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Application Of The Discrete Adjoint Method To Coupled Multidisciplinary Unsteady Flow Problems For Error Estimation And Optimization
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Book Synopsis Application of the Discrete Adjoint Method to Coupled Multidisciplinary Unsteady Flow Problems for Error Estimation and Optimization by : Karthik Mani
Download or read book Application of the Discrete Adjoint Method to Coupled Multidisciplinary Unsteady Flow Problems for Error Estimation and Optimization written by Karthik Mani and published by . This book was released on 2009 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: Adjoint methods have found applications in several key areas of computational fluid dynamics (CFD), namely, shape optimization and goal based adaptive solutions. CFD has become an essential tool in the design process by enabling the rapid testing of multiple designs, and currently it is normal practice to use CFD in conjunction with optimization algorithms for design improvement. In the context of shape optimization problems based on CFD, adjoint methods offer the significant advantage of computing sensitivity derivatives of the optimization cost function with respect to the set of design parameters, at a cost that is essentially independent of the number of design parameters. Adjoint methods reduce the cost of obtaining the complete gradient vector at any point in the design space equivalent to that of a single flow solution at the same point in the design space. This immediately enables the use of all gradient based optimization algorithms and lifts any restrictions on the number of design parameters required for the adequate definition of the optimization problem. Adaptive techniques in CFD constitute the other aspect where adjoint methods have have made great inroads. Typical adaptive solutions of the governing flow equations rely on estimating the local error in an evolving solution to target regions of the computational mesh for increased discrete resolution. The main goal of any adaptive solution method is the overall increase in solution accuracy with minimal increase in computational cost. However, targeting local error in the solution does not translate into efficient use of computational resources, since ultimately it is the accurate estimation of boundary integrated functional quantities such as load coefficients that are of importance to the user. Contrary to local error-based methods, adjoint methods allow the adaptation of the computational mesh specifically for the improvement of functionals such as load coefficients. This is achieved by mathematically establishing a clear relationship between the functional of interest and the regions of the computational mesh that are most relevant to it. The current work extends the use of adjoint methods to multiple governing disciplines that are tightly coupled, and more importantly unsteady in nature. The adjoint method is derived in a very general form for the purpose of computing the gradient vector for use in shape optimization in the context of coupled multidisciplinary unsteady equations. It is shown that computing the gradient vector in unsteady problems involves solving the analysis problem forward in time and then solving the adjoint problem backward in time. While adjoint methods have been used successively to drive spatial mesh adaptation, the current work extends the use of the computed unsteady adjoint variables for estimating temporal discretization error, which is then applied to temporal mesh adaptation. Additionally, the computed adjoint variables are also used for the estimation of algebraic error in the solution arising due to intentional or nonintentional partial convergence of the governing equations. Results indicate that the adaptation of the temporal resolution and convergence tolerance limits using adjoint-based error estimates is able to outperform traditional adaptation methods such as uniform refinement and those based on local error estimates. All of the development is carried out in a fully unstructured mesh framework with dynamic deformation of the computational spatial mesh.
