TY - GEN

T1 - Finite element formulation via the theorem of expended power

T2 - Total energy framework

AU - Har, Jason

AU - Tamma, Kumar K.

PY - 2009/1/1

Y1 - 2009/1/1

N2 - 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 [ε (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.

AB - 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 [ε (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.

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U2 - 10.2514/6.2009-2579

DO - 10.2514/6.2009-2579

M3 - Conference contribution

AN - SCOPUS:84855625164

SN - 9781563479731

T3 - Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference

BT - 17th AIAA/ASME/AHS Adaptive Structures Conf., 11th AIAA Non-Deterministic Approaches Conf., 10th AIAA Gossamer Spacecraft Forum, 5th AIAA Multidisciplinary Design Optimization Specialist Conf., MDO

PB - American Institute of Aeronautics and Astronautics Inc.

ER -