arXiv subjects

Erwin Hernández

Publications and source records attributed to Erwin Hernández.

3 recordsLinked to original sources

arXiv2021

Analysis of an abstract mixed formulation for viscoelastic problems

This study provides an abstract framework to analyze mixed formulations in viscoelasticity, in the classic saddle point form. Standard hypothesis for mixed methods are adapted to the Volterra type equations in order to obtain stability of the proposed problem. Error estimates are derived for suitable finite element spaces. We apply the developed theory to a bending moment formulation for a linear viscoelastic Timoshenko beam and for the Laplace operator with memory terms. For both problems we report numerical results to asses the performance of the methods.

math.NA↗

arXiv2020

A mixed parameter formulation with applications to linear viscoelasticity

In this work we propose and analyze an abstract parameter dependent model written as a mixed variational formulation based on Volterra integrals of second kind. For the analysis, we consider a suitable adaptation to the classic mixed theory in the Volterra equations setting, and prove the well posedness of the resulting mixed viscoelastic formulation. Error estimates are derived, using the available results for Volterra equations, where all the estimates are independent of the perturbation parameter. We consider an application of the developed theory in a viscoelastic Timoshenko beam, and report numerical experiments in order to assess the independence of the perturbation parameter.

math.NA↗

arXiv2017

A locking-free optimal control problem with $L^1$ cost for optimal placement of control devices in Timoshenko beam

The numerical approximation of an optimal control problem with $L^1$-control of a Timoshenko beam is considered and analyzed by using the finite element method. From the practical point of view, inclusion of the $L^1$--norm in the cost functional is interesting in the case of beam vibration model, since the sparsity enforced by the $L^1$--norm is very useful for localizing actuators or control devices. The discretization of the control variables is performed by using piecewise constant functions. The states and the adjoint states are approximated by a \emph{locking free scheme} of linear finite elements. Analogously to the purely $L^2$--norm penalized optimal control, it is proved that this approximation have optimal convergence order, which do not depend on the thickness of the beam.

math.OC↗