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1
Walther, Andrea. "Discrete Adjoints: Theoretical Analysis, Efficient Computation, and Applications." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2008. http://nbn-resolving.de/urn:nbn:de:bsz:14-ds-1214221752009-12115.
Full textAbstract:
The technique of automatic differentiation provides directional derivatives and discrete adjoints with working accuracy. A complete complexity analysis of the basic modes of automatic differentiation is available. Therefore, the research activities are focused now on different aspects of the derivative calculation, as for example the efficient implementation by exploitation of structural information, studies of the theoretical properties of the provided derivatives in the context of optimization problems, and the development and analysis of new mathematical algorithms based on discrete adjoint information. According to this motivation, this habilitation presents an analysis of different checkpointing strategies to reduce the memory requirement of the discrete adjoint computation. Additionally, a new algorithm for computing sparse Hessian matrices is presented including a complexity analysis and a report on practical experiments. Hence, the first two contributions of this thesis are dedicated to an efficient computation of discrete adjoints. The analysis of discrete adjoints with respect to their theoretical properties is another important research topic. The third and fourth contribution of this thesis focus on the relation of discrete adjoint information and continuous adjoint information for optimal control problems. Here, differences resulting from different discretization strategies as well as convergence properties of the discrete adjoints are analyzed comprehensively. In the fifth contribution, checkpointing approaches that are successfully applied for the computation of discrete adjoints, are adapted such that they can be used also for the computation of continuous adjoints. Additionally, the fifth contributions presents a new proof of optimality for the binomial checkpointing that is based on new theoretical results. Discrete adjoint information can be applied for example for the approximation of dense Jacobian matrices. The development and analysis of new mathematical algorithms based on these approximate Jacobians is the topic of the sixth contribution. Is was possible to show global convergence to first-order critical points for a whole class of trust-region methods. Here, the usage of inexact Jacobian matrices allows a considerable reduction of the computational complexity.
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Walther, Andrea. "Discrete Adjoints: Theoretical Analysis, Efficient Computation, and Applications." Doctoral thesis, Technische Universität Dresden, 2007. https://tud.qucosa.de/id/qucosa%3A23715.
Full textAbstract:
The technique of automatic differentiation provides directional derivatives and discrete adjoints with working accuracy. A complete complexity analysis of the basic modes of automatic differentiation is available. Therefore, the research activities are focused now on different aspects of the derivative calculation, as for example the efficient implementation by exploitation of structural information, studies of the theoretical properties of the provided derivatives in the context of optimization problems, and the development and analysis of new mathematical algorithms based on discrete adjoint information. According to this motivation, this habilitation presents an analysis of different checkpointing strategies to reduce the memory requirement of the discrete adjoint computation. Additionally, a new algorithm for computing sparse Hessian matrices is presented including a complexity analysis and a report on practical experiments. Hence, the first two contributions of this thesis are dedicated to an efficient computation of discrete adjoints. The analysis of discrete adjoints with respect to their theoretical properties is another important research topic. The third and fourth contribution of this thesis focus on the relation of discrete adjoint information and continuous adjoint information for optimal control problems. Here, differences resulting from different discretization strategies as well as convergence properties of the discrete adjoints are analyzed comprehensively. In the fifth contribution, checkpointing approaches that are successfully applied for the computation of discrete adjoints, are adapted such that they can be used also for the computation of continuous adjoints. Additionally, the fifth contributions presents a new proof of optimality for the binomial checkpointing that is based on new theoretical results. Discrete adjoint information can be applied for example for the approximation of dense Jacobian matrices. The development and analysis of new mathematical algorithms based on these approximate Jacobians is the topic of the sixth contribution. Is was possible to show global convergence to first-order critical points for a whole class of trust-region methods. Here, the usage of inexact Jacobian matrices allows a considerable reduction of the computational complexity.
3
Hückelheim, Jan Christian. "Discrete adjoints on many cores : algorithmic differentiation of accelerated fluid simulations." Thesis, Queen Mary, University of London, 2017. http://qmro.qmul.ac.uk/xmlui/handle/123456789/24644.
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Simulations are used in science and industry to predict the performance of technical systems. Adjoint derivatives of these simulations can reveal the sensitivity of the system performance to changes in design or operating conditions, and are increasingly used in shape optimisation and uncertainty quantification. Algorithmic differentiation (AD) by source-transformation is an efficient method to compute such derivatives. AD requires an analysis of the computation and its data flow to produce efficient adjoint code. One important step is the activity analysis that detects operations that need to be differentiated. An improved activity analysis is investigated in this thesis that simplifies build procedures for certain adjoint programs, and is demonstrated to improve the speed of an adjoint fluid dynamics solver. The method works by allowing a context-dependent analysis of routines. The ongoing trend towards multi- and many-core architectures such as the Intel XeonPhi is creating challenges for AD. Two novel approaches are presented that replicate the parallelisation of a program in its corresponding adjoint program. The first approach detects loops that naturally result in a parallelisable adjoint loop, while the second approach uses loop transformation and the aforementioned context-dependent analysis to enforce parallelisable data access in the adjoint loop. A case study shows that both approaches yield adjoints that are as scalable as their underlying primal programs. Adjoint computations are limited by their memory footprint, particularly in unsteady simulations, for which this work presents incomplete checkpointing as a method to reduce memory usage at the cost of a slight reduction in accuracy. Finally, convergence of iterative linear solvers is discussed, which is especially relevant on accelerator cards, where single precision floating point numbers are frequently used and the choice of solvers is limited by the small memory size. Some problems that are particular to adjoint computations are discussed.
4
Beigel, Dörte [Verfasser], and Hans Georg [Akademischer Betreuer] Bock. "Efficient goal-oriented global error estimation for BDF-type methods using discrete adjoints / Dörte Beigel ; Betreuer: Hans Georg Bock." Heidelberg : Universitätsbibliothek Heidelberg, 2012. http://d-nb.info/1177148099/34.
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Jando, Dörte [Verfasser], and Hans Georg [Akademischer Betreuer] Bock. "Efficient goal-oriented global error estimation for BDF-type methods using discrete adjoints / Dörte Beigel ; Betreuer: Hans Georg Bock." Heidelberg : Universitätsbibliothek Heidelberg, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:16-heidok-143177.
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Mura, Gabriele Luigi. "Mesh sensitivity investigation in the discrete adjoint framework." Thesis, University of Sheffield, 2017. http://etheses.whiterose.ac.uk/17384/.
