Time Discretization of Structural Problems

Until recently, scientists investigating structural problems were primarily concerned with finding efficient methods for discretizing the spatial component of the structure. Once they constructed an ODE matrix system with a spatial semidiscretization, they simply used standard ODE methods to time-step the system so as to approximate the structural dynamics. Unfortunately these standard ODE methods were often inefficient but most researchers were not concerned with the inefficiencies, because modal bases were considered in many initial structural investigations [7, 8, 13] and the small systems involved often allowed such techniques to be successful. In contrast, the larger systems of fluid dynamics required scientists studying those problems to research efficient temporal solution techniques. More recent investigations of structural problems have illustrated the difficulties associated with modal expansions for complex systems [2, 14] , leading these investigators to develop more sophisticated approximation techniques. The problem is magnified when structural models are coupled with fluid or acoustic equations to model fluid-structure or structural acoustic systems. Upon spatial discretization, such models can yield ODE systems with up to 1000 unknowns. For systems of this size, scientists must find efficient time-marching techniques to permit numerical simulation of system dynamics. The problem is exacerbated in parameter estimation studies which require multiple system evaluations and control implementation which ultimately requires real time system integration. This project addresses the problem of efficient temporal solution of structural problems through a comprehensive study of several commercial ODE packages, as well as the programming and investigation of a method currently employed in fluid applications. The investigation was performed in the context of approximating the dynamics of a thin cylindrical shell. This structure was chosen since it commonly arises in applications and the model is complex enough to yield relatively large ODE systems. To evaluate the various ODE routines, three criteria were considered: preservation of the accuracy of the spatial discretization, amount of required system time, and number of function evaluations. The method must preserve the accuracy of the spatial discretization throughout the time interval of interest. To determine this, the investigation used a Fourier/cubic spline method to discretize the spatial components. As demonstrated by numerical results in [10, 11], this yielded an order O‘ (hx) spatial convergence rate for spatial gridlengths hx. The first test of