Discrete Nonlinear Elasticity: Differential Complexes and Incompressibility

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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.