A Hybridizable Discontinuous Galerkin Method For Modeling Fluid–structure Interaction

Open Access
Sheldon, Jason Paul
Graduate Program:
Engineering Science and Mechanics
Doctor of Philosophy
Document Type:
Date of Defense:
December 11, 2015
Committee Members:
  • Jonathan S Pitt, Dissertation Advisor
  • Scott Miller, Dissertation Advisor
  • Jonathan S Pitt, Committee Chair
  • Scott Miller, Committee Chair
  • Panagiotis Michaleris, Committee Member
  • Francesco Costanzo, Committee Member
  • Paris R Vonlockette, Special Member
  • Hybridizable discontinuous Galerkin
  • Fluid-Structure Interaction
  • Arbitrary Lagrangian-Eulerian Navier-Stokes
  • Elastodynamics
As computational methods have matured and computing power has increased over the years, simulations have grown in complexity by attempting to accurately model both larger and more involved physical systems. Although the computational demand of these simulations has increased, the required accuracy of the solution has not decreased, resulting in simulations that can become prohibitively computationally expensive. New computational tools need to be developed that both maintain solution accuracy while minimizing the ever increasing computational cost in time and resources. This dissertation presents a novel application of the recently developed hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid-structure interaction (FSI) problems. Current applications of the HDG method are reviewed and shown to be limited in scope to single-physics scenarios; however, they do include both solid and fluid problems, which are necessary for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, linear elastodynamics, nonlinear elastodynamics, Eulerian Navier-Stokes, and arbitrary Lagrangian-Eulerian Navier-Stokes are derived. The elasticity formulations are all 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. In past work (Sheldon, 2012; Sheldon et al., 2014), a continuous Galerkin FSI model with a variety of coupling strategies was implemented, which greatly facilitated the process of creating a novel HDG FSI model. HDG FSI modeling comes with its own unique challenges, however, which are discussed and then addressed by modifications to the established component formulations. The resultant HDG FSI model is then presented. 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 numerically validated against benchmark problems proposed by Turek and Hron (Turek and Hron, 2006). The HDG results show increasing accuracy compared to the benchmark’s measured quantities as simulations are refined. Finally, concluding remarks are presented and the future work necessary to turn this HDG FSI model into a production level tool is outlined.