Flexible beam in R3 under large overall motions and Asynchronous Variational Integrators

The theory of discrete variational mechanics has its roots in the optimal control literature of the 1960's. The past ten years have seen a major development of discrete variational mechanics and corresponding numerical integrators, due largely to pioneering work by Jerrold Marsden and his collaborators at Caltech. Discrete mechanics emerged from the interplay of classical theoretical mechanics, numerical analysis, and computer science. Recent years have seen an explosive growth of research in discrete mechanics, discrete exterior calculus, and corresponding integrators capable of preserving geometric structure in mechanical systems. There has been a growing realization that stability of numerical methods can be obtained by methods which are compatible with these structures in the sense that many discrete variational integrators are symplectic-momentum methods. That is, they preserve the symplectic structure of phase space and momentum maps arising from the symmetries and invariants of the system. Moreover asynchronous variational integrators (AVI) conserve energy. Remarkably, to our knowledge, there is no major application of these discrete mechanics techniques to civil engineering. In particular, we are not aware of any application of discrete mechanics to the study of beams and surfaces formed by elements tied by multi-edges, exhibiting sharp corners under constraints and with large deflections. Applying geometric mechanics methods that can more faithfully recapitulate global behavior by decoupling it from constraints for local accuracy permits the development of dynamic modeling and static two-dimensional simulations.