Finite element formulations via the theorem of expended power in the Lagrangian, Hamiltonian and total energy frameworks

Abstract

Traditionally, variational principles and variational methods have been employed in describing finite element formulations for elastodynamics applications. Here we present alternative avenues emanating from the theorem of expended power, using the differential calculus directly. We focus on scalar representations under three distinct frameworks: Lagrangian mechanics, Hamiltonian mechanics, and a new framework involving a built-in measurable quantity, called the total energy in the configuration space. All three frameworks are derivable from each other, since they represent the same physics as Newton’s second law; however, the total energy framework which we advocate inherits features that are comparable and competitive to the usual Newtonian based finite element formulations, with several added advantages ideally suited for conducting numerical discretization. The present approach to numerical space-time discretization in continuum elastodynamics provides physical insight via the theorem of expended power and the differential calculus involving the distinct scalar functions: the Lagrangian L.q;P q/V T Q! R, the Hamiltonian H.p;q/V T Q! R, and the total energy E.q;P q/V T Q! R. We show that in itself the theorem of expended power naturally embodies the weak form in space, and after integrating over a given time interval yields the weighted residual form in time. Hence, directly emanating from the theorem of expended power, this approach yields three differential operators: a discrete Lagrangian differential operator, a Hamiltonian differential operator, and a total energy differential operator. The semidiscrete ordinary differential equations in time derived with our approach can be readily shown to preserve the same physical attributes as the corresponding continuous systems. This contrasts with traditional approaches, where such proofs are nontrivial or are not readily tractable. The modeling of complicated structural dynamical systems such as a rotating bar and the Timoshenko beam are shown for illustration. Variational concepts have long played a significant role in the development of numerical discretization techniques. First, variational principles in physics and mechanics have been used to derive the governing differential equations from which numerical discretizations are routinely conducted. The principles — including Hamilton’s principle in dynamics, in the electromagnetic and gravitational fields, and even in quantum mechanics; the principle of stationary potential energy and the Hu‐Washizu’s variational principle in continuum statics; Fermat’s principle of least time leading to Snell’s law in optics; Maupertuis’ principle of least action, Jacobi and Lagrange’s principle of least action in dynamics again; and Gurtin’s