Finite element formulations using the theorem of power expended: Total energy framework with a differential formulation

Traditional practices involving variational calculus have historically dominated most finite element formulations to-date, and have no doubt served as indispensable tools. Besides these practices, our recent contributions in Acta Mechanica (Har and Tamma, 2009, in press) described new alternatives and developments emanating from Hamilton's Law of Varying Action (HLVA) as a starting point with a measurable built-in scalar function, namely, the Total Energy. The associated framework (in contrast to Lagrangian or Hamiltonian mechanics framework) demonstrated certain new advances, and also provided some fundamental insight into explaining traditional practices of finite element discretization. Here we additionally provide other advances, new directions, and viable alternatives in contrast to all these past practices which routinely employ variational concepts. In particular, focusing on elastodynamics applications, in this paper we provide for the first time finite element formulations stemming instead from a differential formulation and the theorem of power expended with a measurable built-in scalar function, namely, the Total Energy {[\mathcal{E}({\varvec q},\dot{{\varvec q}}):TQ\to \mathbb{R}]}, as a starting point to capitalize on certain added advantages. The autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts: (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 balance laws such as linear momentum or Hamilton's principle are routinely used to derive such equations, but without resorting to any variational concepts, or approaches such as variational principles, (ii) explain naturally how the classical Bubnov-Galerkin weighted-residual form that is customarily employed for discretization can be readily constructed for both space and time, and alternately, (iii) circumvents relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton's law) involving Cauchy's equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above, and instead provides new avenues of discretization for continuum-dynamical systems. For illustration, numerical discretizations are presented for the modeling of complicated structural dynamical systems.