High-Order Stabilized Finite Elements on Dynamic Meshes

The development of dynamic mesh capability for turbulent flow simulations using the Streamlined Upwind Petrov-Galerkin (SUPG) discretization is described. The current work extends previous research to include high-order spatial accuracy, including the satisfaction of the discrete geometric conservation law (GCL) on curved elements. Two closely-related schemes are described and the ability of these schemes to satisfy the GCL, while also maintaining temporal accuracy and conservation is assessed. Studies indicate that although one scheme discretizes the time derivative in conservative form, both schemes exhibit temporal conservation errors that decrease according to the expected design order of accuracy. The source of the temporal conservation errors is examined, and it is demonstrated that many finite-volume and finite-element schemes can also be expected to have difficulty strictly satisfying conservation in time. The effects on conservation are examined and, while present in the simulations, are seen to be negligible for the problems considered

High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

This paper presents robust discontinuous Galerkin methods for the incompressible Navier-Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework for both monolithic as well as projection or splitting-type Navier-Stokes solvers. The framework is flexible, allows implicit and explicit formulations of the convective term, and adaptive time-stepping. The Navier-Stokes equations with ALE transport term are solved on the deformed geometry storing one instance of the mesh that is updated from one time step to the next. Discretization in space is applied to the time discrete equations so that all weak forms and mass matrices are evaluated at the end of the current time step. This design ensures that the proposed formulations fulfill the geometric conservation law automatically, as is shown theoretically and demonstrated numerically by the example of the free-stream preservation test. We discuss the peculiarities related to the imposition of boundary conditions in intermediate steps of projection-type methods and the ingredients needed to preserve high-order accuracy. We show numerically that the formulations proposed in this work maintain the formal order of accuracy of the Navier-Stokes solvers. Moreover, we demonstrate robustness and accuracy for under-resolved turbulent flows

Output Error Control Using r-Adaptation

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143062/1/6.2017-4111.pd

Efficient Output-Based Adaptation Mechanics for High-Order Computational Fluid Dynamics Methods

As numerical simulations are applied to more complex and large-scale problems, solution verification becomes increasingly important in ensuring accuracy of the computed results. In addition, although improvements in computer hardware have brought expensive simulations within reach, efficiency is still paramount, especially in the context of design optimization and uncertainty quantification. This thesis addresses both of these needs through contributions to solution-based adaptive algorithms, in which the discretization is modified through a feedback of solution error estimates so as to improve the accuracy. In particular, new methods are developed for two discretizations relevant to Computational Fluid Dynamics: the Active Flux method and the discontinuous Galerkin method. For the Active Flux method, which is fully-discrete third-order discretization, both the discrete and continuous adjoint methods are derived and used to drive mesh (h) refinement and dynamic node movement, also known as $r$ adaptation. For the discontinuous Galerkin method, which is an arbitrary-order finite-element discretization, efficiency improvements are presented for computing and using error estimates derived from the discrete adjoint, and a new $r$-adaptation strategy is presented for unsteady problems. For both discretizations, error estimate efficacy and adaptive efficiency improvements are shown relative to other strategies.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144065/1/dkaihua_1.pd

Preface to the special issue 'High order methods for CFD problems'

Since a few years, there has been a renew in interest in (very) high order schemes for compressible fluid dynamics for steady and unsteady problems. Within Europe and the US, several conferences deal with these issues, some of them have been launched only recently. Several special issues of recent or future AIAA conferences are specially devoted to that topic. The goal of these researches is to design cheaper and more efficient numerical methods able to handle very large and very complex problems

Methods for Optimal Output Prediction in Computational Fluid Dynamics.

In a Computational Fluid Dynamics (CFD) simulation, not all data is of equal importance. Instead, the goal of the user is often to compute certain critical "outputs" -- such as lift and drag -- accurately. While in recent years CFD simulations have become routine, ensuring accuracy in these outputs is still surprisingly difficult. Unacceptable levels of output error arise even in industry-standard simulations, such as the steady flow around commercial aircraft. This problem is only exacerbated when simulating more complex, unsteady flows. In this thesis, we present a mesh adaptation strategy for unsteady problems that can automatically reduce errors in outputs of interest. This strategy applies to problems in which the computational domain deforms in time -- such as flapping-flight simulations -- and relies on an unsteady adjoint to identify regions of the mesh contributing most to the output error. This error is then driven down via refinement of the critical regions in both space and time. Here, we demonstrate this strategy on a series of flapping-wing problems in two and three dimensions, using high-order discontinuous Galerkin (DG) methods for both spatial and temporal discretizations. Compared to other methods, results indicate that this strategy can deliver a desired level of output accuracy with significant reductions in computational cost. After concluding our work on mesh adaptation, we take a step back and investigate another idea for obtaining output accuracy: adapting the numerical method itself. In particular, we show how the test space of discontinuous finite element methods can be "optimized" to achieve accuracy in certain outputs or regions. In this work, we compute test functions that ensure accuracy specifically along domain boundaries. These regions -- which are vital to both scalar outputs (such as lift and drag) and distributions (such as pressure and skin friction) -- are often the most important from an engineering standpoint.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133418/1/kastsm_1.pd

High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes

We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm. We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gasdynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder and the Saltzman problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as 4.7.Comment: 31 pages, 13 figure

A stabilized finite element dynamic overset method for the Navier-Stokes equations

In terms of mesh resolution requirements, higher-order finite element discretization methods offer a more economic means of obtaining accurate simulations and/or to resolve physics at scales not possible with lower-order schemes. For simulations that may have large relative motion between multiple bodies, overset grid methods have demonstrated distinct advantages over mesh movement strategies. Combining these approaches offers the ability to accurately resolve the flow phenomena and interaction that may occur during unsteady moving boundary simulations. Additionally, overset grid techniques when utilized within a finite element setting mitigate many of the difficulties encountered in finite volume implementations. This research presents the development of an overset grid methodology for use within a streamline/upwind Petrov-Galerkin formulation for unsteady, viscous, moving boundary simulations. A novel hole cutting procedure based on solutions to Poisson equation is introduced and compared to existing techniques. A MPI-based parallel three-dimensional overset grid assembly framework is developed. Order of accuracy is examined via the method of manufactured solutions. The potential benefits of using Adaptive Mesh Refinement (AMR) in overset grid simulations are explored by combining the overset method with an AMR approach. The importance of considering linearization due to the overset boundaries within the preconditioning is studied. Numerical experiments are performed comparing an ILU(k) preconditioner with two proposed modifications referred to as “triangular inter-grid ILU(k)” and “Jacobi inter-grid ILU(k)”. The efficiency gains observed from the proposed modifications are also applicable to general parallel simulations on distributed memory machines, regardless of whether an overset grid approach is used. Overset grid results are presented for several inviscid and viscous, steady-state and time-dependent moving boundary simulations with linear, quadratic, and cubic elements