Academic literature on the topic 'Adjoint discret'
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Journal articles on the topic "Adjoint discret"
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Zhao, Shunliu, Matthew G. Russell, Amir Hakami, Shannon L. Capps, Matthew D. Turner, Daven K. Henze, Peter B. Percell, et al. "A multiphase CMAQ version 5.0 adjoint." Geoscientific Model Development 13, no. 7 (July 2, 2020): 2925–44. http://dx.doi.org/10.5194/gmd-13-2925-2020.
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Abstract. We present the development of a multiphase adjoint for the Community Multiscale Air Quality (CMAQ) model, a widely used chemical transport model. The adjoint model provides location- and time-specific gradients that can be used in various applications such as backward sensitivity analysis, source attribution, optimal pollution control, data assimilation, and inverse modeling. The science processes of the CMAQ model include gas-phase chemistry, aerosol dynamics and thermodynamics, cloud chemistry and dynamics, diffusion, and advection. Discrete adjoints are implemented for all the science processes, with an additional continuous adjoint for advection. The development of discrete adjoints is assisted with algorithmic differentiation (AD) tools. Particularly, the Kinetic PreProcessor (KPP) is implemented for gas-phase and aqueous chemistry, and two different automatic differentiation tools are used for other processes such as clouds, aerosols, diffusion, and advection. The continuous adjoint of advection is developed manually. For adjoint validation, the brute-force or finite-difference method (FDM) is implemented process by process with box- or column-model simulations. Due to the inherent limitations of the FDM caused by numerical round-off errors, the complex variable method (CVM) is adopted where necessary. The adjoint model often shows better agreement with the CVM than with the FDM. The adjoints of all science processes compare favorably with the FDM and CVM. In an example application of the full multiphase adjoint model, we provide the first estimates of how emissions of particulate matter (PM2.5) affect public health across the US.
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Ni, Angxiu. "Backpropagation in hyperbolic chaos via adjoint shadowing." Nonlinearity 37, no. 3 (January 30, 2024): 035009. http://dx.doi.org/10.1088/1361-6544/ad1aed.
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Abstract To generalise the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator S acting on covector fields. We show that S can be equivalently defined as: S is the adjoint of the linear shadowing operator S; S is given by a ‘split then propagate’ expansion formula; S ( ω ) is the only bounded inhomogeneous adjoint solution of ω. By (a), S adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), S also expresses the other part of the linear response, the unstable contribution. By (c), S can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395 690–709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions.
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Capps, S. L., D. K. Henze, A. Hakami, A. G. Russell, and A. Nenes. "ANISORROPIA: the adjoint of the aerosol thermodynamic model ISORROPIA." Atmospheric Chemistry and Physics Discussions 11, no. 8 (August 19, 2011): 23469–511. http://dx.doi.org/10.5194/acpd-11-23469-2011.
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Abstract. We present the development of ANISORROPIA, the discrete adjoint of the ISORROPIA thermodynamic equilibrium model that treats the Na+-SO42−-HSO4−-NH4+-NO3−-Cl−-H2O aerosol system, and we demonstrate its sensitivity analysis capabilities. ANISORROPIA calculates sensitivities of an inorganic species in aerosol or gas phase with respect to the total concentrations of each species present with only a two-fold increase in computational time over the forward model execution. Due to the highly nonlinear and discontinuous solution surface of ISORROPIA, evaluation of the adjoint required a new, complex-variable version of the the model, which determines first-order sensitivities with machine precision and avoids cancellation errors arising from finite difference calculations. The adjoint is verified over an atmospherically relevant range of concentrations, temperature, and relative humidity. We apply ANISORROPIA to recent field campaign results from Atlanta, GA, USA, and Mexico City, Mexico, to characterize the inorganic aerosol sensitivities of these distinct urban air masses. The variability in the relationship between PM2.5 mass and precursor concentrations shown has important implications for air quality and climate. ANISORROPIA enables efficient elucidation of aerosol concentration dependence on aerosol precursor emissions in the context of atmospheric chemical transport model adjoints.
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Hekmat, Mohamad Hamed, and Masoud Mirzaei. "Development of Discrete Adjoint Approach Based on the Lattice Boltzmann Method." Advances in Mechanical Engineering 6 (January 1, 2014): 230854. http://dx.doi.org/10.1155/2014/230854.