Full textAbstract:
Aerodynamic optimisation using gradient-based methods has found a wide range of academic applications in the last 30 years. This framework is also becoming more and more popular in the industrial world where, most of the time, unstructured grids are largely used. In this framework, apart from the need to solve the flow field, there is the need to quickly map the aerodynamic surface in terms of some aerodynamic figure of merits such as the drag coefficient, without being limited by the computational expense related to the grid size. This is a concrete industrial need which requires the efficient computation of the grid sensitivity. A novel method based on the DGM (Delaunay Graph Mapping) mesh movement is proposed to efficiently compute the grid sensitivity required in the discrete adjoint optimisation framework. The method makes use of a one-to-one explicit algebraic mapping between the volume mesh and the solid boundary nodes. This procedure results in a straightforward computation of the gradient without the need to invert a large, sparse and stiff matrix generally associated with implicit mesh movements such as the spring or LE (Linear Elastic) analogy. The method is verified using FDs (Finite Difference) and a thorough comparison in terms of CPU time, formulation against the LE-based mesh movement and adjoint gradient is presented. The DGM-based gradient chain allows to comfortably obtain the gradient with respect to each surface mesh point. Unfortunately, these gradients cannot be used directly because of their inherent poor smoothness feature. In order to address this issue one has to use a parameterisation technique which inevitably sacrifices the design space explorablity. To bridge the gap between the free-nodes and the parameterisation approaches, a novel formulation of the CST (Class Shape Transformation) was developed and termed l-CST (local-CST). The method is based on a simple trigonometric function which works as a cut-off filter on the BPs (Bernstein Polynomials) which are used to enforce a strong on-demand local control. The method is tested on an inverse geometric fitting and its effect on the resulting aerodynamic coefficients and the pressure distribution is also analysed. The DGM-based chain allows the efficient mapping of the entire surface while the l-CST allows the combination of excellent explorablity and surface smoothness. The former is tested within the non-consistent mesh movement and sensitivity framework because there are situations where one method may be preferred over the other based on the grounds that mesh movement is a very different task than mesh sensitivity although strongly related to each other. The latter is instead tested against the free-nodes approach which offers a similar advantage in terms of discrete control although without maintaining a C2 curve unless properly smoothed.
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Schneider, Rene. "Applications of the discrete adjoint method in computational fluid dynamics." Thesis, University of Leeds, 2006. http://etheses.whiterose.ac.uk/1343/.
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The discrete adjoint method allows efficient evaluation of the derivative of a function I(s) with respect to parameters s in situations where I depends on s indirectly, via an intermediate variable w(s), which is computationally expensive to evaluate. In this thesis two applications of this method in the context of computational fluid dynamics are considered. The first is shape optimisation, where the discrete adjoint approach is employed to compute the derivatives with respect to shape parameters for a performance functional depending on the solution of a mathematical flow model which has the form of a discretised system of partial differential equations. In this context particular emphasis is given to efficient solution strategies for the linear systems arising in the discretisation of the flow models. Numerical results for two example problems are presented, demonstrating the utility of the approach. The second application, in adaptive mesh design, allows efficient evaluation of the derivatives of an a posteriori error estimate with respect to the positions of the nodes in a finite element mesh. This novel approach makes additional information available which may be utilised to guide the automatic design of adaptive meshes. Special emphasis is given to problems with anisotropic solution features, for which adaptive anisotropic mesh refinement can deliver significant performance improvements over existing adaptive hrefinement approaches. Two adaptive solution algorithms are presented and compared to existing approaches by applying them to a reaction-diffusion model problem.
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Rothauge, Kai. "The discrete adjoint method for high-order time-stepping methods." Thesis, University of British Columbia, 2016. http://hdl.handle.net/2429/60285.
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This thesis examines the derivation and implementation of the discrete adjoint method for several time-stepping methods. Our results are important for gradient-based numerical optimization in the context of large-scale model calibration problems that are constrained by nonlinear time-dependent PDEs. To this end, we discuss finding the gradient and the action of the Hessian of the data misfit function with respect to three sets of parameters: model parameters, source parameters and the initial condition. We also discuss the closely related topic of computing the action of the sensitivity matrix on a vector, which is required when performing a sensitivity analysis. The gradient and Hessian of the data misfit function with respect to these parameters requires the derivatives of the misfit with respect to the simulated data, and we give the procedures for computing these derivatives for several data misfit functions that are of use in seismic imaging and elsewhere. The methods we consider can be divided into two categories, linear multistep (LM) methods and Runge-Kutta (RK) methods, and several variants of these are discussed. Regular LM and RK methods can be used for ODE systems arising from the semi-discretization of general nonlinear time-dependent PDEs, whereas implicit-explicit and staggered variants can be applied when the PDE has a more specialized form. Exponential time-differencing RK methods are also discussed. The implementation of the associated adjoint time-stepping methods is discussed in detail. Our motivation is the application of the discrete adjoint method to high-order time-stepping methods, but the approach taken here does not exclude lower-order methods. All of the algorithms have been implemented in MATLAB using an object-oriented design and are written with extensibility in mind. For exponential RK methods it is illustrated numerically that the adjoint methods have the same order of accuracy as their corresponding forward methods, and for linear PDEs we give a simple proof that this must always be the case. The applicability of some of the methods developed here to pattern formation problems is demonstrated using the Swift-Hohenberg model.Science, Faculty of
Mathematics, Department of
Graduate
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Roth, Rolf [Verfasser]. "Multilevel Optimization of Turbulent Flows by Discrete Adjoint Techniques / Rolf Roth." München : Verlag Dr. Hut, 2012. http://d-nb.info/1025821424/34.
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Towara, Markus [Verfasser], Uwe [Akademischer Betreuer] Naumann, and Wolfgang [Akademischer Betreuer] Schröder. "Discrete adjoint optimization with OpenFOAM / Markus Towara ; Uwe Naumann, Wolfgang Schröder." Aachen : Universitätsbibliothek der RWTH Aachen, 2018. http://d-nb.info/1187346942/34.
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Le, Moigne Alan. "A discrete Navier-Stokes adjoint method for aerodynamic optimisation of BlendedWing-Body configurations." Thesis, Cranfield University, 2002. http://hdl.handle.net/1826/826.
Full textAbstract:
An aerodynamic shape optimisation capability based on a discrete adjoint solver for Navier-
Stokes flows is developed and applied to a Blended Wing-Body future transport aircraft.
The optimisation is gradient-based and employs either directly a Sequential Quadratic Programming
optimiser or a variable-fidelity optimisation method that combines low- and
high-fidelity models. The shape deformations are parameterised using a B´ezier-Bernstein
formulation and the structured grid is automatically deformed to represent the design changes.
The flow solver at the heart of this optimisation chain is a Reynolds averaged Navier-
Stokes code for multiblock structured grids. It uses Osher’s approximate Riemann solver
for accurate shock and boundary layer capturing, an implicit temporal discretisation and
the algebraic turbulence model of Baldwin-Lomax. The discrete Navier-Stokes adjoint
solver based on this CFD code shares the same implicit formulation but has to calculate
accurately the flow Jacobian. This implies a linearisation of the Baldwin-Lomax model.
The accuracy of the resulting adjoint solver is verified through comparison with finitedifference.
