Nuclear Engineering and Design 106 (1988) 69-85 69 North-Holland, Amsterdam VARIATIONAL APPROACH TO FLUID-STRUCTURE INTERACTION WITH SLOSHING * Wing Kam LIU and Rasim Aziz URAS Northwestern University, Department of Mechanical Engineering~ Evanston, Illinois 60208, USA Received 11 March 1987 A general variational principle for fluid-structure interaction problems with sloshing is presented. Both seismic and body forces are accounted for in the development. It is shown that various fluid-strucutre interaction formulations can be obtained from the developed functional. 1. Introduction A wide variety of engineering problems involve fluid-structure interaction (FSI) with sloshing, in particular, the seismic analysis of liquid-filled tanks and nuclear reactor systems. The simplest approach to these problems is to neglect the coupling among the fluid, structure and free-surface. The fluid and sloshing behavior is determined with a rigid wall assumption and then the structural response is obtained by imposing the dynamic pressure to the structural model (see for example, refs. [1-7], for a discussion). This approach generally yields conservative results since the rigid-wall forces are larger than the flexible wall forces. But an uncoupled analysis underestimates the structural response if the natural frequencies of the coupled system are close to the excitation frequencies [2], which is often the case in the seismic analysis of liquid-filled tanks and nuclear reactor systems [3-4]. Two basic approaches exist for the coupled analysis of FSI systems. In the first approach, the pressure or velocity potential formulation, the fluid is characterized by a single pressure or velocity potential variable at each node of the finite element mesh. The fluid and the structure are coupled through the fluid-structure interface. The coupled set of equations are, in general, unsymmetric. Therefore, a symmetrization of the resulting equations is needed and the implementation of this formulation requires a special-purpose computer code. With this formulation, the sloshing effect cannot be incorporated easily [4,8-151. The second approach is the displacement formulation. The fluid motion is expressed in terms of the finite element nodal displacements. Hence, compatibility and equilibrium along the interface are automati- cally satisfied. The fluid is analyzed like an elastic solid but with a negligible shear modulus. This assumption leads to the appearance of non-physical "circulation" modes. The number of these modes tends to increase with mesh refinement. Different techniques are introduced to remove these non-physical modes [16-20]. The main advantage of this approach is the similarity between the discretized forms of the fluid and the structure. Thus, easy implementation of the fluid equations into existing FEM codes is achieved. In addition, sloshing effects can easily be included in the formulation. Nevertheless, the non-physical "circulation" modes give rise to numerical problems, such as artificial viscosity and huge equation systems, among others. * The support of this research by the National Science Foundation, ECE-8614957, is gratefully acknowledged. The valuable discussions with Professor K.J. Bathe on section 5 are also gratefully acknowledged. This paper is dedicated to Dr. Stanley H. Fistedis, who retired as the Principal Editor of the Journal of Nuclear Engineering and Design. 0029-5493/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)