Duarte Santamarina

A finite element solution of an added mass formulation for coupled fluid-solid vibrations

Summary. A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed.

Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations

This paper deals with the numerical solution of a system of second-order in time partial differential equations modeling the vibrations of a coupling between an elastic solid and an inviscid compressible fluid. Both media are described in terms of their respective displacement fields. The problem is solved in the time domain by combining a Newmarks scheme for time discretization with a finite element method for space discretization. The latter combines standard Lagrangian elements in the solid with lowest-order Raviart-Thomas elements in the fluid. Stability is proved and numerical results showing the good behavior of the method are reported.

Fluid–Structure Acoustic Interaction

This is a survey of numerical methods to compute a particular fluid–solid interaction: the so called “elastoacoustic problem”. It concerns the determination of the motion of an elastic structure in contact with a compressible fluid. In this case displacements are small and, then, we can assume a linear response of the structure.We neglect gravity effects and consider a homogeneous fluid for which its reference density is constant. Other usual simplifications for this kind of problems are that viscous effects are not relevant in the fluid and that velocities are small enough for convective effects to be neglected. We review several alternative formulations of the elastoacoustic vibration problem, which differ from each other in the variables used to describe the fluid: pressure, a displacement potential or both. In all these cases, standard Lagrange finite elements are used for the discretization. We compare the results obtained with all these methods and with the pure displacement formulation, which has to be discretized by Raviart–Thomas elements. Next, we show how to apply a modal synthesis approach based on the displacement potential formulation. Finally, we report how the pure displacement formulation can be used to deal with thin layers of interface acoustic damping material and to solve the elastoacoustic problem in the time–domain.

A dynamic viscoelastic contact problem with normal compliance

A dynamic contact problem between a viscoelastic body and a deformable obstacle is numerically considered in this work. The contact is modeled by using the well-known normal compliance contact condition. The variational formulation of this problem is written in terms of the velocity field and it leads to a parabolic nonlinear variational equation. An existence and uniqueness result is stated. Fully discrete approximations are then introduced by using the finite element method to approximate the spatial variable, and a hybrid combination of the implicit and explicit Euler schemes to discretize the time derivatives. An a priori error analysis is recalled. Then, an a posteriori error analysis is provided extending some results already obtained in the study of the heat equation, other parabolic equations and the quasistatic case. Upper and lower bounds are proved. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy and the behavior of the error estimators.

TWO DISCRETIZATION SCHEMES FOR A TIME-DOMAIN DISSIPATIVE ACOUSTICS PROBLEM

This paper deals with a time-domain mathematical model for dissipative acoustics and is organized as follows. First, the equations of this model are written in terms of displacement and temperature fields and an energy equation is obtained. The resulting initial-boundary value problem is written in a functional framework allowing us to prove the existence and uniqueness of solution. Next, two different time-discretization schemes are proposed, and stability and error estimates are proved for both. Finally, numerical results are reported which were obtained by combining these time-schemes with Lagrangian and Raviart–Thomas finite elements for temperature and displacement fields, respectively.