The aerodynamic shape optimisation chain is applied to an aerofoil drag minimisation problem.
This serves as a test case to try and reduce computing time by simplifying the fidelity
of the model. The simplifications investigated include changing the convergence level of
the adjoint solver, reducing the grid size and modifying the physical model of the adjoint
solver independently or in the entire optimisation process. A feasible optimiser and the use
of a penalty function are also tested. The variable-fidelity method proves to be the most ef-
ficient formulation so it is employed for the three-dimensional optimisations in addition to
parallelisation of the flow and adjoint solvers with OpenMP. A three-dimensional Navier-
Stokes optimisation of the ONERA M6 wing is presented. After describing the concept
of Blended Wing-Body and the studies carried out on this aircraft, several aerodynamic
optimisations are performed on this geometry with the capability developed in this thesis.
12
Balsubramanian, Ravishankar. "Error estimation and grid adaptation for functional outputs using discrete-adjoint sensitivity analysis." Master's thesis, Mississippi State : Mississippi State University, 2002. http://library.msstate.edu/etd/show.asp?etd=etd-10032002-113749.
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Khayatzadeh, Peyman. "Aerodynamic shape optimization of natural laminar flow (NLF) airfoils via discrete adjoint approach." Thesis, McGill University, 2013. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=119529.
Full textAbstract:
In this research the γ-Rẽθt transition model is combined with the k-ω SST turbulence model to predict the transition region for a laminar-turbulent boundary layer and redesign the geometry to achieve lower skin friction drag coefficients. The present work addresses several modifications necessary for a robust transition model and investigates the accuracy of the model for a wide range of angles of attack and Reynolds numbers. The transition model is employed to predict the transition locations and an assessment of the various transition mechanisms, Reynolds number effects, sectional characteristics, and aerodynamic performance for two subsonic airfoils are presented with comparisons to experimental data and numerical solutions. Discrete adjoint equations for the transition and turbulence models are derived and implemented into the design framework. The adjoint-based optimization procedure is employed to optimize the S809 wind turbine profile and NLF(1)-0416 airfoil to postpone the onset of transition and extend the natural laminar region of the transitional flow for minimizing the total drag, while maintaining the lift, or maximizing the lift-to-drag ratio. The obtained results demonstrate the ability of the developed optimization framework to design new natural laminar flow airfoils.Dans cette recherche, le modele de transition γ-Rẽθt est combine avec le modele de turbulence k-ω SST afin de predire la transition vers le regime turbulent et d'obtenir des formes aerodynamiques au frottement visqueux minimal. Nous presentons les modifications necessaires a l'amelioration de la robustesse du modele de transition et etudions la precision du modele pour une large gamme d'angles d'attaque et de nombres de Reynolds. Le modele de transition est utilise pour predire le point de transition, pour evaluer les mecanismes de transition ainsi que les differents effets relies au nombre de Reynolds, et pour etudier les caracteristiques et les performances aerodynamique de deux profils aerodynamiques. Des comparaisons avec des donnees experimentales et des solutions numeriques sont presentees. Les equations adjointes discretes des modeles de transition et de turbulence sont derivees et employees dans un processus d'optimization. Cette procedure d'optimisation est utilisee pour modifier le profil d'eolienne S809 et le profil NLF(1)-0416 afin de retarder la transition et d'etendre la zone laminaire de l'ecoulement, de minimiser le coefficient de trainee totale tout en maintenant la portance, ou de maximiser la finesse aerodynamique. Les resultats obtenus demontrent la capacite du processus d'optimisation developpe} a concevoir de nouveaux profils ecoulement laminaire naturel.
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Le, Moigne Alan. "A discrete Navier-Stokes adjoint method for aerodynamic optimisation of Blended Wing-Body configurations." Thesis, Cranfield University, 2002. http://dspace.lib.cranfield.ac.uk/handle/1826/826.
Full textAbstract:
An aerodynamic shape optimisation capability based on a discrete adjoint solver for Navier-Stokes flows is developed and applied to a Blended Wing-Body future transport aircraft. The optimisation is gradient-based and employs either directly a Sequential Quadratic Programming optimiser or a variable-fidelity optimisation method that combines low- and high-fidelity models. The shape deformations are parameterised using a B´ezier-Bernstein formulation and the structured grid is automatically deformed to represent the design changes. The flow solver at the heart of this optimisation chain is a Reynolds averaged Navier- Stokes code for multiblock structured grids. It uses Osher’s approximate Riemann solver for accurate shock and boundary layer capturing, an implicit temporal discretisation and the algebraic turbulence model of Baldwin-Lomax. The discrete Navier-Stokes adjoint solver based on this CFD code shares the same implicit formulation but has to calculate accurately the flow Jacobian. This implies a linearisation of the Baldwin-Lomax model. The accuracy of the resulting adjoint solver is verified through comparison with finitedifference. The aerodynamic shape optimisation chain is applied to an aerofoil drag minimisation problem. This serves as a test case to try and reduce computing time by simplifying the fidelity of the model. The simplifications investigated include changing the convergence level of the adjoint solver, reducing the grid size and modifying the physical model of the adjoint solver independently or in the entire optimisation process. A feasible optimiser and the use of a penalty function are also tested. The variable-fidelity method proves to be the most efficient formulation so it is employed for the three-dimensional optimisations in addition to parallelisation of the flow and adjoint solvers with OpenMP. A three-dimensional Navier-Stokes optimisation of the ONERA M6 wing is presented. After describing the concept of Blended Wing-Body and the studies carried out on this aircraft, several aerodynamic optimisations are performed on this geometry with the capability developed in this thesis.
15
Valicov, Petru. "Problèmes de placement, de coloration et d'identification." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2012. http://tel.archives-ouvertes.fr/tel-00801982.
