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University of Michigan

1.
Kast, Steven Michael.
Methods for *Optimal* Output Prediction in Computational Fluid Dynamics.

Degree: PhD, Aerospace Engineering, 2016, University of Michigan

URL: http://hdl.handle.net/2027.42/133418

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.
*Advisors/Committee Members: Fidkowski, Krzysztof J. (committee member), Krasny, Robert (committee member), Bui-Thanh, Tan (committee member), Roe, Philip L (committee member).*

Subjects/Keywords: Unsteady adjoint; Output error estimation; Deforming domains; Discontinuous Galerkin; Discontinuous Petrov-Galerkin; Optimal test functions; Aerospace Engineering; Computer Science; Engineering

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Kast, S. M. (2016). Methods for Optimal Output Prediction in Computational Fluid Dynamics. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/133418

Chicago Manual of Style (16^{th} Edition):

Kast, Steven Michael. “Methods for Optimal Output Prediction in Computational Fluid Dynamics.” 2016. Doctoral Dissertation, University of Michigan. Accessed January 24, 2021. http://hdl.handle.net/2027.42/133418.

MLA Handbook (7^{th} Edition):

Kast, Steven Michael. “Methods for Optimal Output Prediction in Computational Fluid Dynamics.” 2016. Web. 24 Jan 2021.

Vancouver:

Kast SM. Methods for Optimal Output Prediction in Computational Fluid Dynamics. [Internet] [Doctoral dissertation]. University of Michigan; 2016. [cited 2021 Jan 24]. Available from: http://hdl.handle.net/2027.42/133418.

Council of Science Editors:

Kast SM. Methods for Optimal Output Prediction in Computational Fluid Dynamics. [Doctoral Dissertation]. University of Michigan; 2016. Available from: http://hdl.handle.net/2027.42/133418

2. Chan, Jesse L. A DPG method for convection-diffusion problems.

Degree: PhD, Computational and Applied Mathematics, 2013, University of Texas – Austin

URL: http://hdl.handle.net/2152/21417

Over the last three decades, CFD simulations have become commonplace as a tool in the engineering and design of high-speed aircraft. Experiments are often complemented by computational simulations, and CFD technologies have proved very useful in both the reduction of aircraft development cycles, and in the simulation of conditions difficult to reproduce experimentally. Great advances have been made in the field since its introduction, especially in areas of meshing, computer architecture, and solution strategies. Despite this, there still exist many computational limitations in existing CFD methods; in particular, reliable higher order and hp-adaptive methods for the Navier-Stokes equations that govern viscous compressible flow. Solutions to the equations of viscous flow can display shocks and boundary layers, which are characterized by localized regions of rapid change and high gradients. The use of adaptive meshes is crucial in such settings – good resolution for such problems under uniform meshes is computationally prohibitive and impractical for most physical regimes of interest. However, the construction of "good" meshes is a difficult task, usually requiring a-priori knowledge of the form of the solution. An alternative to such is the construction of automatically adaptive schemes; such methods begin with a coarse mesh and refine based on the minimization of error. However, this task is difficult, as the convergence of numerical methods for problems in CFD is notoriously sensitive to mesh quality. Additionally, the use of adaptivity becomes more difficult in the context of higher order and hp methods. Many of the above issues are tied to the notion of robustness, which we define loosely for CFD applications as the degradation of the quality of numerical solutions on a coarse mesh with respect to the Reynolds number, or nondimensional viscosity. For typical physical conditions of interest for the compressible Navier-Stokes equations, the Reynolds number dictates the scale of shock and boundary layer phenomena, and can be extremely high – on the order of 10⁷ in a unit domain. For an under-resolved mesh, the Galerkin finite element method develops large oscillations which prevent convergence and pollute the solution. The issue of robustness for finite element methods was addressed early on by Brooks and Hughes in the SUPG method, which introduced the idea of residual-based stabilization to combat such oscillations. Residual-based stabilizations can alternatively be viewed as modifying the standard finite element test space, and consequently the norm in which the finite element method converges. Demkowicz and Gopalakrishnan generalized this idea in 2009 by introducing the Discontinous Petrov-Galerkin (DPG) method with optimal test functions, where test functions are determined such that they minimize the discrete linear residual in a dual space. Under the ultra-weak variational formulation, these test functions can be computed locally to yield a symmetric, positive-definite system. The main…
*Advisors/Committee Members: Demkowicz, Leszek (advisor), Moser, Robert deLancey (advisor).*

Subjects/Keywords: Finite element methods; Discontinuous Galerkin; Petrov-Galerkin; Optimal test functions; Minimum residual methods; Convection-diffusion; Compressible flow

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chan, J. L. (2013). A DPG method for convection-diffusion problems. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/21417

Chicago Manual of Style (16^{th} Edition):

Chan, Jesse L. “A DPG method for convection-diffusion problems.” 2013. Doctoral Dissertation, University of Texas – Austin. Accessed January 24, 2021. http://hdl.handle.net/2152/21417.

MLA Handbook (7^{th} Edition):

Chan, Jesse L. “A DPG method for convection-diffusion problems.” 2013. Web. 24 Jan 2021.

Vancouver:

Chan JL. A DPG method for convection-diffusion problems. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2013. [cited 2021 Jan 24]. Available from: http://hdl.handle.net/2152/21417.

Council of Science Editors:

Chan JL. A DPG method for convection-diffusion problems. [Doctoral Dissertation]. University of Texas – Austin; 2013. Available from: http://hdl.handle.net/2152/21417