EAGER: Structure-Preserving Discretization of Elasticity Using Geometric Ideas

这个探索性研究早期资助（EAGER）奖的目的是为离散弹性的几何理论奠定基础。该项目使用一种新颖的自洽几何和力学离散化来系统地构建几何结构保持数值格式。该项目没有离散连续统控制方程，而是在不涉及任何连续统量的情况下“从头开始”；构型空间将被直接离散化。以不可压缩非线性弹性为例，将保容微分同态空间离散化，而不是将不可压缩性作为内部约束。现有的数值方法采用离散场插值后的本构方程。在这个项目中，本构方程的几何结构及其结构保持离散化将被仔细研究。研究活动将有可能导致一个几何离散弹性理论，它将统一所有现有的数值方法，并将使建立新的和更健壮的数值方案成为可能，这些方案以其控制方程、守恒定律和内部约束的形式反映相应的连续体模型。这种方法的社会效益将是修改/改进现有的分析代码，使它们更加可靠。更可靠的分析工具将带来更好的设计（避免结构的灾难性失效），同时，将允许制造更轻、更高效的结构。在教育方面，PI将吸引研究生参与计算力学和微分几何之间的多学科研究。

The objective of this EArly Grant for Exploratory Research (EAGER) award is to lay the foundations of a geometric theory of discrete elasticity. The project uses a novel self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes. Instead of discretizing the continuum governing equations, the project starts "ab initio" with no reference to any continuum quantity; the configuration space will be directly discretized. As an example, in the case of incompressible nonlinear elasticity, instead of imposing incompressibility as an internal constraint, the space of volume-preserving diffeomorphisms will be discretized. The existing numerical methods use the constitutive equations after interpolation of discrete fields. In this project, the geometric structure of constitutive equations and their structure-preserving discretizations will be carefully investigated.The research activities will potentially lead to a geometric discrete elasticity theory that will unify all the existing numerical methods, and will make it possible to build new and more robust numerical schemes that mirror the corresponding continuum models in the form of their governing equations, conservation laws, and internal constraints. The social benefit of this approach will be in modifying/improving the existing analysis codes and making them more reliable. More reliable analysis tools will lead to better designs (avoiding catastrophic failures of structures) and, at the same time, will allow lighter and more efficient structures to be manufactured. On the educational side, the PI will engage graduate students in a multidisciplinary research at the interface between computational mechanics and differential geometry.