Geometric Numerical Discretizations of Gauge Field Theories and Interconnected Systems

实用的工程系统，如人体假肢中的机电系统，通常涉及由耦合的、不同的物理理论描述的组件。该研究项目开发了通过将简单的模型连接在一起来轻松建模复杂系统的方法，这使得工程社区更容易使用复杂的数学和建模工具。这将减少构建高保真数学模型的障碍，这些模型可用于在合理设计过程中进行快速原型设计，从而将概念设计更好、更便宜、更快速地转化为商业上可实现的产品。除了建模，研究还涉及到为互联系统设计控制器。这与自主空中和水下航行器的分布式网络控制有关，这些网络应用于搜索和救援，海洋勘探以及灾害监测和国土安全的分布式传感器网络。为了进一步促进这项研究在工程界的传播，研究人员和他的合作者将编写一本两卷本的研究生教材，供广大技术读者使用。狄拉克和多狄拉克力学和几何提供了一个统一的数学框架来描述拉格朗日和哈密顿力学和场论，以及简并的、相互联系的和非完整的系统。变分积分器产生几何结构保留数值方法，自动保留辛形式和动量映射，并表现出优异的长期能量稳定性。该项目研究了对协变规范理论（如电磁学和广义相对论）和互连系统（如电路和多体系统）的离散化变分积分器方法的深远推广。这包括：(i)构造和分析拉格朗日偏微分方程的多重狄拉克变分积分器的系统框架；（ii）利用时空有限元外演算和洛伦兹度量值测地线有限元研究协变规范理论的离散协变和瞬时公式与群等变有限元空间之间的联系；（iii）狄拉克机械系统的离散互连理论及其相关的控制理论。

Practical engineered systems, such as electro-mechanical systems in human prosthetics, typically involve components described by coupled, distinct physical theories. This research project develops methods to easily model complicated systems by connecting simpler models together, which allows sophisticated mathematical and modeling tools to be more readily accessible to the engineering community. This will reduce the barrier to constructing high-fidelity mathematical models that can be used for rapid prototyping in the rational design process, leading to better, cheaper, and faster translation of conceptual designs into commercially realizable products. In addition to modeling, the research also involves designing controllers for interconnected systems. This is relevant to the control of distributed networks of autonomous aerial and underwater vehicles, which have applications to search and rescue, ocean exploration, and distributed sensor networks for both disaster monitoring and homeland security. To further facilitate the dissemination of this research to the engineering community, the investigator and his collaborators will develop a two-volume graduate textbook that will be accessible to a broad technical audience.Dirac and multi-Dirac mechanics and geometry provide a unified mathematical framework for describing Lagrangian and Hamiltonian mechanics and field theories, as well as degenerate, interconnected, and nonholonomic systems. Variational integrators yield geometric structure-preserving numerical methods that automatically preserve the symplectic form and momentum maps, and exhibit excellent long-time energy stability. This project studies far reaching generalizations of the variational integrator approach for discretizing covariant gauge theories, such as electromagnetism and general relativity, and interconnected systems, such as electrical circuits and multibody systems. This involves: (i) a systematic framework for constructing and analyzing multi-Dirac variational integrators for Lagrangian partial differential equations; (ii) the connection between discrete covariant and instantaneous formulations of covariant gauge theories, and group-equivariant finite-element spaces using spacetime finite-element exterior calculus and Lorentzian metric-valued geodesic finite-elements; (iii) discrete interconnection theory for Dirac mechanical systems and their associated control theory.