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Dans cette thèse, nous nous intéressons à trois problèmes issus de l'informatique théorique, à savoir le placement de formes rectangulaires dans un conteneur (OPP), la coloration dite "forte" d'arêtes des graphes et les codes identifiants dans les graphes. L'OPP consiste à décider si un ensemble d'items rectangulaires peut être placé sans chevauchement dans un conteneur rectangulaire et sans dépassement des bords de celui-ci. Une contrainte supplémentaire est prise en compte, à savoir l'interdiction de rotation des items. Le problème est NP-difficile même dans le cas où le conteneur et les formes sont des carrés. Nous présentons un algorithme de résolution efficace basé sur une caractérisation du problème par des graphes d'intervalles, proposée par Fekete et Schepers. L'algorithme est exact et utilise les MPQ-arbres - structures de données qui encodent ces graphes de manière compacte tout en capturant leurs propriétés remarquables. Nous montrons les résultats expérimentaux de notre approche en les comparant aux performances d'autres algorithmes existants. L'étude de la coloration forte d'arêtes et des codes identifiants porte sur les aspects structurels et de calculabilité de ces deux problèmes. Dans le cas de la coloration forte d'arêtes nous nous intéressons plus particulièrement aux familles des graphes planaires et des graphes subcubiques. Nous montrons des bornes optimales pour l'indice chromatique fort des graphes subcubiques en fonction du degré moyen maximum et montrons que tout graphe planaire subcubique sans cycles induits de longueur 4 et 5 est coloriable avec neuf couleurs. Enfin nous confirmons la difficulté du problème de décision associé, en prouvant qu'il est NP-complet dans des sous-classes restreintes des graphes planaires subcubiques. La troisième partie de la thèse est consacrée aux codes identifiants. Nous proposons une caractérisation des graphes identifiables dont la cardinalité du code identifiant minimum est n − 1, où n est l'ordre du graphe. Nous étudions la classe des graphes adjoints et nous prouvons des bornes inférieures et supérieures serrées pour la cardinalité du code identifiant minimum dans cette classe. Finalement, nous montrons qu'il existe un algorithme linéaire de calcul de ce paramètre dans la classe des graphes adjoints L(G) où G a une largeur arborescente bornée par une constante. En revanche nous nous apercevons que le problème est NP-complet dans des sous-classes très restreintes des graphes parfaits.
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Dubuisson, Clement. "Etude du spectre discret de perturbations d'opérateurs de la physique mathématique." Thesis, Bordeaux, 2014. http://www.theses.fr/2014BORD0127/document.
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Le but de cette thèse est d’obtenir des informations sur le spectre discret d’opérateurs non auto-adjoints définis par des perturbations relativement compactes d’opérateurs auto-adjoints. Ces opérateurs auto-adjoints sont choisis parmi les opérateurs classiques de mécanique quantique. Il s’agit des opérateurs de Dirac, de Klein-Gordon et le laplacien fractionnaire qui généralise l’opérateur de Schrödinger habituellement considéré pour de tels problèmes. La principale méthode utilisée ici relève d’un théorème d’analyse complexe donnant une condition de type Blaschke sur les zéros d’une fonction holomorphe du disque unité. Cette condition traduit lecomportement des valeurs propres de l’opérateur perturbé sous forme d’inégalités de type Lieb-Thirring. Une autre méthode venant d’analyse fonctionnelle a été employée pour obtenir de telles inégalités et les deux méthodes sont comparées entre ellesThe topic of this thesis concerns the discrete spectrum of non-selfadjoint operators defined by relatively compact perturbation of selfadjoint operators. These selfadjoint operators are choosen among classical operators of quantum mechanics. These areDirac operator, Klein-Gordon operator, and the fractional Laplacian who generalize the Schrödinger operator. The main method is based on a theorem of complex analysis which gives Blaschke-type condition on the zeros of a holomorphic function on the unit disc. This Blaschke condition gives the information on the behaviour of eigenvalues of the perturbed operator by mean of Lieb-Thirring-type inequalities. Another method using functional analysis is also used to obtain these kind of inequalities and both methods are compared to each other
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Nielsen, Eric John. "Aerodynamic Design Sensitivities on an Unstructured Mesh Using the Navier-Stokes Equations and a Discrete Adjoint Formulation." Diss., Virginia Tech, 1998. http://hdl.handle.net/10919/29459.
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A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated, and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences and a complex-variable approach.
Several simplifying approximations to the complete linearization of the residual are also presented. A first-order approximation to the dependent variables is implemented in the adjoint and design equations, and the effect of a "frozen" eddy viscosity and neglecting mesh sensitivity terms is also examined. The resulting derivatives from these approximations are all shown to be inaccurate and often of incorrect sign. However, a partially-converged adjoint solution is shown to be sufficient for computing accurate sensitivity derivatives, yielding a potentially large cost savings in the design process. The convergence rate of the adjoint solver is compared to that of the flow solver. For inviscid adjoint solutions, the cost is roughly one to four times that of a flow solution, whereas for turbulent computations, this ratio can reach as high as ten. Sample optimizations are performed for inviscid and turbulent transonic flows over an ONERA M6 wing, and drag reductions are demonstrated.Ph. D.
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Wang, Yang. "Robust and stable discrete adjoint solver development for shape optimisation of incompressible flows with industrial applications." Thesis, Queen Mary, University of London, 2017. http://qmro.qmul.ac.uk/xmlui/handle/123456789/24870.
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This thesis investigates stabilisation of the SIMPLE-family discretisations for incompressible flow and their discrete adjoint counterparts. The SIMPLE method is presented from typical \prediction-correction" point of view, but also using a pressure Schur complement approach, which leads to a wider class of schemes. A novel semicoupled implicit solver with velocity coupling is proposed to improve stability. Skewness correction methods are applied to enhance solver accuracy on non-orthogonal grids. An algebraic multi grid linear solver from the HYPRE library is linked to flow and discrete adjoint solvers to further stabilise the computation and improve the convergence rate. With the improved implementation, both of flow and discrete adjoint solvers can be applied to a wide range of 2D and 3D test cases. Results show that the semi-coupled implicit solver is more robust compared to the standard SIMPLE solver. A shape optimisation of a S-bend air flow duct from a VW Golf vehicle is studied using a CAD-based parametrisation for two Reynolds numbers. The optimised shapes and their flows are analysed to con rm the physical nature of the improvement. A first application of the new stabilised discrete adjoint method to a reverse osmosis (RO) membrane channel flow is presented. A CFD model of the RO membrane process with a membrane boundary condition is added. Two objective functions, pressure drop and permeate flux, are evaluated for various spacer geometries such as open channel, cavity, submerged and zigzag spacer arrangements. The flow and the surface sensitivity of these two objective functions is computed and analysed for these geometries. An optimisation with a node-base parametrisation approach is carried out for the zigzag con guration channel flow in order to reduce the pressure drop. Results indicate that the pressure loss can be reduced by 24% with a slight reduction in permeate flux by 0.43%.
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Militão, Damiano da Silva. "Desenvolvimento de um método espectronodal livre de erros de truncamento espacial para problemas adjuntos de transporte de partículas neutras monoenergéticas na formulação de ordenadas discretas em geometria unidimensional." Universidade do Estado do Rio de Janeiro, 2011. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=3210.
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Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorUm método numérico nodal livre de erros de truncamento espacial é desenvolvido para problemas adjuntos de transporte de partículas neutras monoenergéticas em geometria unidimensional com fonte fixa na formulação de ordenadas discretas (SN). As incógnitas no método são os fluxos angulares adjuntos médios nos nodos e os fluxos angulares adjuntos nas fronteiras dos nodos, e os valores numéricos gerados para essas quantidades são os obtidos a partir da solução analítica das equações SN adjuntas. O método é fundamentado no uso da convencional equação adjunta SN discretizada de balanço espacial, que é válida para cada nodo de discretização espacial e para cada direção discreta da quadratura angular, e de uma equação auxiliar adjunta não convencional, que contém uma função de Green para os fluxos angulares adjuntos médios nos nodos em termos dos fluxos angulares adjuntos emergentes das fronteiras dos nodos e da fonte adjunta interior. Resultados numéricos são fornecidos para ilustrarem a precisão do método proposto.
