Collaborative: MSPA-MCS: Computational and Mathematical Foundations for the Synthesis of Multiresolution Representations with Variational Integrators and Discrete Geometry

从非常小到非常大的尺度影响着物理现象，如风暴系统的演变或事故中汽车的结构变形。通过在计算机模拟中解析最精细的尺度来准确预测这些是非常昂贵的。研究人员正在研究细尺度信息如何影响粗尺度行为，反之亦然。实际上，“总结”这些关系使我们能够准确有效地模拟粗尺度效应，而无需在计算中明确地解决最细的尺度。这项研究的关键在于将这些现象的数学模型（本质上具有无限分辨率）中的结构仔细转移到具有有限分辨率和有限计算资源的计算领域。正在开发的方法将允许快速评估总体影响，并能够在需要额外细节的地方进行计算“深入”。物理系统通常用一组连续方程来描述，使用几何力学和微分几何的工具来分析和捕获它们的属性。为了计算的目的，必须推导出基本方程的离散（在空间和时间上）表示。理论从一开始就是离散的（而不是事后离散的），内置了关键的几何属性，可以更容易地产生对潜在连续系统真实的鲁棒数值模拟：它们在离散计算领域中精确地保留了连续系统的不变量。到目前为止，这些方法还没有考虑到跨尺度的影响。然而，物理和数值计算都需要这样的多分辨率策略。这个研究项目是为离散变分方法和离散微分几何开发一个多分辨率理论，并将其应用于薄壳和流体建模。主要的科学创新在于在计算尺度上保持对称性的技术。

AbstractEffects ranging from very small to very large scales govern physical phenomena such as the evolution of a storm system or the structural deformation of an automobile during an accident. Accurately predicting these by resolving the finest scales in a computer simulation is prohibitively expensive. The investigators are studying how fine scale information impacts coarse scale behavior and vice versa. In effect, "summarizing" these relationships allow us to model coarse scale effects accurately and efficiently without the need to explicitly resolve the finest scales in a computation. A key to this study lies in the careful transfer of structures present in the mathematical models of these phenomena (which in essence have infinite resolution) to the computational realm with its finite resolution and finite computational resources. The methods being developed will allow rapid assessment of overall effects with the ability "to drill down" computationally where additional detail is required.Physical systems are typically described by a set of continuous equations using tools from geometric mechanics and differential geometry to analyze and capture their properties. For purposes of computation one must derive discrete (in space and time) representations of the underlying equations. Theories which are discrete from the start (rather than discretized after the fact), with key geometric properties built in, can more readily yield robust numerical simulations which are true to the underlying continuous systems: they exactly preserve invariants of the continuous systems in the discrete computational realm. So far these methods have not accounted for effects across scales. Yet both physics and numerical computation require such multi-resolution strategies. This research project is developing a multi-resolution theory for discrete variational methods and discrete differential geometry with applications to thin-shell and fluid modeling. The principal scientific innovation lies in techniques to conserve symmetries across computational scales.