Implicit time integration of a class of constrained hybrid formulations-Part I: Spectral stability theory

Incomplete field formulations have recently been the subject of intense research because of their potential in coupled analysis of independently modeled substructures, adaptive refinement, domain decomposition and parallel processing. This paper presents a spectral stability theory for the differential/algebraic dynamic systems associated with these formulations, discusses the design and analysis of suitable time-integration algorithms, and emphasizes the treatment of the inter-subdomain linear constraint equations. These constraints are shown to introduce a destabilizing effect in the dynamic system that can be analyzed by investigating the behavior of the time-integration algorithm at infinite and zero frequencies. Three different approaches for constructing penalty-free unconditionally stable second-order accurate solution procedures for this class of hybrid formulations are presented, analyzed and illustrated with numerical examples. In particular, the advantages of the Hilber-Hughes-Taylor (HHT) method and its generalized version (Generalized a) are highlighted. The family of problems discussed in this paper can also be viewed as model problems for the more general case of hybrid formulations with non-linear constraints. For example, it is shown numerically in this paper that the theoretical results predicted by the spectral stability theory also apply to non-linear multibody dynamics formulations. Therefore, some of the algorithms outlined in this work are important alternatives to the popular technique consisting of transforming differential/algebraic equations into ordinary differential equations via the introduction of a stabilization term that depends on arbitrary constants, and that influences the computed solution.