Theorem of Expended Power and Finite Element Formulation: Hamiltonian Mechanics Framework

Abstract

Within the Hamiltonian mechanics framework described here, we have developed in this paper a general numerical discretization via the finite element method for continuum-elastodynamics applications directly stemming from the theorem of expended power involving a built-in scalar function: namely, the autonomous Hamiltonian (H(p, q): T * Q → R). To provide alternative viewpoints and new avenues, a differential formulation is proposed here rather than resorting to classical variational formulations in which, historically, most finite element developments dealing with elastodynamics have been traditionally described. We place particular emphasis upon the notion that the scalar function that represents the autonomous Hamiltonian of the continuous-body dynamical systems can be a crucial mathematical function and physical quantity that is a constant of motion in conservative systems; this is in contrast to vector quantities customarily employed with Newton-based formulations for which physical quantities are more difficult to associate. As such, the scalar representation enables one to readily capitalize on theorems such as Noether's to establish symmetry properties so that proof of satisfaction of the discrete system as that of the continuous system can be readily established, unlike traditional practices emanating from Newton-based representations. The proposed concepts emanating from the theorem of expended power inherently involving the scalar function (namely, the Hamiltonian) naturally embody the weak form in space that can be described by a discrete Hamiltonian differential operator, and integrating over a given time interval yields the weighted residual in time statement. Thereby, a novel yet simple space-discrete Hamiltonian formulation proposed here only needs to employ the discrete Hamiltonian differential operator, which provides new and alternate avenues and directly yields the semidiscrete ordinary differential equations in time that can be readily shown to preserve the same physical attributes as the continuous systems for continuum-dynamical applications. Theoretical formulations are shown for selected structural elements for illustration of the basic concepts.