Numerical procedure with analytic derivative for unsteady fluid–structure interaction

Abstract

SUMMARYThe unsteady interaction between an incompressible fluid and a deformable elastic structure is analyzed. An implicit numerical method is proposed. At each time step, the stresses at the fluid-structure interface are determined as a solution of an optimization problem. The modal decomposition of the structure equations leads to a problem to be solved with a reduced number of unknowns. The analytic gradient of the cost function was derived. Numerical tests validate the analytic derivative and show the behavior of a two-dimensional Navier-Stokes equations with plate-like model interaction.

Conjugate Gradient Method to Solve Fluid Structure Interaction Problem

“…We take parameters for fluid and structure in [3] [4]. (Tables 1-3) The Table 3 shows that, if we take the tolerance , …”

Controllability Approach for a Fluid Structure Interaction Problem

“…Our numerical results are presented in Table 1 wave, the fluid flow and the displacement of the structure are on good behavior. On the one hand, the advantage of the method compared to Mbaye and Murea [9,10] is that we don't need to approximate the pressure on the interface by a linear combination of functions and compute the analytic solution of the structure equation. The fact that we don't need to increase the number of controls in order to obtain good results Murea [10], is another advantage of this method on the other hand.…”

“…The wall of the vessel is described by the beam equation and the fluid is the blood which is modeled by the Stokes equations. The parameter values of the fluid and the structure are Mbaye and Murea [9,10]:…”

A fast method for solving fluid–structure interaction problems numerically

“…This technique was successfully employed in [16,17], where implicit algorithms are presented. We will use the same least square method based on the BFGS method in order to impose the continuity of the stress at the interface.…”

“…The gradient of the cost function is approached by finite differences in [16] where it is proved the superiority of the Broyden, Fletcher, Goldforb, Shano (BFGS) method in comparison with the modified Newton method (Newton with line search), when moderate time step is used. The analytic formula of the gradient of the cost function is presented in [17].…”