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The purpose of this research is to present a general procedure with low implementation cost to develop the discrete adjoint approach for solving optimization problems based on the LB method. Initially, the macroscopic and microscopic discrete adjoint equations and the cost function gradient vector are derived mathematically, in detail, using the discrete LB equation. Meanwhile, for an elementary case, the analytical evaluation of the macroscopic and microscopic adjoint variables and the cost function gradients are presented. The investigation of the derivation procedure shows that the simplicity of the Boltzmann equation, as an alternative for the Navier-Stokes (NS) equations, can facilitate the process of extracting the discrete adjoint equation. Therefore, the implementation of the discrete adjoint equation based on the LB method needs fewer attempts than that of the NS equations. Finally, this approach is validated for the sample test case, and the results gained from the macroscopic and microscopic discrete adjoint equations are compared in an inverse optimization problem. The results show that the convergence rate of the optimization algorithm using both equations is identical and the evaluated gradients have a very good agreement with each other.
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Larour, Eric, Jean Utke, Anton Bovin, Mathieu Morlighem, and Gilberto Perez. "An approach to computing discrete adjoints for MPI-parallelized models applied to Ice Sheet System Model 4.11." Geoscientific Model Development 9, no. 11 (November 1, 2016): 3907–18. http://dx.doi.org/10.5194/gmd-9-3907-2016.
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Abstract. Within the framework of sea-level rise projections, there is a strong need for hindcast validation of the evolution of polar ice sheets in a way that tightly matches observational records (from radar, gravity, and altimetry observations mainly). However, the computational requirements for making hindcast reconstructions possible are severe and rely mainly on the evaluation of the adjoint state of transient ice-flow models. Here, we look at the computation of adjoints in the context of the NASA/JPL/UCI Ice Sheet System Model (ISSM), written in C++ and designed for parallel execution with MPI. We present the adaptations required in the way the software is designed and written, but also generic adaptations in the tools facilitating the adjoint computations. We concentrate on the use of operator overloading coupled with the AdjoinableMPI library to achieve the adjoint computation of the ISSM. We present a comprehensive approach to (1) carry out type changing through the ISSM, hence facilitating operator overloading, (2) bind to external solvers such as MUMPS and GSL-LU, and (3) handle MPI-based parallelism to scale the capability. We demonstrate the success of the approach by computing sensitivities of hindcast metrics such as the misfit to observed records of surface altimetry on the northeastern Greenland Ice Stream, or the misfit to observed records of surface velocities on Upernavik Glacier, central West Greenland. We also provide metrics for the scalability of the approach, and the expected performance. This approach has the potential to enable a new generation of hindcast-validated projections that make full use of the wealth of datasets currently being collected, or already collected, in Greenland and Antarctica.
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Wu, Hangkong, Xuanlong Da, Dingxi Wang, and Xiuquan Huang. "Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver." Aerospace 10, no. 2 (January 21, 2023): 106. http://dx.doi.org/10.3390/aerospace10020106.
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This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated codes are manually adjusted to reduce memory and CPU time consumption. This manual adjustment is performed by replacing the automatically generated low-efficient differentiated codes with manually developed ones. To demonstrate the effectiveness of the proposed approach, the single-stage transonic compressor–NASA Stage 35 and the 1.5-stage Aachen turbine–are used. The solution information exchange at a rotor-stator/stator-rotor interface is achieved by a conservative, non-reflective, and robust discrete adjoint mixing plane method. The results show that the discrete adjoint solver developed by hybrid automatic and manual differentiation is more economical in computational cost than that developed purely by an AD tool and has higher sensitivity accuracy than the adjoint solver with the constant eddy viscosity (CEV) assumption. Moreover, the multi-row turbomachinery design optimizations can be efficiently performed by the discrete adjoint solver developed by the hybrid automatic and manual differentiation.
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Towara, Markus, Michel Schanen, and Uwe Naumann. "MPI-Parallel Discrete Adjoint OpenFOAM." Procedia Computer Science 51 (2015): 19–28. http://dx.doi.org/10.1016/j.procs.2015.05.181.
