Total energy framework, finite elements, and discretization via Hamilton's law of varying action

Within the total energy framework which we introduce here for the first time (in contrast to Lagrangian or Hamiltonian mechanics framework), we provide an alternative and have developed in this paper a general numerical discretization for continuum-elastodynamics directly stemming from Hamilton’s law of varying action (HLVA) involving a measurable built-in scalar function, namely, Total Energy $${[{\mathcal{E}}\left({{\boldsymbol{q}},\dot{{\boldsymbol{q}}}}\right): TQ\rightarrow {\mathbb{R}}]}$$ . The Total Energy we use herein for enabling the space discretization is defined as the kinetic energy plus the potential energy for N-body systems, and the kinetic energy plus the total potential energy for continuum-body systems. It thereby provides a direct measure and sound physical interpretation naturally, while enabling this framework to permit general numerical discretizations such as with finite elements. In the variational formulation proposed here, we place particular emphasis upon the notion that the scalar function which represents the autonomous total energy of the continuum/N-body dynamical systems can be a crucial mathematical function and physical quantity which is a constant of motion in conservative systems. In addition, we prove that the autonomous total energy possesses the three invariant properties and can be viewed as the so-called total energy version of Noether’s theorem; therefore, the autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts directly emanating from HLVA inherently involving the scalar function, namely, total energy: (i) can be shown to yield the same governing mathematical model equations of motion that are continuous in space and time together with the natural boundary conditions just as Hamilton’s principle (HP) is routinely used to derive such equations, but without obvious inconsistency via such a principle as explained in the paper; (ii) explain naturally the Bubnov–Galerkin weighted-residual form that is customarily employed for discretization for both space and time, and alternately, (iii) circumvent relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton’s second law) involving Cauchy’s equations of motion (governing equations) arising from continuum mechanics or via (i) and (ii) above if one chooses this option, and instead provides new avenues of discretization for continuum-dynamical systems. The present developments naturally embody the weak form in space and time that can be described by a discrete Total Energy Differential Operator (TEDO). Thereby, a novel yet simple, space-discrete Total Energy formulation proposed here only needs to employ the discrete TEDO which provides new avenues and directly yields the semi-discrete ordinary differential equations in time which can be readily shown to preserve the same physical attributes as the continuous systems for continuum-dynamical applications unlike traditional practices. The modeling of complicated structural dynamical systems such as Euler-Bernoulli beams and Reissner–Mindlin plates is particularly shown here for illustration.