AF: Large: Eulerian Computational Mechanics through Variational Principles

离散力学是一种相对较新的技术，用于推导出高效且可预测的时空积分器，这些积分器忠实于它们要模拟的连续世界。由于这些积分器是基于机械原理，他们继承，在计算层面上，大多数机械系统的变分性质，从粒子力学连续介质力学，具有惊人的效率和鲁棒性，并与任意精度。我们的研究目标是进一步发展这种技术，并引入一类全新的基于欧拉的变分积分偏微分方程。通过设计，我们的基于几何的时间积分器将展示变分积分器的所有经典特征。因此，我们的研究，包含交织在一起的理论和实践的组成部分，旨在产生简单的和预测性的仿真工具，广泛的现象，在流体动力学，电磁学，磁流体力学，等离子体物理学和反应CFD.While的离散化变分公式的力学的想法是标准的椭圆问题，直到最近才被用来推导变分时步算法的机械系统。我们希望利用力学中拉格朗日和哈密顿原理的大量文献来开发离散的、基于欧拉的变分积分器，并进一步扩展过去几年开发的几何力学积分器的领域。所寻求的计算和数学基础，沿着产生的积分器，将提供新的见解，在理解欧拉模拟中的离散化和数值的基本作用。因此，这些新的计算工具将在广泛的具有挑战性的物理问题，如磁流体力学，等离子体物理学，带电和复杂的流体中推进最先进的。我们的团队，在多学科计算项目和学术产业合作的现有优势，将开发一个创新的数学计算课程，以培养未来的科学家在理论和计算问题。除了为我们解决复杂，多尺度物理现象的长期研究目标提供垫脚石外，该项目中力学和几何对象的协同混合有望成为基础和计算重要方法的丰富来源，以及物理科学，数学和信息技术之间的跨学科交互。

Discrete mechanics is a relatively recent technique to derive efficient and predictive space-time integrators that are faithful to the continuous world they are meant to simulate. As these integrators are based on mechanical principles, they inherit, at the computational level, the variational nature of most mechanical systems, from particle mechanics to continuum mechanics, with surprising efficiency and robustness, and with arbitrary accuracy. Our research goal is to further develop this technique and introduce an entirely new class of Eulerian-based variational integrators of partial differential equations. By design, our geometry-based time integrators will exhibit all of the classic hallmarks of variational integrators. Consequently our research, containing intertwined theoretical and practical components, is aimed at producing simple and predictive simulation tools for a wide range of phenomena in fluid dynamics, electromagnetism, magnetohydrodynamics, plasma physics, and reactive CFD.While the idea of discretizing variational formulations of mechanics is standard for elliptic problems, only recently was it used to derive variational time-stepping algorithms for mechanical systems. We wish to leverage the large body of literature on Lagrangian and Hamiltonian principles in mechanics to develop discrete, Eulerian-based variational integrators and further extend the realm of geometric mechanical integrators developed over the past few years. The computational and mathematical foundations sought after, along with the resulting integrators, will provide fresh insight in understanding the fundamental role of discretization and numerics in Eulerian simulations. These new computational tools will consequently advance the state-of-the-art in a wide range of challenging physical problems such as magnetohydronamics, plasma physics, charged, and complex fluids. Our team, with existing strengths in multidisciplinary computational projects and academic-industry collaborations, will develop an innovative curriculum in mathematical computing to train future scientists in theoretical and computational matters. Besides providing a stepping stone for our long-term research goal of solving complex, multiscale physical phenomena, the synergetic mixture of mechanics and geometric objects in this project promises to be a rich source of fundamental and computationally important methods, as well as interdisciplinary interactions between physical sciences, mathematics, and information technology.