Discrete Nonlinear Elasticity: Differential Complexes and Incompressibility

这项补助金的研究目标是奠定基础的几何理论的离散弹性使用技术从微分几何，代数拓扑，和离散外部演算。这个数学框架将是至关重要的，在建立新的数值方法，是免费的虚假的数值工件，如数值耗散和体积锁定，同时反映相应的连续模型，例如，他们保存能量和线性和角动量。该项目的结果有可能彻底改变离散化在计算力学中的理解和实现方式。特别是，我们制定不可压缩的弹性不使用拉格朗日乘子，我们极值化的体积保持运动的流形上的弹性作用。我们引入了一个非线性弹性复合体，它可以用于离散不可压缩非线性弹性的严格收敛分析。如果成功，所提出的研究活动将导致一个几何离散弹性理论，将统一所有现有的数值方法，并将有可能建立新的和更强大的数值方案，反映相应的连续介质模型在其控制方程的形式，守恒定律和内部约束。 这个跨学科的项目将从根本上影响计算力学，通过展示几何技术的重要性和在为不同的离散领域选择正确的离散空间时获得的新见解。

The research objective of this grant is to lay the foundations of a geometric theory of discrete elasticity using techniques from differential geometry, algebraic topology, and discrete exterior calculus. This mathematical framework will be critical in building new numerical methods that are free of spurious numerical artifacts, e.g. numerical dissipation and volume locking, while mirroring the corresponding continuum models, e.g. they conserve energy and linear and angular momenta. The results of this project have the potential to transformatively change the way discretization is understood and implemented in computational mechanics. In particular, we formulate incompressible elasticity without using Lagrange multipliers; we extremize the elasticity action over the manifold of volume-preserving motions. We introduce a nonlinear elasticity complex that can be used in a rigorous convergence analysis of discrete incompressible nonlinear elasticity.If successful, the proposed research activities will 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. This interdisciplinary project will fundamentally impact computational mechanics by demonstrating the importance of geometric techniques and the new insights gained in choosing the correct discrete spaces for different discrete fields.