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Nonlinear transient response and its sensitivity using finite elements in time

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Abstract

The bilinear formulation proposed earlier by Peters and Izadpanah to develop finite elements in time to solve undamped linear systems, is extended (and found to be readily amenable) to develop time finite elements to obtain transient responses of both linear and nonlinear, and damped and undamped systems. The formulation is used in the h-, p- and hp-versions. The resulting linear and nonlinear algebraic equations are differentiated to obtain the sensitivity of the transient response with respect to various design parameters. The present developments were tested on a series of linear and nonlinear examples and were found to yield, when compared with results obtained using other methods, excellent results for both the transient response and its sensitivity to system parameters. Mostly, the results were obtained using the Legendre polynomials as basis functions, though, in some cases other orthogonal polynomials namely, the Hermite, the Chebyshev, and integrated Legendre polynomials were also employed (but to no great advantage). A key advantage of the time finite element method, and the one often overlooked in its past applications, is the ease with which the sensitivity of the transient response with respect to various system parameters can be obtained.

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Abbreviations

B:

bilinear term in the variational formulation

a:

linear term in the variational formulation

δ:

variational operator

T:

kinetic energy

V:

potential energy

Q i :

nonconservative forces not included in the variational operation

\(s_{_t } ,\dot s_{_t }\) :

generalized coordinates: displacement, velocity

T 0 :

initial time

T f :

final time

p k :

design parameters

q j :

generalized coordinates

{}:

column vector

<>:

row vector

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Authors and Affiliations

  1. Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, 24061-0203, Blacksburg, VA, USA

    R. K. Kapania & S. Park

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  1. R. K. Kapania
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  2. S. Park
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Additional information

Communicated by S. N. Alturi, 25 September 1995

Portions of this research were supported by a grant from Army Research Office, Grant No. DAALO3-90-G-0134, with Dr. Gary Anderson as the grant monitor.

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Kapania, R.K., Park, S. Nonlinear transient response and its sensitivity using finite elements in time. Computational Mechanics 17, 306–317 (1996). https://doi.org/10.1007/BF00368553

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