On the Numerical Discretization in Space and Time: Part 1 - Hamilton's Law of Varying Action Involving Lagrangian/Hamiltonian/Total Energy Framework

Abstract

The focus in Part 1 of this exposition is strictly restricted to holonomic-sceleronomic systems, and the applications of interest are elastodynamics, which are routinely encountered in a wide class of structural dynamics problems in engineering. Restricting attention to these considerations, new and different perspectives and equivalences are described which deal with finite element space discretization aspects employing three distinctly different frameworks with scalar formalism. It encompasses the Lagrangian mechanics, the Hamiltonian mechanics, and as an alternative, the framework with a built-in measurable quantity based on the Total Energy. Historically, traditional practices routinely employ the weighted residual method or equivalently the principle of virtual work in dynamics for developing finite element formulations. In contrast, the present developments stem from Hamilton's law of varying action (HLVA) as a starting point and involve distinctly different scalar descriptive functions (the Lagrangian [ℒ( q , [qdot] ) : TQ →ℝ], the Hamiltonian [ℋ( p , q ) : T* Q →ℝ], or the Total Energy [ℰ( q , [qdot] ):TQ →ℝ]). These developments naturally embody the weak form in space and the statement of the weighted residual in time. Complicated structural dynamical systems such as a rotating bar and the Timoshenko beam are particularly shown here simply for illustration. In Part 2, we describe the satisfaction of conservation properties of the fully discretized equations of motion in space and time with particular attention to the Total Energy framework (in contrast to the Lagrangian and the Hamiltonian), primarily because it is very natural and is computationally attractive and meaningful to conducting the time discretization [1] process. Instead of starting from the continuous form of representations, the particular focus is upon directly employing the discrete formulations for enabling algorithmic designs for the class of LMS methods for linear dynamic problems for illustration purposes. This is a necessary first step in the design of algorithms for integrating the equations of motion, and they serve as the parent algorithms for subsequent extensions to nonlinear dynamic problems.