1,153 research outputs found
On Convergence of a Three-layer Semi-discrete Scheme for the Nonlinear Dynamic String Equation of Kirchhoff-type with Time-dependent Coefficients
This paper considers the Cauchy problem for the nonlinear dynamic string
equation of Kirchhoff-type with time-varying coefficients. The objective of
this work is to develop a temporal discretization algorithm capable of
approximating a solution to this initial-boundary value problem. To this end, a
symmetric three-layer semi-discrete scheme is employed with respect to the
temporal variable, wherein the value of a nonlinear term is evaluated at the
middle node point. This approach enables the numerical solutions per temporal
step to be obtained by inverting the linear operators, yielding a system of
second-order linear ordinary differential equations. Local convergence of the
proposed scheme is established, and it achieves quadratic convergence
concerning the step size of the discretization of time on the local temporal
interval. We have conducted several numerical experiments using the proposed
algorithm for various test problems to validate its performance. It can be said
that the obtained numerical results are in accordance with the theoretical
findings.Comment: 31 pages, 13 figures, 11 table
Torsional instability in suspension bridges: the Tacoma Narrows Bridge case
All attempts of aeroelastic explanations for the torsional instability of
suspension bridges have been somehow criticised and none of them is unanimously
accepted by the scientific community. We suggest a new nonlinear model for a
suspension bridge and we perform numerical experiments with the parameters
corresponding to the collapsed Tacoma Narrows Bridge. We show that the
thresholds of instability are in line with those observed the day of the
collapse. Our analysis enables us to give a new explanation for the torsional
instability, only based on the nonlinear behavior of the structure
A discrete geometric approach for simulating the dynamics of thin viscous threads
We present a numerical model for the dynamics of thin viscous threads based
on a discrete, Lagrangian formulation of the smooth equations. The model makes
use of a condensed set of coordinates, called the centerline/spin
representation: the kinematical constraints linking the centerline's tangent to
the orientation of the material frame is used to eliminate two out of three
degrees of freedom associated with rotations. Based on a description of twist
inspired from discrete differential geometry and from variational principles,
we build a full-fledged discrete viscous thread model, which includes in
particular a discrete representation of the internal viscous stress.
Consistency of the discrete model with the classical, smooth equations is
established formally in the limit of a vanishing discretization length. The
discrete models lends itself naturally to numerical implementation. Our
numerical method is validated against reference solutions for steady coiling.
The method makes it possible to simulate the unsteady behavior of thin viscous
jets in a robust and efficient way, including the combined effects of inertia,
stretching, bending, twisting, large rotations and surface tension
An optimal control problem for a Kirchhoff-type equation
In this paper we study a control problem for a Kirchhoff-type equation. The method to obtain first order necessary optimality conditions is the Dubovitskii-Milyoutin formalism because the classical arguments do not work. We obtain a characterization of the optimal control by a partial differential system which is solved numerically.PROCAD/CASADINHOConselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de Economía y Competitivida
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