Book Synopsis Parallel Computational Fluid Dynamics by : Rupak Biswas
Download or read book Parallel Computational Fluid Dynamics written by Rupak Biswas and published by DEStech Publications, Inc. This book was released on 2010 with total page 703 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Tangent and Adjoint Problems in Partially Converged Flows by : Emmett M. Padway
Download or read book Tangent and Adjoint Problems in Partially Converged Flows written by Emmett M. Padway and published by . This book was released on 2020 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: Adjoint methods see widespread use in computational fluid dynamics (CFD) in two important domains of the field: shape optimization and output-based refinement.CFD has become more widespread as computing power has become more available and algorithms have become both more advanced and efficient. This allows for the use of cheap, low-fidelity methods in the conceptual design stage, and more accurate high-fidelity methods later in the detailed design stage. The high fidelity methods coupled with optimization toolboxes for automated design have a significant place in the detailed design process, mainly to better inform the use of expensive wind tunnel testing. Furthermore, the use of adjoint methods in shape optimization allows for cheaper sensitivity evaluation for gradient-based design approaches, allowing for scaling independent of the number of design variables. As a result highly refined/parameterized design optimizations are possible with negligible cost increases. Adjoint methods are also utilized in the context of adaptive mesh refinement as they can be used to create output-based error estimates. Adjoint-based methods form an error estimate that calculates error in the output-of-interest, and are desirable for their efficacy in tailoring/adapting meshes for maximum accuracy in the engineering quantity of interest. This can allow for coarsening meshes in areas of little engineering interest and refining in areas of high interest,thus resulting in a maximally useful mesh for engineering quantities with a given number of degrees of freedom. The adjoint systems used in these analyses require that the nonlinear problem be solved to machine precision --that the discretized form of the governing equations be satisfied. However, as the field has attempted more difficult simulations either due to the accuracy of spatial discretization of the simulation, the geometry of the model, or the increasing push to unsteady flow, it has become increasingly difficult to satisfy this constraint. Some members of thefield have demonstrated that this can lead to difficult to converge adjointproblems that provide sensitivities highly dependent on the state atwhich the simulation is terminated. Additionally, this can lead to issues with theerror estimation process by returning inaccurate error estimates and poor refinement patterns. This work develops a novel methodology for adjoint based optimization and errorestimation for unconverged simulations through linearization of the nonlinearsolution process. It contains results for various different nonlinear solversand applications of the adjoint system to design optimization and errorestimation. As CFD has been used for more complex simulations (unsteady,high-order, complex geometry) the ability to get a useful adjoint solution thatis guaranteed to not diverge is tremendously desirable. The goal of this work isto show a hierarchy of different linearizations with different approximationsand apply them to these partially converged problems. This work presents tangentand adjoint formulations for each of the various nonlinear solution strategiesused in the test cases, including the solution strategies required to avoid flowdivergence for problems with strong shocks. It also shows the varying linearsolver technologies, used in the nonlinear process, the mesh deformationprocess, and the tangent and adjoint processes. Included also are threedifferent error estimation techniques for the steady state adjoint and theiranalogues for the pseudo-time accurate adjoint. These error estimationtechniques are used to drive mesh adaptation and the algorithms for the adaptivemesh refinement package are included as well. A detailed error analysis ofapproximate linearization of the nonlinear solution process for both the tangentand adjoint modes is shown as well. The end result is a series of methods andalgorithms guaranteed to provide either adjoint based sensitivities or adjointbased error estimates that are insensitive to the specific state of terminationof the solution process and can be useful for simulations which do not fullyconverge, where the calculation of accurate sensitivities has been a majorstumbling block to date.
Book Synopsis Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics by : Li Wang
Download or read book Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics written by Li Wang and published by . This book was released on 2009 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of high-order discontinuous Galerkin (DG) discretizations has become more widespread over the last decade for solving convection-dominated computational fluid dynamics problems. The appeal of these methods relates to their favorable asymptotic accuracy properties, combined with compact stencils and favorable scalability properties on parallel computing architectures. This work covers advances in several areas of high-order DG discretizations, including the development of implicit solvers, discrete adjoint methods for shape optimization, and output-based error estimation and mesh and time-step adaptation. For time-dependent problems, high-order implicit time-integration schemes are considered exclusively to avoid the stability restrictions of explicit methods, with particular emphasis on balancing spatial and temporal accuracy of the overall approach. In order to make the high-order schemes competitive, efficient solution techniques consisting of a p -multigrid approach driven by element Jacobi smoothers are investigated and developed to accelerate convergence of the non-linear systems, in which the results demonstrate h independent convergence rates, while remaining relatively insensitive to time-step sizes. A framework based on discrete adjoint sensitivity analysis has also been developed for applications in shape optimization and goal-oriented error estimation. An adaptive discontinuous Galerkin algorithm driven by an adjoint-based error estimation procedure has been developed, which incorporates both h-, p- and combined hp -adaptive schemes, for producing accurate simulations at optimal cost in the objective functional of interest. Current results show superior performance of these adaptive schemes over uniform mesh refinement methods, as well as the potential of the hp refinement approach to capture strong shocks without limiters. Finally, the adjoint-based error estimation strategy is successfully extended to unsteady flow problems, where the time-dependent flow solution is solved in a forward manner in time but the corresponding unsteady adjoint solution is evaluated as a backward time integration. Results demonstrate that this methodology provides accurate global temporal error prediction, and may be employed to drive an adaptive time-step refinement strategy for improving the accuracy of specified time-dependent functionals of interest.
