Thesis Summary:

The goal of this dissertation is to develop and analyze efficient numerical tools to deal with vibration problems for coupled systems involving elastic structures and dissipative fluids.We will consider the vibration problem of a clamped Timoshenko beam with variable cross section, an elasticity eigenproblem, an interaction problem between two heterogeneous dissipative fluids and a dissipative elastoacustic problem. In order to approximate the solutions of these problems, we use numerical methods based on the classical finite element method (FEM) and the discontinuous Galerkin method (DG). In the first part of this dissertation, we analyze a low-order finite element method to approximate the natural frequencies and the vibration modes of a non-homogeneous Timoshenko beam. We consider a formulation in which the bending moment is introduced as an additional unknown for the source problem. Optimal order error estimates are proved for displacements, rotations, shear stress and bending moment of the vibration modes, as well a double order of convergence for the vibration frequencies. These estimates are independent of the beam thickness, which leads us to the conclusion that the method is locking free with respect to this parameter. For the implementation of the numerical method, we show that the elimination of the displacements and rotations leads to a well posed generalized matrix eigenvalue problem whose solutions are comparable to the one obtained with other classical formulations in terms of computational cost. Some numerical experiments are presented to assess the performance of the method. In the second chapter, we address the interaction problem between two dissipative fluids. The presence of dissipation leads us to the study of a quadratic eigenvalue problem. A rigorous mathematical analysis of the spectral problem is performed to establish a spectral characterization of the associated solution operator. We prove that the solution operator admits an essential spectrum that is well separated from the physical spectrum. We use the lowest order Raviart-Thomas elements to discretize the problem. We observe that the presence of viscosity leads to new difficulties, since the solution operator is non-regularizing and therefore non compact. We prove that our proposed method is convergent and spurious modes free and that the corresponding eigenfunctions and eigenvalues converge with the expected order. The theoretical results are validated with some numerical experiments. In the third part we consider the dual mixed formulation for the elasticity equations written in terms of the stress and the rotation tensors. Our aim is to use a DG method to compute the lowest frequencies of the resulting mixed spectral problem. To this end, we approximate the stress tensor with polynomials of degree $k$ and the rotations with polynomials of degree $k-1$. We endow the DG spaces with their natural mesh dependent norms and adapt the non-compact operator theory to prove that our DG method does not introduce spurious modes for a small enough meshsize and a large enough stabilization parameter. We report some numerical examples to assess the performance of the method in relation with the stabilisation parameter, the meshsize and the polynomial degree. Finally, we present an elastoacustic problem, where a dissipative fluid contained in a rigid cavity is considered. The presence of dissipation leads to a quadratic eigenvalue problem. As in the previous chapters, the fluid is modeled with the Stokes equations and the solid with the linear elasticity equations. A continuous spectral formulation written in terms of the displacements of the fluid and the solid is presented and analyzed. The solution operator associated to the eigenvalue problem is introduced and its spectrum is characterized. A finite element method is introduced, where the displacement of the fluid is approximated with $ H(div)$ elements and the solid displacement with $ H^1$ elements. This particular choice of finite element spaces leads to a non-conforming method. The analysis of convergence, error estimates and spurious free results are obtained as in the second part of this dissertation. Some numerical experiments are presented to asses the performance of the method.

ISI Publications from the Thesis

Felipe LEPE, Salim MEDDAHI, David MORA, Rodolfo RODRíGUEZ: Mixed discontinuous Galerkin approximation of the elasticity eigenproblem. Numerische Mathematik, vol. 142, 3, pp. 749-786, (2019).

Felipe LEPE, Salim MEDDAHI, David MORA, Rodolfo RODRíGUEZ: Acoustic vibration problem for dissipative fluids. Mathematics of Computation, vol. 88, 315, pp. 45-71, (2019).

Felipe LEPE, David MORA, Rodolfo RODRíGUEZ: Finite element analysis of a bending moment formulation for the vibration problem of a non-homogeneous Timoshenko beam. Journal of Scientific Computing, vol. 66, 2, pp. 825-848, (2016)