Gradient Adaptive Transfinite Elements (GATE) offer a generic method to implement arbitrary high-order numerical solutions to existing finite element formulations. GATE are a family of elements that are formulated using transfinite interpolation, a generic blending-based interpolation technique that can construct optimal interpolation functions about areas and volumes. The arbitrary nature of transfinite elements allows for the creation of elements that can serve a variety of useful purposes. Elements can be constructed to transition from a region using high-order representation to low-order representation. High-order representation is important in capturing large gradients, especially in regions near geometric boundaries. Without such capability, one is either left with the choice of paying a high cost to implement high-order representation across the entire domain, or excessively refining the low-order representation near the high gradient region. The high order nature of GATE is well suited for a Direct Numerical Simulation (DNS) of the Navier Stokes equations. In this present work, we apply GATE to find DNS solutions of the Navier Stokes equations and coupled Fluid-Structure Interaction (FSI) problems.
The high-order representation provided by GATE is not limited to the representation of the solution but can be used in the same manner for the geometry. This is typically done using isoparametric mapping, a technique that maps the geometry to an element reference domain using the same basis functions used for the representation of the field. GATE can also be used as a parametric mapping with a high-order representation or exact representation used to represent the boundary geometry. The ability of GATE to transition between high-order and low-order regions is well suited for parametric mapping where a high degree of mapping accuracy is needed at the boundary and low-order mapping accuracy is acceptable in the field.
GATE family elements can also be used for general C1 conforming interpolation to solve higher-order governing equations or provide convenience in the evaluation of quantities that depend on the gradient of the solution. This has an added benefit in reducing the total number of degrees of freedom in the numerical representation of the system while still retaining high-order accuracy.
GATE is well suited for the numerical representation of FSI systems. FSI involves the coupling between fluid and structure domains. The quality of the results of an FSI simulation depends on the accurate representation of the coupling across the fluid-structure interface. A monolithic formulation of the FSI system couples the fluid and structural domains together as a single interdependent set of equations. Such a formulation has an advantage over other methods that partition the fluid and structural domains and solve them independently. The solution of a coupled monolithic system exactly represents the coupled system, there is no place to introduce a bias to one domain or misrepresentation of the interface between the domains. Additionally, the coupling can be accounted for in the linearization of the system and improve convergence when an iterative nonlinear solver is employed. Using GATE on the fluid and structure interface offers a convenient means to ensure conformity between the two domains.
We developed a research tool to implement GATE for a variety of simulations performed for this dissertation. This was done because existing finite element software draws from libraries with preexisting element formulations based on traditional methods. Many aspects of existing software depend on the layout of nodes in the elements for the assembly required to construct the complete finite element system and thus would require significant changes to incorporate the flexibility of GATE. With the lessons learned from this effort, the implementation of GATE into existing software will be less of a challenge and would be beneficial so GATE could benefit from a performance optimized software implementation.
This dissertation is organized into three major sections representing specific aims to support the GATE implementation of FSI systems. The first aim is to formulate the governing equations for an FSI system and the monolithic finite element formulation to represent the equations numerically. The second aim is to formulate the GATE family of elements and test their performance using manufactured solution methods and test cases with incompressible and compressible flows. Lastly, GATE is used to find solutions to two types of FSI systems. One system is a classical pitching and heaving airfoil model, and the other is a flexible strip trailing a square block benchmark problem.