Book Synopsis Design Optimization of Periodic Flows Using a Time-spectral Discrete Adjoint Method by : Ki Hwan Lee
Download or read book Design Optimization of Periodic Flows Using a Time-spectral Discrete Adjoint Method written by Ki Hwan Lee and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Standard methods for unsteady optimization carry heavy computational costs and large storage requirements, mostly due to the lengthy time integration involved in the unsteady flow simulations. Such difficulties limit its practical application to cases where the time integration is performed over only a smaller segment of the entire period. The result is a loss of accuracy in the representation of the physical model. For certain unsteady flows with periodicity, a dramatic reduction in both compu- tational cost and required storage is realized through implementing the Time Spectral method. Furthermore, by introducing an adjoint-based method as an alternative way of obtaining gradient information, computational cost is further reduced. This combi- nation of Time-Spectral and adjoint-based methodology therefore allows for unsteady optimization within a reasonable time frame while maintaining accuracy. In this dissertation, the Discrete Adjoint method is implemented and applied to unsteady flows with periodicity, in the context of the Time Spectral Method. The acquired adjoint gradient information is fed into an optimizer and truly unsteady optimization work is carried out for the first time on a realistic test case. The devel- opment and implementation of necessary boundary conditions prove crucial for the successful implementation of the Discrete Adjoint method. As a simple test case, the NACA 0012 airfoil is selected for simulation in steady inviscid, unsteady inviscid, steady viscous, and unsteady viscous flows. In each case, the resulting gradient information obtained from both the adjoint and finite difference method is compared. Upon completion of the airfoil test case, the adjoint-based method is applied to a helicopter blades, UH60, for both steady and unstaedy inviscid flows. The gradient information obtained by the adjoint-based method shows good agreement with the conventional, Finite Difference gradient information. The design methodology was developed for a single processor, however, multi- processor capability is also implemented. In order to accommodate realistic meshes, multi-block capability is added as well. With all of the necessary components im- plemented, optimization is carried out on the UH60 helicopter blade. The objective function is time-averaged torque over all time instances and the optimized result shows an improvement of 5 % over the current configuration. Stanford University Multi-block (SUmb), while implementing the unsteady Reynolds-Averaged Navier Stokes equations with multi-block and multi-processor algorithms, is the chosen flow solver. PETSc is employed as the adjoint solver. Successful implementation of the Discrete Adjoint method to unsteady fluids with periodicity provides the gradient information more easilty than the traditional finite difference method which is hindered by its heavy computational cost and large stor- age requirements. This research establishes a new optimization methodology which utilizes Discrete Adjoint gradient information derived from flow solutions, obtained using the Time Spectral method.
Book Synopsis Improved Accuracy and Uncertainty Management for High-Speed and Unsteady Flows Using Optimization Techniques by :
Download or read book Improved Accuracy and Uncertainty Management for High-Speed and Unsteady Flows Using Optimization Techniques written by and published by . This book was released on 2010 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this project was to demonstrate the potential of adjoint methods for improving the accuracy of current simulation capabilities and to enable more effective uncertainty quantification techniques. A modular adjoint approach was developed in this work and has been applied to the following problems: optimized mesh deformation techniques, hypersonic discretization error reduction through h-p adaptive techniques, real-gas hypersonic simulations, unsteady transonic flow simulations, and unsteady fully coupled aeroelastic problems. In addition to formulating and solving the adjoint problem for all these cases, this work has demonstrated the potential of adjoint-produced sensitivities for efficiently solving time-dependent optimization problems, including fully coupled aeroelastic problems, as well as the potential for adjoint methods for providing error estimates which can be used to drive adaptive meshing or time-stepping procedures and/or for performing uncertainty quantification.
Book Synopsis AIAA Journal by : American Institute of Aeronautics and Astronautics
Download or read book AIAA Journal written by American Institute of Aeronautics and Astronautics and published by . This book was released on 2005 with total page 868 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Engineering Design Optimization by : Joaquim R. R. A. Martins
Download or read book Engineering Design Optimization written by Joaquim R. R. A. Martins and published by Cambridge University Press. This book was released on 2021-11-18 with total page 653 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on course-tested material, this rigorous yet accessible graduate textbook covers both fundamental and advanced optimization theory and algorithms. It covers a wide range of numerical methods and topics, including both gradient-based and gradient-free algorithms, multidisciplinary design optimization, and uncertainty, with instruction on how to determine which algorithm should be used for a given application. It also provides an overview of models and how to prepare them for use with numerical optimization, including derivative computation. Over 400 high-quality visualizations and numerous examples facilitate understanding of the theory, and practical tips address common issues encountered in practical engineering design optimization and how to address them. Numerous end-of-chapter homework problems, progressing in difficulty, help put knowledge into practice. Accompanied online by a solutions manual for instructors and source code for problems, this is ideal for a one- or two-semester graduate course on optimization in aerospace, civil, mechanical, electrical, and chemical engineering departments.
