A fictitious domain/distributed Lagrange multiplier based fluid–structure interaction scheme with hierarchical B-Spline grids

Abstract

We present a numerical scheme for fluid–structure interaction based on hierarchical B-Spline grids and fictitious domain/distributed Lagrange multipliers. The incompressible Navier–Stokes equations are solved over a Cartesian grid discretised with B-Splines. The fluid grid near the immersed solids is refined locally using hierarchical B-Splines. The immersed solid is modelled as geometrically-exact beam discretised with standard linear Lagrange shape functions. The kinematic constraint at the fluid–solid interface is enforced with distributed Lagrange multipliers. The unconditionally-stable and second-order accurate generalised-α method is used for integration in time for both the fluid and solid domains. A fully-implicit and fully-coupled solution scheme is developed by using the Newton–Raphson method to solve the non-linear system of equations obtained with Galerkin weak formulation. First, the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder. Then, the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme. The results obtained with the present scheme are compared with the reference values