#### Vol. 4, No. 3, 2009

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

### Jason Har and Kumar K. Tamma

Vol. 4 (2009), No. 3, 475–508
##### 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 $\mathsc{ℒ}\left(q,\stackrel{̇}{q}\right):TQ\to ℝ$, the Hamiltonian $\mathsc{ℋ}\left(p,q\right):{T}^{\ast }Q\to ℝ$, and the total energy $\mathsc{ℰ}\left(q,\stackrel{̇}{q}\right):TQ\to ℝ$. 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.

##### Keywords
Lagrangian, Hamiltonian, total energy, finite elements, theorem of expended power, $N$-body problem, continuous bodies, elastodynamics