A numerical nodal method that is free from all spatial truncation errors is developed for one-speed slab-geometry discrete ordinates (SN) fixed-source adjoint neutral particle transport problems. The unknown in the method are the node-edge and the node-average adjoint angular fluxes, and the numerical values obtained for these quantities are those of the analytic solution of the adjoint SN equations. The method is based on the use of the standard spatially discretized SN balance adjoint equation, which holds in each spatial node and for each discrete ordinates direction, and a nonstandard adjoint auxiliary equation that contains a Greens function for the node-average adjoint angular fluxes in terms of the exiting adjoint angular fluxes from the node edges and the adjoint interior source. Numerical results are given to illustrate the methods accuracy.
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Bourasseau, Sébastien. "Contribution à une méthode de raffinement de maillage basée sur le vecteur adjoint pour le calcul de fonctions aérodynamiques." Thesis, Nice, 2015. http://www.theses.fr/2015NICE4138/document.
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L’adaptation de maillage est un outil puissant pour l’obtention de simulations aérodynamiques précises à coût limité. Dans le cas particulier des simulations visant au calcul de fonctions aérodynamiques (efforts, moments, rendements...), plusieurs méthodes dites de raffinement ciblé (ou, en anglais, « goal-oriented ») basées sur le vecteur adjoint de la fonction d’intérêt ont été proposées. L’objectif de la thèse est l’extension d’une méthode de ce type basée sur la dérivée totale dJ/dX de la grandeur aérodynamique d’intérêt, J, par rapport aux coordonnées du maillage volumique, X. Les trois méthodes usuelles de calcul de gradient discret – la méthode de différentiation directe, la méthode adjointe-"paramètres" et la méthode adjointe-"maillage" évaluant dJ/dX – ont tout d’abord été étudiées et codées dans le logiciel elsA de l’ONERA pour des maillages non-structurés, pour des écoulements compressibles de fluide parfait et des écoulements laminaires. La seconde étape du travail a consisté à créer un senseur local θ basé sur dJ/dX qui identifie les zones du maillage volumique où la position des nœuds a une forte incidence sur l’évaluation de la fonction J. Ce senseur sert d’indicateur pour l’adaptation de différents maillages, pour différents régimes d’écoulement (subsonique, transsonique, supersonique), pour des configurations d’aérodynamique interne (aube et tuyère) et externe (profil d’aile). La méthode proposée est comparée à une méthode de raffinement ciblée très populaire (Venditti et Darmofal, 2001) et à une méthode de raffinement basée sur les caractéristiques de l’écoulement (ou, en anglais, « feature-based ») ; elle conduit à des résultats très satisfaisantsMesh adaptation is a powerful tool to obtain accurate aerodynamic simulations with limited cost. In the specific case of computation of aerodynamic functions (forces, moments, efficiency ...), goal-oriented methods based on the adjoint vector have been proposed. The aim of the thesis is the extension of a method of this type based on the total derivative dJ/dX of the aerodynamic output of interest, J, with respect to the volume mesh coordinates, X. The three common methods for calculating discrete gradient – the direct differentiation method, the parameter-adjoint method and mesh-adjoint method evaluating dJ/dX – have been studied first and coded in the elsA ONERA software for unstructured grids, for compressible inviscid and laminar flows. The second part of this work was has been to define a local sensor θ based on dJ/dX in order to identify zones where the volume mesh nodes position has a strong impact on the evaluation of the function J. This sensor is the selected indicator for different mesh adaptations for different flow regimes (subsonic, transonic, supersonic) for internal (blade and nozzle) and external (wing profile) aerodynamic configurations. The proposed method is compared to a well-known goal-oriented method (Darmofal and Venditti, 2001) and to a feature-based method ; it leads to very consistent results. very consistent results
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Nguyen-Dinh, Maxime. "Qualification des simulations numériques par adaptation anisotropique de maillages." Phd thesis, Université Nice Sophia Antipolis, 2014. http://tel.archives-ouvertes.fr/tel-00987202.
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La simulation numérique est largement utilisée pour évaluer les performances aérodynamiques des aéronefs ainsi qu'en optimisation de forme. Ainsi l'objectif de ces simulations est souvent le calcul de fonctions aérodynamiques. L'objet de cette thèse est d'étudier des méthodes d'adaptation de maillages basées sur la dérivée totale de ces fonctions par rapport aux coordonnées du maillage (notée dJ/dX). Celle-ci pouvant être calculée par la méthode adjointe discrète. La première partie de cette étude concerne l'application de méthodes d'adaptation de maillages appliquées à des écoulements de fluides parfaits. Le senseur qui détecte les zones de maillage à raffiner s'appuie sur la norme de cette dérivée pour adapter des maillages pour le calcul d'une fonction J. La seconde partie du travail est la construction et l'étude de critères plus fiables basés sur dJ/dX pour d'une part adapter des maillages et d'autre part estimer si un maillage est bien adapté ou non pour le calcul de la fonction J. De plus une méthode de remaillage plus efficace basée sur une EDP elliptique est aussi présentée. Cette nouvelle méthode est appliquée pour des écoulements bidimensionnels de fluides parfaits ainsi que pour un écoulement décrit par les équations RANS. La dernière partie de l'étude est consacrée à l'application de la méthode proposée à des cas tridimensionnels d'écoulement RANS sur des géométries d'intérêt industriel.
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Marcelet, Meryem. "Etude et mise en oeuvre d'une méthode d'optimisation de forme couplant simulation numérique en aérodynamique et en calcul de structure." Phd thesis, Paris, ENSAM, 2008. http://tel.archives-ouvertes.fr/tel-00367508.