Book Synopsis An Efficient Numerical Framework for Sensitivity Analysis for Unsteady Flows by : Liu Yang
Download or read book An Efficient Numerical Framework for Sensitivity Analysis for Unsteady Flows written by Liu Yang and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "The adjoint method is an efficient approach to computing sensitivities, and has been successfully applied in many fields. However, this approach has not seen widespread acceptance for unsteady problems primarily due to very large storage requirements of present algorithms. The fundamental challenge in unsteady adjoint approaches is the need to integrate the equations in reverse time, which requires all previous flow solutions to be available during the backwards integration. The straightforward treatment of storing all previous flow solutions is prohibitive for simulations with a large number of grid points and time steps. To alleviate this challenge, we propose to compress the flow solutions and store only the reduced bases and expansion coefficients. The flow solutions are recovered when needed in solving the unsteady adjoint equations. In this work, an unsteady discrete adjoint-based approach has been developed. Both continuous and discrete forms of the unsteady adjoint are presented and an adjoint solver based on the latter is validated against finite-difference based sensitivities. A novel adaptive-multi-window compression algorithm has been proposed, where the reduced bases are generated by solving a Proper Orthogonal Decomposition (POD) problem of a matrix consisting flow solutions within a time window. We propose an innovative algorithm that employs an error indicator to determine whether the current bases are sufficient to represent the solution at new time steps; thus saving both computational and storage cost. We apply the unsteady adjoint-based framework to evaluate sensitivity of functionals for turbulent flows in hydraulic turbine draft tubes. We conduct this research within a numerical framework based on the compressible Reynolds-averaged Navier-Stokes equations coupled with the stiffened gas equation of state to model incompressible flow, and a low-speed preconditioner to accelerate convergence. A secondary novel contribution, is an extended eddy-preserving limiter scheme to capture strong turbulent vortices within draft tubes. The numerical framework is first verified on three-dimensional vortex advection cases and then applied to flow simulations in a bulb turbine draft tube. Finally, sensitivities computed with the use of full and compressed flow solutions are compared to demonstrate the feasibility of the approach"--
Book Synopsis Viscosity Stabilized Adjoint Method for Unsteady Compressible Navier-Stokes Equations by : Chaitanya Anil Talnikar
Download or read book Viscosity Stabilized Adjoint Method for Unsteady Compressible Navier-Stokes Equations written by Chaitanya Anil Talnikar and published by . This book was released on 2018 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: Design optimization methods are a popular tool in computational fluid dynamics for designing components or finalizing the flow parameters of a system. The adjoint method accelerates the design process by providing gradients of the design objective with respect to the system parameters. But, typically, adjoint-based design optimization methods have used low fidelity simulations like Reynolds Averaged Navier-Stokes (RANS). To reliably capture the complex flow phenomena like turbulent boundary layers, turbulent wakes and fluid separation involved in high Reynolds number flows, high fidelity simulations like large eddy simulation (LES) are required. Unfortunately, due to the chaotic dynamics of turbulence, the adjoint method for LES diverges and produces incorrect gradients. In this thesis, the adjoint method for unsteady flow equations is modified by adding artificial viscosity to the adjoint equations. The additional viscosity stabilizes the adjoint solution and maintains reasonable accuracy of the gradients obtained from it. The accuracy of the method is assessed on multiple turbulent flow problems, including subsonic flow over a cylinder and transonic flow over a gas turbine vane. The utility of the method is then tested in performing shape optimization of the trailing edge of a transonic turbine vane. The optimal design, found using a modified gradient-based Bayesian optimization algorithm, shows approximately 15% better aero-thermal performance than the baseline design. Such design optimizations are possible due to the availability of massively parallel supercomputers. Designing high performance fluid flow solvers for the next generation supercomputers is a challenging task. In this thesis, a two-level computational graph method for writing optimized distributed flow solvers on heterogeneous architectures is presented. A checkpoint-based automatic differentiation method is used to derive the corresponding adjoint flow solver in this framework.
