Finite Element Formulation Via the Theorem of Expended Power: Total Energy Framework

A general computational procedure and developments for the numerical discretization of continuum-elastodynamics problemns directly stemming from the theorem of expended power (TEP) involving a built-in scalar function, namely, Total Energy [E (q, q̇) : TQ → R], is presented within the total energy framework. This is in contrast to classical Lagrangian or Hamiltonian mechanics framework, and is a viable alternative. The proposed concepts emanating from the TEP inherently involving the scalar function, namely, total energy: 1) 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 resorting any variational approach such as the variational principles or the variational methods; 2) explains naturally how the Bubnov-Galerkin weighted-residual form that is customarily employed for discretization arises for both space and time, and alternately, 3) circumvent 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, and instead provides new avenues of discretization for continuum-dynamical systems. Applicability to nonlinear problems and nonholonomic-scleronomic constraints as well such as those involving contacts are also addressed. The modeling of complex structural dynamical systems, namely, Euler-Bernoulli beams and Reissner-Mindlin plates are illustrated. I