AF: Large: Eulerian Computational Mechanics through Variational Principles

离散力学是一种相对较新的技术，它推导出高效和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。由于这些积分器基于力学原理，它们在计算水平上继承了大多数力学系统的变分性质，从粒子力学到连续介质力学，具有惊人的效率和鲁棒性，并且具有任意的精度。我们的研究目标是进一步发展这一技术，并引入一类全新的基于欧拉的偏微分方程变分积分器。通过设计，我们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，包括相互交织的理论和实践组成部分，旨在为流体动力学，电磁学，磁流体动力学，等离子体物理学和反应性CFD中的广泛现象提供简单和预测的模拟工具。虽然离散力学变分公式的思想是椭圆问题的标准，但直到最近才被用于导出力学系统的变分时间步算法。我们希望利用力学中关于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机械积分器领域。计算和数学基础，以及由此产生的积分器，将为理解离散化和数值在欧拉模拟中的基本作用提供新的见解。因此，这些新的计算工具将在磁流体力学、等离子体物理、带电流体和复杂流体等一系列具有挑战性的物理问题中推进最先进的技术。我们的团队，凭借在多学科计算项目和学术与产业合作方面的现有优势，将开发数学计算方面的创新课程，以培养未来的理论和计算问题科学家。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踏脚石之外，该项目中力学和几何物体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互动。

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.