UC Berkeley

In this work, we develop and analyze a partitioned implicit time integration method designed for fluid-structure interaction (FSI) modelling using discontinuous Galerkin (DG) discretizations and fully implicit Runge-Kutta (IRK) methods. The discontinuous Galerkin method is a high-order finite element method used for fluid simulations on unstructured meshes. We take a partitioned approach, modelling the structures separately and limiting communication between structure and fluid by use of a prediction-correction framework. We verify the high-order accuracy, the stability, and the performance of our method on simple model problems including a one-dimensional sprung piston and a two-dimensional pitching airfoil with prescribed vertical motion and a torsional restoring force, both with structures of two degrees of freedom. Finally we apply our method to the standard problem of a cantilever beam shedding vortexes with two sets of parameters, and a vibrating tuning fork problem. Both these final applications consist of fluids modelled on fully unstructured meshes of high-order triangular elements deformed by radial basis functions according to the structure motion, and structures modeled using a neo-Hookean formulation and discretized using standard continuous finite elements.

Our scheme fully decouples the implicit solutions of the structure and the fluid, with communication limited to boundary conditions and mesh deformation. We present our method with two variations, one using explicit methods for predicted quantities and another using implicit. We implement our method up to seventh order, and compare its cost with standard Diagonally Implicit Runge-Kutta methods where possible. Our findings are that this new method can be significantly cheaper, in particular for the more complex cantilever problem, and it also has better stability properties.