A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction

Abstract

This work presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid-structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian-Eulerian NavierStokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modeling is coupling together their disparate mathematics on the fluid-solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.

An Overset Mesh Framework for an Isentropic ALE Navier-Stokes HDG Formulation

“…There have been previous HDG formulations for moving and deforming domains via an ALE perspective of the Navier-Stokes equations in Nguyen and Peraire [18] and Fidkowski [19]. The HDG ALE formulation has also been used to build an HDG FSI formulation in Sheldon et al [1]. The HDG method has versatility and adaptability to enable straightforward implementation for a variety of different physics, which makes it ideal for complex multiphysics applications like FSI.…”

“…In order to capture large changes in the computational domain, which include solid deformations and/or rigid translation modes of motion, it is necessary to first study an arbitrarily Lagrangian-Eulerian (ALE) formulation of the governing fluid equations. This particular building block is of critical importance because we will employ a monolithic solver for FSI simulations [1], which requires that the governing fluid equations and governing solid equations be cast in the same reference frame. Further, an ALE formulation automatically allows for dynamic bodies, and therefore dynamic meshes to be incorporated.…”

Hybridized Discontinuous Galerkin Methods for Wave Propagation

“…Finally, an original Green strain-displacement-velocity formulation was proposed in [105] for the purpose of solving fluid-structure interaction problems. Observed convergence rates were k + 1 for the approximate displacements and velocities, but only k for the approximate strains.…”

An energy‐based discontinuous Galerkin method for coupled elasto‐acoustic wave equations in second‐order form

“…9 The coupled viscoelastic-acoustic problem approximated by high order DG on polygonal and polyhedral meshes has been studied in the work of Antonietti et al 10 Hybridizable discontinuous Galerkin methods have also been used for acoustics and elastodynamics 11,12 and for the coupled problem. 13 Spectral elements have been used to simulate the coupled problem in the time domain in the works of Komatitsch et al 14 and Mönkölä. 15 The latter work also shows how controllability methods can be used to solve the frequency domain problem by finding time-periodic solutions to the transient problem.…”