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.
The objective of this EAGER was 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 mirror the corresponding continuum models, e.g. they conserve energy and momenta. Intellectual Merit: The proposed computational geometric mechanics uses a novel self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes. Instead of discretizing continuum governing equations, we will start ab initio with no reference to any continuum quantity and will directly discretize the configuration space of the problem. In the case of incompressible linear elasticity, instead of imposing incompressibility as an internal constraint, we discretized the space of volume-preserving diffeomorphisms. The existing numerical methods use the constitutive equations after interpolation of discrete fields. Broader Impact: The proposed 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. In parallel, the problems studied in this project will motivate new research directions, e.g. discretization of bundle-valued differential forms. Project Outcomes: In the past century, physical theories such as relativity and string theory were formulated using the language of topology and differential geometry. Successful implementation of these theories motivated the idea of using these powerful mathematical methods in more practical engineering theories such as electromagnetism and continuum mechanics. For continuum mechanics and, in particular, nonlinear elasticity, certain rich geometrical structures have been found. However, these ideas are mostly used for introducing basic concepts and obtaining general governing equations; there hasn’t been any efforts to actually use such geometrical techniques to numerically solve the governing equations - the most important part of each engineering theory. On the other hand, it has been observed that the traditional numerical schemes that use classical methods of discretization such as finite elements and finite volumes are subject to numerical artifacts. The locking phenomenon that occurs in modeling incompressible elastic bodies is an example of such numerical errors. Interestingly, to this date there is no robust numerical scheme for nonlinear elasticity. Our idea is to use geometrical structure of continuous media and in particular nonlinear elasticity not only to formulate governing equations, but also to actually solve these equations. We believe that the reason for such numerical errors is the lack of correct methods of discretization compatible to the geometrical structures. During the course of this investigation, we extended the smooth geometrical structure of elastic bodies to the discrete models suitable for numerical computations. We presented a discrete geometric structure-preserving numerical scheme for incompressible linearized elasticity. We proved that the governing equations of finite and linearized incompressible elasticity can be obtained using Hamilton's principle and Hodge decomposition theorem without using Lagrange multipliers. We used ideas from algebraic topology, exterior calculus, and discrete exterior calculus to develop a discrete geometric theory for linearized elasticity. We considered the discrete displacement field as our primary unknown and characterized the space of divergence-free discrete displacements as the solution space. Therefore, we preserve the geometric structure of the smooth problem by considering discrete quantities that have the same geometric structure as their smooth counterparts. Finally, motivated by the Lagrangian structure of the smooth case, we defined a discrete Lagrangian and used Hamilton's principle in the space of discrete divergence-free displacement fields to obtain the governing equations of the discrete theory. We used a discrete Laplace-Beltrami operator to determine the pressure field, which is assumed to be a dual 0-form. We then considered some numerical examples and observed that our discretization scheme is free of numerical artifacts, e.g. checkerboarding of pressure. Based on the rate of convergence of the results of the numerical examples, our method is comparable with finite element mixed formulations. Contrary to the most mixed formulations that are designed for near-incompressible elasticity, our method is specifically designed for incompressible elasticity. Applications to fluid mechanics and finite elasticity and studying convergence issues will be the subjects of future research.
