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

这个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，从模拟烟和火与运动物体的相互作用，到病毒表面在流体中的演变。这种方法充分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，该方程将动量和能量守恒作为基本对称性和不变性的结果。然而，目前用于模拟的数值技术很少捕捉到在数学和物理中已经被证明是正确的动力学行为的基础的几何原理。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方法和欧拉方法，这两种常用的(但很少组合的)运动表示。主要应用包括薄壳模拟、碰撞处理和流固耦合。这些教育工作充分利用了图形模拟演示对学生的吸引力。教育活动的主要目的是激励学生，包括代表性不足的群体和K-12学生，不仅在计算机科学方面，而且在物理、数学和机械工程方面进行培训，以面对日益增长的跨学科性质的科学。由此产生的计算工具和实际应用(包括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)将通过在期刊和会议上发表的出版物进行传播。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).