Jonathan S. Pitt

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 Navier–Stokes 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 Adaptive h-Refinement Algorithm for Local Damage Models

An adaptive mesh refinement strategy is proposed for local damage models that often arise from internal state variable based continuum damage models. The proposed algorithm employs both the finite element method and the finite difference method to integrate the equations of motion of a linear elastic material with simple isotropic microcracking. The challenges of this problem include the time integration of coupled partial differential equations with time-dependent coefficients, and the proper choice of solution spaces to yield a stable finite element formulation. Discontinuous elements are used for the representation of the damage field, as it is believed that this reduction in regularity is more consistent with the physical nature of evolving microcracking. The adaptive mesh refinement algorithm relies on custom refinement indicators, two of which are presented and compared. The two refinement indicators we explore are based on the time rate of change of the damage field and on the energy release rate, respectively, where the energy release rate measures the energy per unit volume available for damage to evolve. We observe the performance of the proposed algorithm and refinement indicators by comparing the predicted damage morphology on different meshes, hence judging the capability of the proposed technique to address, but not eliminate, the mesh dependency present in the solutions of the damage field.

Coupled FE/BE for periodic acoustic systems

While coupled finite element and boundary element (FE/BE) codes are used regularly in acoustics—particularly structural acoustics—and periodic boundary conditions are common, the combination of the three is rare. However, such calculations have been performed in the electricity and magnetism community for quite some time. This talk presents a fully coupled FE/BE framework for acoustics with doubly-periodic boundary conditions, accomplished by way of the Ewald transformation of the appropriate periodic Green’s function. This method is likely of primary interest to the acoustic metamaterials community and it is used to analyze an example problem drawn from this field. Both the internal acoustic field within the FE mesh as well as the system’s radiation characteristics are examined.

Simulation of coupled structural-acoustic response with dynamic damage evolution

A novel time-domain method for simulating dynamic damage evolution in a coupled structural-acoustic system is presented. The system is derived via the theory of continuum damage mechanics, and incorporates standard damage evolution models, but is readily extendible to more exotic formulations. The overall solution method is staggered, solving for the dynamic damage evolution first with an explicit step, and then using the new values in the coupled computation of the structural-acoustic system. The spatial domain is discretized using a mixed finite element method, and the temporal domain is discretized with a higher-order implicit time discretization scheme. Efforts toward fully coupled verification of the solution algorithm are presented, as are validation studies for cases without evolving damage. Applications with evolving damage are presented, and present a first principles study of changes in the structural acoustic response to dynamically evolving damage in the structure. Special attention is given t...