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

影响范围从非常小到非常大的尺度支配物理现象，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糙的尺度效应，而无需在计算中显式地解决最精细的尺度。这项研究的关键在于仔细转移的结构存在于这些现象的数学模型（本质上有无限的分辨率）的计算领域，其有限的分辨率和有限的计算资源。正在开发的方法将允许快速评估总体影响，并在需要额外细节的情况下通过计算“向下钻取”。物理系统通常由一组连续方程描述，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。从一开始就离散的理论（而不是事后离散化的理论），具有内置的关键几何性质，可以更容易地产生鲁棒的数值模拟，这些模拟对底层的连续系统是真实的：它们在离散计算领域中精确地保持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，物理学和数值计算都需要这样的多分辨率策略。本研究计画为离散变分法与离散微分几何发展多解析度理论，并应用于薄壳与流体模型。主要的科学创新在于跨计算尺度保持对称性的技术。

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.