CAREER: Theory and Practice of Space-Time Variational Integrators for Simulation and Animation

该研究项目的主要目标是开发新的数值模拟方法，提供广泛的应用，从模拟烟雾和火焰与移动物体的相互作用，到浸入流体中的病毒表面的演变。这种方法利用了对过去一个世纪发展起来的连续运动方程的几何理解，它将动量和能量保存作为基本对称性和不变性的结果。然而，目前的数值模拟技术很少捕捉到数学和物理学中显示的对适当的动力学行为至关重要的几何原理。为了开发在数值上尊重复杂系统的时间和空间结构的仿真算法，本研究的主要部分需要开发一种新的框架来结合拉格朗日和欧拉方法，这是两种常见的（但很少结合）运动表示。关键应用包括薄壳模拟、碰撞处理和流固耦合。教育方面的努力利用了图形模拟演示对学生的吸引力。教育活动的主要目标是激励学生，包括代表性不足的群体和K-12学生，不仅要在计算机科学方面进行培训，还要在物理、数学和机械工程方面进行培训，以面对科学日益跨学科的本质。由此产生的计算工具和实际应用（包括，例如，生物计量学中的变形模型、蛋白质折叠和对接以及地下水模拟）将通过期刊和会议的出版物传播。项目结果可在项目网站（http://www.cse.msu.edu/~ytong/SpaceTimeIntegrator.html）上找到。

The primary goal of this research project is to develop novel numerical simulation methods offering a wide range of applications, ranging from the simulation of smoke and fire interacting with moving objects, to the evolution of virus surfaces immersed in fluids. The approach leverages geometric understanding of the continuous equations of motion developed over the past century, which frames momentum and energy preservation as consequences of basic symmetries and invariance. However, current numerical techniques for simulation rarely capture the geometric principles that have been shown in mathematics and physics to be fundamental to proper dynamical behavior. To develop simulation algorithms that numerically respect both temporal and spatial structures of complex systems, a major part of this research requires the development of a novel framework to combine Lagrangian and Eulerian methods, two common (but rarely combined) representations of motion. Key applications include thin-shell simulation, collision handling, and fluid-structure interaction. The educational efforts leverage the appeal that graphical simulation demos have to students. The main objective of the educational activities is to motivate students, including underrepresented groups and K-12 students, to train not just in computer science, but also in physics, mathematics, and mechanical engineering to face the increasingly interdisciplinary nature of science.The resulting computational tools and practical applications (including, e.g., deformation model in biometrics, protein folding and docking, and groundwater simulation) will be disseminated through publications in journals and conferences. Project results can be found at the project website (http://www.cse.msu.edu/~ytong/SpaceTimeIntegrator.html).