EAGER: Structure-Preserving Discretization of Elasticity Using Geometric Ideas

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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.