Full textAbstract:
L'objet de ce travail a principalement consisté en l'étude et la mise en oeuvre d'une méthode de calcul des gradients des fonctions aérodynamiques par rapport à des paramètres géométriques pour un système aéroélastique soumis à un écoulement lointain stationnaire. Dans un premier temps, une méthodologie de calcul de l'équilibre aéroélastique statique a tout d'abord été développée. Dans ce cadre, le comportement du fluide peut être modélisé par les équations d'Euler ou par les équations de Navier-Stokes moyennées (RANS). Celles-ci sont numériquement résolues par elsA - code de simulation numérique pour la mécanique des fluides développé à l'ONERA. Le comportement de la structure est, quant à lui, prédit par la théorie des poutres et les équations d'Euler-Bernoulli. Le chargement aérodynamique est transmis à la structure par l'intermédiaire de la matrice des coefficients d'influence également appelée matrice de flexibilité. Seuls les efforts de torsion et de flexion sont transmis de manière consistante à la structure, dont seuls les mouvements induits de torsion et de flexion sont calculés sous l'hypothèse des petits déplacements. La déformation résultante sur le maillage du domaine fluide est prédite analytiquement par analogie avec la mécanique du solide. Enfin, le système aéroélastique couplé est résolu selon un processus itératif inspiré de la méthode du point fixe. Dans un deuxième temps, un cadre de calcul, pour le système aéroélastique décrit précédemment, des gradients des fonctions d'intérêt (objectif et contraintes) par rapport à un vecteur de paramètres géométriques de la forme solide a été mis en oeuvre. Les gradients peuvent être calculés par la méthode de l'équation linéarisée discrète ou par la méthode du vecteur adjoint discret. Ces méthodes reposent sur la résolution de systèmes linéaires couplés, effectuée, dans le cadre de cette étude, par un processus itératif doublement retardé. Pour finir, ces développements ont été appliqués au calcul des gradients des coefficients aérodynamiques de traînée et de portance par rapport à un ensemble de paramètres de forme pour trois configurations aérodynamiques de complexité croissante: équations d'Euler résolues sur un maillage multibloc coïncident, équations RANS résolues sur un maillage monobloc, et, finalement, équations RANS résolues sur un maillage multibloc non-coïncident. La validité des résultats a été établie par comparaison aux gradients calculés par différences finies. Une dernière partie du travail a été consacrée à l'évaluation des performances de quatre modèles réduits non physiques dans le cadre d'un processus d'optimisation de forme d'une configuration bidimensionnelle de turbomachine.
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Pham, Chi-Tuân. "Linéarisation du flux visqueux des équations de navier-stokes et de modèles de turbulence pour l'optimisation aérodynamique en turbomachines." Phd thesis, Paris, ENSAM, 2006. http://pastel.archives-ouvertes.fr/pastel-00002058.
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Le calcul de gradients de fonctions aérodynamiques par rapport à des paramètres géométriques de la forme solide est une sous-discipline de la simulation numérique en mécanique des fluides. Cette dérivation par rapport à deux variables dépendantes, le champ aérodynamique et le maillage, liés par les équations discrètes de la mécanique des fluides, nécessite l'inversion d'un système linéaire dont la matrice est la matrice jacobienne des équations de la mécanique des fluides discrétisées par rapport au champ aérodynamique (méthode de l'équation linéarisée) ou la transposée de cette matrice jacobienne (méthode adjointe). La précision du calcul de cette matrice jacobienne fait débat lorsque les équations de la mécanique des fluides sont modélisées avec l'approche RANS. L'objectif de cette thèse est de déterminer le degré d'exactitude de la linéarisation d'un flux visqueux discret et des équations discrètes de certains modèles de turbulence, nécessaire à l'obtention de gradients précis de fonctions métiers des concepteurs de turbomachines par rapport à des paramètres géométriques d'une aube. L'écriture de linéarisations approchées du flux visqueux (avec ou sans approche dite de "couche mince") et de deux modèles de turbulence (modèle algébrique de Michel et al. et modèle à deux équations de transport k-e de Launder-Sharma) est détaillée. Pour le modèle de Michel et al., plusieurs approximations pour la linéarisation des équations du modèle ont été testées et comparées. Des résultats (valeurs de gradients de fonctions aérodynamiques, sensibilités d'écoulement pour la méthode linéarisée) sont présentés pour la tuyère de Déléry dite C, l'aile ONERA M6 et deux configurations d'aube isolée de turbine. Des recommandations sont formulées pour le calcul de gradients pour des configurations de machine tournante avec modélisation RANS.
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Walker, Scottie. "Spectrally-matched neutron detectors designed using computational adjoint SN for plug-in replacement of Helium-3." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/49093.
Full textAbstract:
Neutron radiation detectors are an integral part of the Department of Homeland Security (DHS) efforts to detect the illicit trafficking of radioactive or special nuclear materials into the U.S. In the past decade, the DHS has deployed a vast network of radiation detection systems at various key positions to prevent or to minimize the risk associated with the malevolent use of these materials. The greatest portion of this detection burden has been borne by systems equipped with 3He because of its highly desirable physical and nuclear properties. However, a dramatic increase in demand and dwindling supply, combined with a lack of oversight for the existing 3He stockpile has produced a critical shortage of this gas which has virtually eliminated its viability for detector applications. A number of research efforts have been undertaken to develop suitable 3He replacements; however, these studies have been solely targeted toward simple detection cases where the overall detection efficiency is the only concern. For these cases, an insertion of additional detectors or materials can produce reaction rates that are sufficient, because the neutron spectral response is essentially irrelevant. However, in applications such as safeguards, non-proliferation efforts, and material control and accountability programs (MC&A), a failure to use detectors that are spectrally matched to 3He can potentially produce dire consequences. This is because these more difficult detection scenarios are associated with fissile material assessments for 239Pu and other actinides and these analyses have almost universally been calibrated to an equivalent 3He response. In these instances, a “simple” detector or material addition approach is neither appropriate nor possible, due to influences resulting from the complex nature of neutron scattering in moderators, cross sections, gas pressure variations, geometries, and surrounding structural interference. These more challenging detection cases require a detailed computational transport analysis be performed for each specific application.
A leveraged approach using adjoint transport computations that are validated by forward transport and Monte Carlo computations and laboratory measurements can address these more complex detection cases and this methodology was utilized in the execution of the research. The initial task was to establish the fidelity of a computational approach by executing radiation transport models for existing BF3 and 3He tubes and then comparing the modeling results to laboratory measurements made using these identical devices. Both tubes were 19.6 cm in height, 1-inch in diameter, and operated at 1 and 4 atm pressure respectively. The models were processed using a combination of forward Monte Carlo and forward and adjoint 3-D discrete ordinates (SN) transport methods. The computer codes MCNP5 and PENTRAN were used for all calculations of a nickel-shielded plutonium-beryllium (PuBe) source term that provided a neutron output spectra equivalent to that of weapons-grade plutonium (WGPu).
Once the computational design approach was validated, the adjoint SN method was used to iteratively identify six distinct plug-in models that matched the neutron spectral response and reaction rate of a 1-inch diameter 3He tube with a length of 10 cm and operating at 4 atm pressure. The equivalent designs consist of large singular tubes and dual tubes containing BF3 gas, 10B linings, and/or 10B-loaded polyvinyl toluene (PVT). The reaction rate for each plug-in design was also verified using forward PENTRAN and MCNP5 calculations. In addition to the equivalent designs, the adjoint method also yielded various insights into neutron detector design that can lead to additional designs using a combination of different detector materials such as BF3/10B-loaded PVT, 10B-lined tubes/10B-loaded PVT, etc.
