Thesis Summary:

The aim of this thesis is to analyze different problems involving thin structures and their discretization by finite element methods. We study three problems, namely: • The computation of the vibration modes of a Timoshenko curved rod with arbitrary geometry; • The approximation of the vibration modes of a laminated plate modeled by Reissner-Mindlin equations; • A finite element method for stiffened plates composed by a ReissnerMindlin plate and a Timoshenko rod. In the first problem, we prove optimal order error estimates for displacements, rotations and shear stresses and a double order of convergence for the vibration frequencies, all of these estimates independent of the thickness of the rod. We present numerical experiments that confirm the theoretical results and the free-locking character of the method. In the second problem, we study the convergence of the proposed method. We prove an adequate a-priori estimate for the associated load problem and obtain optimal order error estimates for the in-plane and transverse displacements and the rotations in L2 and H1 norms and double order of convergence for the vibration frequencies. All of these estimates are again independent of the thickness of the plate. Numerical tests which confirm that the method is locking-free are presented. In the last work, we prove that the resulting problem is well posed and study the case in that the stiffener is located concentrically with respect to the plate. The problem is decoupled into two problems as for standard Reissner-Mindlin plates. The stiffened in-plane problem results in a standard analysis and not depending on the plate thickness. The stiffened bending problem is more challenging. We show that the solution is bounded above and below independently on the plate thickness. Optimal error estimates are proved for displacements, rotations and shear stresses for the plate and the stiffener. Finally numerical experiments demonstrate the lockingfree character of the method.

ISI Publications from the Thesis

Ricardo DURáN, Rodolfo RODRíGUEZ, Frank SANHUEZA: Numerical analysis of a finite element method to compute the vibration modes of a Reissner-Mindlin laminated plate. Mathematics of Computation, vol. 80, 275, pp. 1239-1264, (2011).

Erwin HERNáNDEZ, Enrique OTáROLA, Rodolfo RODRíGUEZ, Frank SANHUEZA: Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 180-207, (2009)