Numerical analysis of dissipative dynamical systems in solid and fluid mechanics, with a special emphasis on coupled problems.

摘要

The goals of this dissertation are the formulation, analysis and implementation of efficient time-stepping algorithms that optimally represent the dissipative properties of a number of systems in solid and fluid mechanics, with special emphasis on coupled problems.; The first part focuses on coupled thermomechanical problems in solid mechanics, ranging from linearized thermoelasticity to finite strain thermoplasticity. A-priori stability estimates are derived for the continuum problems, identifying the dissipative structure of these systems. Given their coupled character, computational efficiency requires fractional-step (staggered) methods, characterized by a modular and low-cost implementation. However, standard staggered schemes are only conditionally stable. The analysis presented here in the context of the classical theory of operator splits points out the cause of this limitation; namely, the previously identified dissipative structure is broken by the split. Moreover, this approach allows the formulation and analysis of a new class of nonlinear unconditionally stable staggered time-stepping algorithms, the first of this kind. The extension to other coupled systems in solid mechanics is indicated.; The second part concentrates on incompressible Navier-Stokes and coupled magnetohydrodynamics (MHD). These systems possess an asymptotic long-term dissipative structure characterized by an absorbing set and a global attractor. Time-stepping algorithms are analyzed in this context. Rigorous nonlinear stability and long-term analyses are presented for the time semidiscrete and the fully discrete mixed finite element systems. Direct and projection methods (that is, fractional-step methods designed to enforce the incompressibility constraint) are considered. A class of time-stepping schemes presenting nonlinear stability and optimally dissipative long-term properties is identified which, in addition, is linear within the time step. The existence of an algorithmic attractor is proven for a characteristic member of this class. The combination of these results with the strategy outlined above for thermomechanical problems leads to the formulation of new monolithic and staggered algorithms for coupled MHD that exhibit these same properties.; The efficiency of all these time-stepping algorithms is demonstrated via a number of numerical simulations that employ both standard and non-standard finite element methods.