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Bashtova, Kateryna. "Modélisation et identification de paramètres pour les empreintes des faisceaux de haute énergie." Thesis, Université Côte d'Azur (ComUE), 2016. http://www.theses.fr/2016AZUR4112/document.
Full textAbstract:
Le progrès technologique nécessite des techniques de plus en plus sophistiquées et précises de traitement de matériaux. Nous étudions le traitement de matériaux par faisceaux de haute énergie : un jet d’eau abrasif, une sonde ionique focalisée, un laser. L’évolution de la surface du matériau sous l’action du faisceau de haute énergie est modélisée par une EDP. Cette équation contient l’ensemble des coefficients inconnus - les paramètres de calibration de mo- dèle. Les paramètres inconnus peuvent être calibrés par minimisation de la fonction coût, c’est-à-dire, la fonction qui décrit la différence entre le résultat de la modélisation et les données expérimentales. Comme la surface modélisée est une solution du problème d’EDP, cela rentre dans le cadre de l’optimisation sous contrainte d’EDP. L’identification a été rendue bien posée par la régularisation du type Tikhonov. Le gradient de la fonction coût a été obtenu en utilisant les deux méthodes : l’approche adjointe et la différen- ciation automatique. Une fois la fonction coût et son gradient obtenus, nous avons utilisé un minimiseur L-BFGS pour réaliser la minimisation.Le problème de la non-unicité de la solution a été résolu pour le problème de traitement par le jet d’eau abrasif. Des effets secondaires ne sont pas inclus dans le modèle. Leur impact sur le procédé de calibration a été évité. Ensuite, le procédé de calibration a été validé pour les données synthétiques et expérimentales. Enfin, nous avons proposé un critère pour distinguer facilement entre le régime thermique et non- thermique d’ablation par laserThe technological progress demands more and more sophisticated and precise techniques of the treatment of materials. We study the machining of the material with the high energy beams: the abrasive waterjet, the focused ion beam and the laser. Although the physics governing the energy beam interaction with material is very different for different application, we can use the same approach to the mathematical modeling of these processes.The evolution of the material surface under the energy beam impact is modeled by PDE equation. This equation contains a set of unknown parameters - the calibration parameters of the model. The unknown parameters can be identified by minimization of the cost function, i.e., function that describes the differ- ence between the result of modeling and the corresponding experimental data. As the modeled surface is a solution of the PDE problem, this minimization is an example of PDE-constrained optimization problem. The identification problem was regularized using Tikhonov regularization. The gradient of the cost function was obtained both by using the variational approach and by means of the automatic differentiation. Once the cost function and its gradient calculated, the minimization was performed using L-BFGS minimizer.For the abrasive waterjet application the problem of non-uniqueness of numerical solution is solved. The impact of the secondary effects non included into the model is avoided as well. The calibration procedure is validated on both synthetic and experimental data.For the laser application, we presented a simple criterion that allows to distinguish between the thermal and non-thermal laser ablation regimes
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Günnel, Andreas. "Adaptive Netzverfeinerung in der Formoptimierung mit der Methode der Diskreten Adjungierten." Master's thesis, Universitätsbibliothek Chemnitz, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201000390.
Full textAbstract:
Formoptimierung bezeichnet die Bestimmung der Geometrischen Gestalt eines Gebietes auf dem eine partielle Differentialgleichung (PDE) wirkt, sodass bestimmte gegebene Zielgrößen, welche von der Lösung der PDE abhängen, Extrema annehmen. Bei der Diskret Adjungierten Methode wird der Gradient einer Zielgröße bezüglich einer beliebigen Anzahl von Formparametern mit Hilfe der Lösung einer adjungierten Gleichung der diskretisierten PDE effizient ermittelt. Dieser Gradient wird dann in Verfahren der numerischen Optimierung verwendet um eine optimale Lösung zu suchen. Da sowohl die Zielgröße als auch der Gradient für die diskretisierte PDE ermittelt werden, sind beide zunächst vom verwendeten Netz abhängig. Bei groben Netzen sind sogar Unstetigkeiten der diskreten Zielfunktion zu erwarten, wenn bei Änderungen der Formparameter sich das Netz unstetig ändert (z.B. Änderung Anzahl Knoten, Umschalten der Konnektivität). Mit zunehmender Feinheit der Netze verschwinden jedoch diese Unstetigkeiten aufgrund der Konvergenz der Diskretisierung. Da im Zuge der Formoptimierung Zielgröße und Gradient für eine Vielzahl von Iterierten der Lösung bestimmt werden müssen, ist man bestrebt die Kosten einer einzelnen Auswertung möglichst gering zu halten, z.B. indem man mit nur moderat feinen oder adaptiv verfeinerten Netzen arbeitet. Aufgabe dieser Diplomarbeit ist es zu untersuchen, ob mit gängigen Methoden adaptiv verfeinerte Netze hinreichende Genauigkeit der Auswertung von Zielgröße und Gradient erlauben und ob eventuell Anpassungen der Optimierungsstrategie an die adaptive Vernetzung notwendig sind. Für die Untersuchungen sind geeignete Modellprobleme aus der Festigkeitslehre zu wählen und zu untersuchenShape optimization describes the determination of the geometric shape of a domain with a partial differential equation (PDE) with the purpose that a specific given performance function is minimized, its values depending on the solution of the PDE. The Discrete Adjoint Method can be used to evaluate the gradient of a performance function with respect to an arbitrary number of shape parameters by solving an adjoint equation of the discretized PDE. This gradient is used in the numerical optimization algorithm to search for the optimal solution. As both function value and gradient are computed for the discretized PDE, they both fundamentally depend on the discretization. In using the coarse meshes, discontinuities in the discretized objective function can be expected if the changes in the shape parameters cause discontinuous changes in the mesh (e.g. change in the number of nodes, switching of connectivity). Due to the convergence of the discretization these discontinuities vanish with increasing fineness of the mesh. In the course of shape optimization, function value and gradient require evaluation for a large number of iterations of the solution, therefore minimizing the costs of a single computation is desirable (e.g. using moderately or adaptively refined meshes). Overall, the task of the diploma thesis is to investigate if adaptively refined meshes with established methods offer sufficient accuracy of the objective value and gradient, and if the optimization strategy requires readjustment to the adaptive mesh design. For the investigation, applicable model problems from the science of the strength of materials will be chosen and studied
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Walther, Andrea [Verfasser]. "Discrete adjoints: theoretical analysis, efficient computation and applications / von Andrea Walther." 2008. http://d-nb.info/989843572/34.
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Lee, Ki Hwan Alonso Juan José Jameson Antony MacCormack R. W. "Design optimization of periodic flows using a time-spectral discrete adjoint method." 2010. http://purl.stanford.edu/zf671jy0354.
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Anil, N. "Optimal Control Of Numerical Dissipation In Modified KFVS (m-KFVS) Using Discrete Adjoint Method." Thesis, 2008. http://hdl.handle.net/2005/710.
