Lagrangian finite element modelling of dam–fluid interaction: Accurate absorbing boundary conditions

Abstract

The dynamic dam–fluid interaction is considered via a Lagrangian approach, based on a fluid finite element (FE) model under the assumption of small displacement and inviscid fluid. The fluid domain is discretized by enhanced displacement-based finite elements, which can be considered an evolution of those derived from the pioneering works of Bathe and Hahn [Bathe KJ, Hahn WF. On transient analysis of fluid–structure system. Comp Struct 1979;10:383–93] and of Wilson and Khalvati [Wilson EL, Khalvati M. Finite element for the dynamic analysis of fluid–solid system. Int J Numer Methods Eng 1983;19:1657–68]. The irrotational condition for inviscid fluids is imposed by the penalty method and consequentially leads to a type of micropolar media. The model is implemented using a FE code, and the numerical results of a rectangular bidimensional basin (subjected to horizontal sinusoidal acceleration) are compared with the analytical solution. It is demonstrated that the Lagrangian model is able to perform pressure and gravity wave propagation analysis, even if the gravity (or surface) waves are dispersive. The dispersion nature of surface waves indicates that the wave propagation velocity is dependent on the wave frequency. For the practical analysis of the coupled dam–fluid problem the analysed region of the basin must be reduced and the use of suitable asymptotic boundary conditions must be investigated. The classical Sommerfeld condition is implemented by means of a boundary layer of dampers and the analysis results are shown for the cases of sinusoidal forcing. The classical Sommerfeld condition is highly efficient for pressure-based FE modelling, but may not be considered fully adequate for the displacement-based FE approach. In the present paper a high-order boundary condition proposed by Higdom [Higdom RL. Radiation boundary condition for dispersive waves. SIAM J Numer Anal 1994;31:64–100] is considered. Its implementation requires the resolution of a multifreedom constraint problem, defined in terms of incremental displacements, in the ambit of dynamic time integration problems. The first- and second-order Higdon conditions are developed and implemented. The results are compared with the Sommerfeld condition results, and with the analytical unbounded problem results. Finally, a number of finite element results are presented and their related features are discussed and critically compared.