Thesis Summary:

The goal of this thesis is to develop and analyze, theoretically and computationally, numerical methods for the computation of the free vibration modes of coupled systems. We consider an elastoacoustic problem involving interaction between a compressible fluid and a thin structure. Convergence and optimal in order and regularity error estimates are proved for the spectral acoustic problem on a curved non-convex domain. We study two different formulations of the problem. The first one is a pressure formulation which is approximated using standard piecewise linear continuous elements, and the other oner is a displacement formulation approximated using Raviart—Thomas elements. For the thin structure, we use the classical Naghdi model over a reference domain. To avoid numerical locking, we consider a low-order method of the so called MITC (Mixed Interpolation of Tensorial Component) family on quadrilateral meshes. In the case of a plate, under mild assumptions, we obtain convergence and error estimates involving constants independent of the thickness. Finally, we consider the coupled problem using hexahedal Raviart—Thomas elements in the fluid and a non conforming coupling on the fluid—solid interface. We obtain optimal order error estimates for the computed eigenfunctions, as well as a double order for the eigenvalues. We report several numerical experiments to assess the performance of the methods, even in some cases not covered by the developed theory.

ISI Publications from the Thesis

Erwin HERNáNDEZ, Rodolfo RODRíGUEZ: Finite element approximation of spectral acoustic problems on curved domains. Numerische Mathematik, vol. 97, 1, pp. 131-158, (2004)

Ricardo DURáN, Erwin HERNáNDEZ, Luis HERVELLA-NIETO, Elsa LIBERMAN, Rodolfo RODRíGUEZ: Error estimates for low-order isoparametric quadrilateral finite elements for plates. SIAM Journal on Numerical Analysis, vol. 41, 5, pp. 1751-1772, (2003)

Erwin HERNáNDEZ, Rodolfo RODRíGUEZ: Finite element approximation of spectral problems with Neumann boundary conditions on curved domains. Mathematics of Computation, vol. 72, 243, pp. 1099-1115, (2003)

Erwin HERNáNDEZ, Luis HERVELLA-NIETO, Rodolfo RODRíGUEZ: Computation of the vibration modes of plates and shells by low-order MITC quadrilateral finite elements. Computers and Structures, vol. 81, pp. 615-628, (2003)

No-ISI Publications from the Thesis

Erwin HERNáNDEZ, Luis HERVELLA-NIETO, Rodolfo RODRíGUEZ: Computation of the vibration modes of plates and shells coupled with a fluid. Mecanica Computacional, XXI, S.R. Idelsohn et al., eds., pp. 2153-2162, AMCA, Santa Fe, 2002

Other Publications (ISI)

Erwin HERNáNDEZ: Approximation of the vibration modes of a plate coupled with a fluid by low-order isoparametric finite elements. Mathematical Modelling and Numerical Analysis, vol. 38, 6, pp. 1055-1070, (2004)

Gabriel N. GATICA, Erwin HERNáNDEZ, Mario E. MELLADO: A domain decomposition method for linear exterior boundary value problems. Applied Mathematics Letters, vol. 11, 6, pp. 1-9, (1998)