Full textAbstract:
The kinetic schemes, also known as Boltzmann schemes are based on the moment-method-strategy, where an upwind scheme is first developed at the Boltzmann level and after taking suitable moments we arrive at an upwind scheme for the governing Euler or Navier-Stokes equations. The Kinetic Flux Vector Splitting (KFVS)scheme, which belongs to the family of kinetic schemes is being extensively used to compute inviscid as well as viscous flows around many complex configurations of practical interest over the past two decades. To resolve many flow features accurately, like suction peak, minimising the loss in stagnation pressure, shocks, slipstreams, triple points, vortex sheets, shock-shock interaction, mixing layers, flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order KFVS method even though is very robust suffers from the problem of having much more numerical diffusion than required, resulting in very badly smearing of the above features. However, numerical dissipation can be reduced considerably by using higher order kinetic schemes. But they require more points in the stencil and hence consume more computational time and memory. The second order schemes require flux or slope limiters in the neighbourhood of discontinuities to avoid spurious and physically meaningless wiggles or oscillations in pressure, temperature or density. The limiters generally restrict the residue fall in second order schemes while in first order schemes residue falls up to machine zero. Further, pressure and density contours or streamlines are much smoother for first order accurate schemes than second order accurate schemes. A question naturally arises about the possibility of constructing first order upwind schemes which retain almost all advantages mentioned above while at the same time crisply capture the flow features.
In the present work, an attempt has been made to address the above issues by developing yet another kinetic scheme, known as the low dissipative modified KFVS (m-KFVS) method based on modified CIR (MCIR) splitting with molecular velocity dependent dissipation control function. Different choices for the dissipation control function are presented. A detailed mathematical analysis and the underlying physical arguments behind these choices are presented. The expressions for the m-KFVS fluxes are derived. For one of the choices, the expressions for the split fluxes are similar to the usual first order KFVS method. The mathematical properties of 1D m-KFVS fluxes and the eigenvalues of the corresponding flux Jacobians are studied numerically. The analysis of numerical dissipation is carried out both at Boltzmann and Euler levels. The expression for stability criterion is derived. In order to be consistent with the interior scheme, modified solid wall and outer boundary conditions are derived by extending the MCIR idea to boundaries.
The cell-centred finite volume method based on m-KFVS is applied to several standard test cases for 1D, 2D and 3D inviscid flows. In the case of subsonic flows, the m-KFVS method produces much less numerical entropy compared to first order KFVS method and the results are comparable to second order accurate q-KFVS method. In transonic and supersonic flows, m-KFVS generates much less numerical dissipation compared to first order KFVS and even less compared to q-KFVS method. Further, the m-KFVS method captures the discontinuities more sharply with contours being smooth and near second order accuracy has been achieved in smooth regions, by still using first order stencil. Therefore, the numerical dissipation generated by m-KFVS is considerably reduced by suitably choosing the dissipation control variables. The Euler code based on m-KFVS method almost takes the same amount of computational time as that of KFVS method.
Although, the formal accuracy is of order one, the m-KFVS method resolves the flow features much more accurately compared to first order KFVS method but the numerical dissipation generated by m-KFVS method may not be minimal. Hence, the dissipation control vector is in general not optimal. If we can find the optimal dissipation control vector then we will be able to achieve the minimal dissipation. In the present work, the above objective is attained by posing the minimisation of numerical dissipation in m-KFVS method as an optimal control problem. Here, the control variables are the dissipation control vector. The discrete form of the cost function, which is to be minimised is considered as the sum of the squares of change in entropy at all cells in the computational domain. The number of control variables is equal to the total number of cells or finite volumes in the computational domain, as each cell has only one dissipation control variable.
In the present work, the minimum value of cost function is obtained by using gradient based optimisation method. The sensitivity gradients of the cost function with respect to the control variables are obtained using discrete adjoint approach. The discrete adjoint equations for the optimisation problem of minimising the numerical dissipation in m-KFVS method applied to 2D and 3D Euler equations are derived. The method of steepest descent is used to update the control variables. The automatic differentiation tool Tapenade has been used to ease the development of adjoint codes.
The m-KFVS code combined with discrete adjoint code is applied to several standard test cases for inviscid flows. The test cases considered are, low Mach number flows past NACA 0012 airfoil and two element Williams airfoil, transonic and supersonic flows past NACA 0012 airfoil and finally, transonic flow past Onera M6 wing. Numerical results have shown that the m-KFVS-adjoint method produces even less numerical dissipation compared to m-KFVS method and hence results in more accurate solution. The m-KFVS-adjoint code takes more computational time compared to m-KFVS code.
The present work demonstrates that it is possible to achieve near second order accuracy by formally first order accurate m-KFVS scheme while retaining advantages of first order accurate methods.
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Rumpfkeil, Markus Peer. "Airfoil Optimization for Unsteady Flows with Application to High-lift Noise Reduction." Thesis, 2008. http://hdl.handle.net/1807/17241.
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The use of steady-state aerodynamic optimization methods in the computational
fluid dynamic (CFD) community is fairly well
established. In particular, the use of adjoint methods has proven to be very
beneficial because their cost is independent of the number of design variables.
The application of numerical optimization to airframe-generated noise, however, has not received as much attention, but with the significant
quieting of modern engines, airframe noise now competes with engine noise.
Optimal control techniques for unsteady flows are needed in order to be able to reduce airframe-generated noise.
In this thesis, a general framework is formulated to calculate the gradient of a cost function in a nonlinear unsteady flow environment
via the discrete adjoint method. The unsteady optimization algorithm developed in this work
utilizes a Newton-Krylov approach since the gradient-based optimizer uses the quasi-Newton method BFGS, Newton's method is applied to the
nonlinear flow problem, GMRES is used to solve the resulting linear problem inexactly, and last but not least the linear adjoint problem
is solved using Bi-CGSTAB. The flow is governed by the unsteady two-dimensional
compressible Navier-Stokes equations in conjunction with a one-equation turbulence model, which are discretized using
structured grids and a finite difference approach. The effectiveness of the unsteady optimization algorithm is demonstrated
by applying it to several problems of interest including shocktubes,
pulses in converging-diverging nozzles, rotating cylinders, transonic buffeting, and an unsteady trailing-edge flow.
In order to address radiated far-field noise, an acoustic wave propagation program based on
the Ffowcs Williams and Hawkings (FW-H) formulation is implemented and validated. The general framework is then used
to derive the adjoint equations for a novel hybrid URANS/FW-H optimization algorithm
in order to be able to optimize the shape of airfoils based on their calculated far-field pressure fluctuations.
Validation and application results for this novel hybrid URANS/FW-H optimization algorithm show that it is possible
to optimize the shape of an airfoil in an unsteady flow environment to minimize
its radiated far-field noise while maintaining good aerodynamic performance.