Formulation and analysis of dissipative algorithms for nonlinear elastodynamics

In this dissertation we describe the formulation and analysis of a new class of time integration schemes for nonlinear problems of elastodynamics. More specifically, we describe its application in the context of continuum elastodynamics and nonlinear models of shells and rods.; These algorithms allow controllable and unconditional numerical dissipation of the mechanical energy in the full nonlinear range where the elastic problems of interest are formulated. This property holds for any type of hyperelastic constitutive law and the conservation of energy can be recovered as a particular case. The control over the evolution of the energy avoids the appearance of numerical instabilities which typically result in uncontrollable energy growth. The (controllable) numerical dissipation handles the high frequency part of the discrete solution which is often responsible for the blow up of the computation of numerically stiff problems, such as the ones under consideration. As a result of these conservation/dissipation properties, the algorithms presented are very robust for the numerical integration of stiff problems of elastodynamics.; All the elastic models considered in this dissertation are particular instances of Hamiltonian systems with symmetries. The algorithms proposed can in fact be extended to a wider class of elastic Hamiltonian problems in a straightforward manner. For all these problems, the time-stepping methods proposed preserve unconditionally the characteristic conservation laws of momentum and the relative equilibria. As a result, the long term numerical solutions obtained give an accurate picture of the dynamical behavior of these mechanical systems.; The numerical implementation of these methods is described in detail. The practical aspect of their computational cost is carefully evaluated and an iterative implementation is suggested whose cost is comparable to the cost of commonly used integrators.; Numerical examples are presented in each of the elastic frameworks described. They illustrate the performance of the proposed time-stepping algorithms in the context of stiff numerical problems, with particular emphasis in the stability and long term behavior. For each example, the new methods are compared with existing time-stepping algorithms.