A monolithic fluid–structure interaction approach using mixed LSFEM with high-order time integration

This contribution deals with the solution of a new monolithically coupled fluid–structure interaction approach using mixed least-squares (LS) stress–velocity (SV) formulations with implicit time discretization schemes and adaptive time stepping. The variational approach for the fluid is based on the incompressible Navier–Stokes equations in Arbitrary-Lagrangian–Eulerian (ALE) description to consider a deforming fluid domain. For the deformation of the fluid background mesh, a pseudo-material with linear elastic behavior and local hardening using mesh-Jacobian-based stiffening is presented. The proposed LS solid formulations are based on linear and hyperelastic material behavior, and are likewise expressed in terms of stresses and velocities. In combination with conforming finite element spaces in H(div) and H1 for the spatial discretization of the unknowns, an inherent fulfillment of the coupling conditions in FSI problems is achieved. Time discretization is performed using SDIRK methods with different orders and the Houbolt method. Before solving the coupled problem, the accuracy of various high-order time discretizations is investigated when solving dynamic flow and solid problems with SV formulations. Two numerical examples with exact solution are used, i.e. an unsteady Taylor–Green vortex flow and a linear elastic vibrating plate, to evaluate the temporal convergence order of different schemes. The coupled LS FSI approach is tested by solving the benchmark problem, flow around a cylinder with attached flag, which is characterized by large deformations of the solid flag and hence the fluid domain. A major focus is on the investigation and comparison of embedded Runge–Kutta methods with adaptive time step control in terms of efficiency and performance.