Book Synopsis A Hybrid Adjoint Approach for Systems of Arbitrarily Complex Partial Differential Equations by : Thomas William Richard Taylor
Download or read book A Hybrid Adjoint Approach for Systems of Arbitrarily Complex Partial Differential Equations written by Thomas William Richard Taylor and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The adjoint method was first developed for fluid dynamics and aerospace applications in the 1970s and 1980s, and has since been applied to an increasingly varied set of problems. Adjoint-based techniques can provide the sensitivity of an objective function to any number of parameters of a simulation inexpensively at roughly the cost of a single additional flow calculation. This information can be used to perform sensitivity analyses, aerodynamic shape optimization and to estimate the error in the objective function due to numerical discretization. Existing approaches to derive adjoint formalisms involve the so-called discrete and continuous methods, which differ in the order of performing the discretization and linearization steps. In this work, an alternative hybrid adjoint approach is proposed with the aim of combining the relative advantages of the continuous and discrete methods, and thus making it easier to apply the adjoint approach to complex problems. The adjoint method can, in general, be derived via a Lagrange multiplier approach, where the governing equations of the primal problem are enforced in the objective function through the adjoint variables. The new hybrid approach developed in this dissertation enforces the flow conservation equations in a continuous manner and additional models, such as those handling turbulence or combustion, discretely. This hybrid approach has been developed with application to both quasi-one-dimensional Euler flow with a simple combustion model and Reynolds-Averaged Navier-Stokes flow with a general turbulence model.
Download or read book Applied Mechanics Reviews written by and published by . This book was released on 1993 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Fluid-Structure Interaction by : Hans-Joachim Bungartz
Download or read book Fluid-Structure Interaction written by Hans-Joachim Bungartz and published by Springer Science & Business Media. This book was released on 2007-06-24 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume in the series Lecture Notes in Computational Science and Engineering presents a collection of papers presented at the International Workshop on FSI, held in October 2005 in Hohenwart and organized by DFG's Research Unit 493 "FSI: Modeling, Simulation, and Optimization". The papers address partitioned and monolithic coupling approaches, methodical issues and applications, and discuss FSI from the mathematical, informatics, and engineering points of view.
Download or read book Optimal Shape Design written by B. Kawohl and published by Springer Science & Business Media. This book was released on 2000-11-16 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimal Shape Design is concerned with the optimization of some performance criterion dependent (besides the constraints of the problem) on the "shape" of some region. The main topics covered are: the optimal design of a geometrical object, for instance a wing, moving in a fluid; the optimal shape of a region (a harbor), given suitable constraints on the size of the entrance to the harbor, subject to incoming waves; the optimal design of some electrical device subject to constraints on the performance. The aim is to show that Optimal Shape Design, besides its interesting industrial applications, possesses nontrivial mathematical aspects. The main theoretical tools developed here are the homogenization method and domain variations in PDE. The style is mathematically rigorous, but specifically oriented towards applications, and it is intended for both pure and applied mathematicians. The reader is required to know classical PDE theory and basic functional analysis.
Book Synopsis Least-Squares Finite Element Methods by : Pavel B. Bochev
Download or read book Least-Squares Finite Element Methods written by Pavel B. Bochev and published by Springer Science & Business Media. This book was released on 2009-04-28 with total page 669 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since their emergence, finite element methods have taken a place as one of the most versatile and powerful methodologies for the approximate numerical solution of Partial Differential Equations. These methods are used in incompressible fluid flow, heat, transfer, and other problems. This book provides researchers and practitioners with a concise guide to the theory and practice of least-square finite element methods, their strengths and weaknesses, established successes, and open problems.
Book Synopsis International Aerospace Abstracts by :
Download or read book International Aerospace Abstracts written by and published by . This book was released on 1999 with total page 1048 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Evaluating Derivatives by : Andreas Griewank
Download or read book Evaluating Derivatives written by Andreas Griewank and published by SIAM. This book was released on 2008-11-06 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: This title is a comprehensive treatment of algorithmic, or automatic, differentiation. The second edition covers recent developments in applications and theory, including an elegant NP completeness argument and an introduction to scarcity.
