所有上述问题都是相似的，因为相关现象在很大程度上受到界面上的表面张力(在材料科学背景下称为“表面能”)的影响。因此，可以用一些常见的分析和计算工具来研究它们。

9404310 Lowengrub Large-scale computation and computational mathematics are now primary tools for studying those physical processes characterized by randomness and strong nonlinearity. These processes include complex pattern formation and growth problems in diffusion dominated systems, and wave interactions on the ocean's free surface. Closely related are problems with singular or nearly singular behavior, such as flows undergoing changes in topological type, or water wave breaking. The main objective of this proposal is to investigate such problems in fluids and materials by developing and applying state-of-the-art numerical methods to large-scale computation, and understanding important constituent processes through analysis and modelling. All these problems, while arising in quite disparate physical settings, are strongly mediated by the presence of surface tension or energy, and often share mathematical structure. In this proposal, we bring together mathematical and numerical analysis, modelling, and large-scale scientific computation to study certain fundamental problems in fluid dynamics and materials science. One such problem is understanding pattern formation in diffusion dominated systems. This has important implications in the production of metallic alloys. The complex patterns that are observed to form within the alloys during their production, strongly affect the bulk properties of the resulting alloys, such as their strength and toughness. Other problems involve the motions of free surfaces in fluid flows. Two important instances are the interactions of waves on the ocean's surface, and the dynamics of flows undergoing changes in topological type, say as characterized by the production of bubbles through the mixing of two fluids. All of the above problems are similar in the sense that the relevant phenomena is heavily influenced by surface tension at the interface (called "surface energy" in the materials science context). As such, they can be studied by some common analytical and computational tools.