The Finite Element Method Applied to Fluid-Structure Interaction Using the Arbitrary Lagrangian-Eulerian and the Semi-Lagrangian Methods

Abstract

The current study aims at simulating flows and their interaction with solid structures. The discretization of mass and momentum equations, using the Finite Element Analysis (FEA), is presented. Also the semi-Lagrangian (SL) technique, where unconditional stability is successfully achieved for the numerical solution in different geometries, is developed. For the solid domain, a two-dimensional approach was used with a quadratic triangular element mesh in a daynamic simulation. As for the fluid domain, a second order special convergence is assured for velocity fields, since a quadratic + linear pair of triangular mesh elements is used, fulfilling the well-known LBB condition. Then, results are compared with the literature, as a model verification, and also other cases are evaluated. Benchmark validation for the finite element method applied to fluid mechanics and membrane solid cases are provided, as well as a modal analysis. First a static model for the solid domain is compared with the literature, where a clamped beam is considered for its maximum deflection. The dynamic case is used to simulate vibration on that same beam, analyzing the motion through time and its modes of vibration. Then, a fluid-structure interaction is presented, where a beam is now subjected to the fluid’s forces and the solid provides a physical boundary to the fluid’s solution. Pressure and shear stresses are calculated in order to be used as boundary forces to the solid structure. Thus, both domains are evaluated and their influence on each other is studied using the Arbitrary Lagrangian-Eulerian, also known as ALE, technique, which is an intermediate form between the Lagrangian and Eulerian methods, allowing moving boundaries. One of the challenges is to reduce computational cost, since finite element matrices need to be recalculated at each step. A final case is presented, where a beam is subjected to shear and bending stresses due to the flow of the fluid it is submerged in. Von Mises stress is used to evaluate the efforts imposed to the solid body.