Mathematical Sciences: Computations in Fluids and Materials

小行星9404554 大规模计算和计算数学是研究具有随机性和强非线性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模式的形成和增长问题，以及海洋自由表面上的波浪相互作用。 与之密切相关的是具有奇异或近似奇异行为的问题，例如流动经历拓扑类型的变化或水波破碎。 该提案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，虽然产生于完全不同的物理环境中，但都是由表面张力或能量的存在强烈介导的，并且通常具有相同的数学结构。 在这个提议中，我们将数学和数值分析，建模和大规模科学计算结合起来，研究流体动力学和材料科学中的某些基本问题。其中一个问题是理解扩散主导系统中的模式形成。这在金属合金的生产中具有重要意义。 在合金生产过程中观察到的合金内形成的复杂图案强烈影响所得合金的整体性质，例如其强度和韧性。 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波浪的相互作用，以及在拓扑类型上发生变化的流动动力学，例如通过混合两种流体产生气泡。 所有上述问题都是相似的，因为相关现象受到界面表面张力（在材料科学背景下称为“表面能”）的严重影响。 因此，它们可以通过一些常见的分析和计算工具来研究。

9404554 Shelley 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 c an be studied by some common analytical and computational tools.