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

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/88080/1/3224_ftp.pd

An Unsteady Entropy Adjoint Approach for Adaptive Solution of the Shallow-Water Equations

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90693/1/AIAA-2011-3694-887.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

Development and Applications of Adjoint-Based Aerodynamic and Aeroacoustic Multidisciplinary Optimization for Rotorcraft

Urban Air Mobility (UAM) is one of the most popular proposed solutions for alleviating traffic problems in populated areas. In this context, the proposed types of vehicles mainly consist of rotors and propellers powered by electric motors. However, those rotary-wing components can contribute excessively to noise generation. Therefore, a significant noise concern emerges due to urban air vehicles in or around residential areas. Reducing noise emitted by air vehicles is critically important to improve public acceptance of such vehicles for operations in densely populated areas. Two main objectives of the present dissertation are: (1) to expand the multidisciplinary optimization to utilize adjoint-based aeroacoustic and aerodynamic sensitivities; (2) to optimize the shape of proprotor blades to improve the overall performance of selected rotorcraft from both aerodynamic and aeroacoustic perspectives. This dissertation reports on the development and application of an unsteady discrete adjoint solver for aerodynamic and aeroacoustic coupling to obtain an improved design for quieter rotorcraft. The optimization framework developed through this dissertation can be utilized for multiple flight conditions, multiple receivers, and multiple optimization objectives within the same design process. SU2-based code development involves the implementation of aeroacoustic analysis, adjoint computations, and integrations into a multidisciplinary rotorcraft optimization suite. A computational aeroacoustics tool is embedded into the SU2-suite to predict the propagation of the emitted noise from the moving sources with high fidelity. Capabilities of the developed computational aeroacoustics tool are demonstrated for a range of rotor, propeller, and proprotor applications, and they are verified by comparing with wind tunnel data whenever it is available. The aeroacoustic tool also computes sensitivities with respect to the conserved variables and grid coordinates by employing the algorithmic differentiation method. Integration of an acoustic solver into the discrete adjoint solver and related modifications enable the code to compute aeroacoustic sensitivities with respect to the design variables. Applying the developed optimization framework for a proprotor aims to reduce the noise radiation without sacrificing the required aerodynamic performance value. As an outcome of the optimization during forward-flight and hover, the reshaped blade design emits and propagates lower noise levels as perceived by multiple observers. The major contributions are: (1) a multidisciplinary optimization framework that presents an optimized rotorcraft design for better aeroacoustics and aerodynamics; (2) a novel adjoint-based formulation for aeroacoustic sensitivities with respect to design variables; (3) single acoustic objective function including multiple flight conditions and multiple microphone positions; (4) implementation of Farassat 1A formulation into opensource software, SU2, to compute noise propagation emitted from moving sources. In summary, this dissertation provides the results with high fidelity, a well-integrated and rapidly converging optimization tool to improve the rotorcraft\u27s aeroacoustic performance while retaining or improving the aerodynamic performance. Among the conclusions are the following: (1) Computational fluid dynamics analyses (SU2-CFD) can produce accurate results for various rotorcraft applications. (2) The developed aeroacoustic code predicts noise propagation emitted from propellers, rotors, and proprotors with high-fidelity. (3) The acoustic interaction between propeller and wing components can be assessed by employing the aeroacoustic solver. (4) The multidisciplinary optimization framework successively reduces noise level emitted by a proprotor in multiple flight configurations. (5) The optimized design improves emitted noise radiation while satisfying the given aerodynamic constraint(s)

Output-Based Error Estimation and Model Reduction for Chaotic Flows

Turbulent flows are characterized by chaotic variations in state variables and are commonly found in many applications such as jet engine mixing and flow over bluff bodies. Large Eddy Simulations (LES) of these chaotic flows have already proven to be useful to the design process. However, LES is resource and time-intensive. Application of output-based methods for error estimation and mesh adaptation would decrease the cost of these chaotic simulations while still retaining their overall accuracy. However, a direct application of unsteady adjoint-based methods is not possible due to the flows’ inherent sensitivity to the initial conditions and the exponential growth of the corresponding adjoint solutions. This dissertation proposes the Hyper-Reduced Order Modeling-Least Squares Shadowing (HROM-LSS) method, which combines model reduction principles with adjoint sensitivity techniques for chaotic flows to calculate accurate adjoints that are cheaper to solve for than the Least Squares Shadowing (LSS) method on its own. All primal solutions are solved using the discontinuous Galerkin finite element method. Results of the HROM-LSS method for the Kuramoto-Sivashinsky equation and the NACA 0012 airfoil at high Reynolds numbers show promise for this combined method and have been shown to outperform the LSS method when calculating the effect of the discretization errors on the output. In particular, the average CPU times for the HROM-LSS method are reduced by as much as 97.44% for short time simulations and as much as 64% for longer simulations, making the HROM-LSS method a more practical option to calculate adjoint for chaotic flows in order to perform output-based error estimation for turbulent flows.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149954/1/ykmizu_1.pd

A coupled discrete adjoint method for optimal design with dynamic non-linear fluid structure interactions

Incorporating high-fidelity analysis methods in multidisciplinary design optimization necessitates efficient sensitivity evaluation, which is particularly important for time-accurate problems. This thesis presents a new discrete adjoint formulation suitable for fully coupled, non-linear, dynamic FSI problems. The solution includes time-dependent adjoint variables that arise from grid motion and chosen time integration methods for both the fluid and structural domains. Implemented as a generic multizone discrete adjoint solver for time-accurate analysis in the open-source multiphysics solver SU2, this provides a flexible framework for a wide range of applications. Design optimization of aerodynamic structures need accurate characterization of the coupled fluid-structure interactions (FSI). Incorporating high-fidelity analysis methods in the multidisciplinary design optimization (MDO) necessitates efficient sensitivity evaluation, which is particularly important for time-accurate problems. Adjoint methods are well established for sensitivity analysis when large number of design variables are needed. The use of discrete adjoint method through algorithmic differentiation enables the evaluation of sensitivities using an approximation of the Jacobian of the coupled problem, thus enabling this approach to be applied for multidisciplinary analysis. This thesis presents a new discrete adjoint formulation suitable for fully coupled, non-linear, dynamic FSI problems. A partitioned approach is considered with finite volume for the fluid and finite elements for the solid domains. The solution includes the time-dependent adjoint variables that arise from the grid motion and chosen time integration methods for both the fluid and structural domains. Implemented as a generic multizone discrete adjoint solver for timeaccurate analysis in the open-source multiphysics solver SU2, this provides a flexible framework for a wide range of applications. The partitioned FSI solver approach has been leveraged to extend the dynamic FSI capabilities to low speed flows through the introduction of a densitybased unsteady incompressible flow solver. The developed methodology and implementation are demonstrated using a range of numerical test cases. Optimal design for steady, coupled FSI problems are firstly presented before moving to the building blocks of dynamic coupled problems using single domain analysis, for both structural and fluid domains in turn. The new unsteady incompressible fluid solver, for both the primal and adjoint analysis, are verified against a range of well-known benchmark test cases, including problems with grid motion. Finally, applications of coupled dynamic problems are presented to verify both the unsteady incompressible solver for FSI as well as the successful verification of the discrete adjoint sensitivities for the transient response of a transonic compliant airfoil for a variety of both aerodynamic and structural objective functions.Open Acces