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Niwa, Yosuke, Hirofumi Tomita, Masaki Satoh, Ryoichi Imasu, Yousuke Sawa, Kazuhiro Tsuboi, Hidekazu Matsueda, et al. "A 4D-Var inversion system based on the icosahedral grid model (NICAM-TM 4D-Var v1.0) – Part 1: Offline forward and adjoint transport models." Geoscientific Model Development 10, no. 3 (March 17, 2017): 1157–74. http://dx.doi.org/10.5194/gmd-10-1157-2017.
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Abstract. A four-dimensional variational (4D-Var) method is a popular algorithm for inverting atmospheric greenhouse gas (GHG) measurements. In order to meet the computationally intense 4D-Var iterative calculation, offline forward and adjoint transport models are developed based on the Nonhydrostatic ICosahedral Atmospheric Model (NICAM). By introducing flexibility into the temporal resolution of the input meteorological data, the forward model developed in this study is not only computationally efficient, it is also found to nearly match the transport performance of the online model. In a transport simulation of atmospheric carbon dioxide (CO2), the data-thinning error (error resulting from reduction in the time resolution of the meteorological data used to drive the offline transport model) is minimized by employing high temporal resolution data of the vertical diffusion coefficient; with a low 6-hourly temporal resolution, significant concentration biases near the surface are introduced. The new adjoint model can be run in discrete or continuous adjoint mode for the advection process. The discrete adjoint is characterized by perfect adjoint relationship with the forward model that switches off the flux limiter, while the continuous adjoint is characterized by an imperfect but reasonable adjoint relationship with its corresponding forward model. In the latter case, both the forward and adjoint models use the flux limiter to ensure the monotonicity of tracer concentrations and sensitivities. Trajectory analysis for high CO2 concentration events are performed to test adjoint sensitivities. We also demonstrate the potential usefulness of our adjoint model for diagnosing tracer transport. Both the offline forward and adjoint models have computational efficiency about 10 times higher than the online model. A description of our new 4D-Var system that includes an optimization method, along with its application in an atmospheric CO2 inversion and the effects of using either the discrete or continuous adjoint method, is presented in an accompanying paper Niwa et al.(2016).
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Agarwal, Ravi P., Safi S. Rabie, and Samir H. Saker. "On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete ℓp-Classes." Fractal and Fractional 7, no. 3 (March 14, 2023): 261. http://dx.doi.org/10.3390/fractalfract7030261.
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In this paper, we study some relations between different weights in the classes Bp,Bp*,Mp and Mp* that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space Λp. These results will be proven by using the properties of classes Bp,Bp*,Mp and Mp*, including the self-improving properties and also the properties of the generalized Hardy operator Hp, the adjoint operator Sq and some fundamental relations between them connecting their composition to their sum.
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Cao, Junying, Zhongqing Wang, and Ziqiang Wang. "A Uniform Accuracy High-Order Finite Difference and FEM for Optimal Problem Governed by Time-Fractional Diffusion Equation." Fractal and Fractional 6, no. 9 (August 28, 2022): 475. http://dx.doi.org/10.3390/fractalfract6090475.
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In this paper, the time fractional diffusion equations optimal control problem is solved by 3−α order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are used to make the space variables discrete, and obtain the semidiscrete scheme of the state and adjoint state. The priori error estimates for the semidiscrete scheme for state and adjoint state equation are established. Furthermore, the 3−α order uniform accuracy scheme is used to make the time variable discrete in the semidiscrete scheme and construct the full discrete scheme for the control problems based on the first optimal condition and ‘first optimize, then discretize’ approach. The fully discrete scheme’s stability and truncation error are analyzed. Finally, two numerical examples are denoted to show that the theoretical analysis are correct.
Dissertations / Theses on the topic "Adjoint discret"
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Dittmann, Florian. "Study and Optimisation of Supersonic Ejectors for Heat Recovery Refrigeration Cycles." Electronic Thesis or Diss., Université Paris sciences et lettres, 2024. http://www.theses.fr/2024UPSLM029.
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Des éjecteurs supersoniques pour des cycles de réfrigération à récupération de chaleur sont modélisés, étudiés et optimisés à l’aide de la mécanique des fluides numérique et la méthode de l’état adjoint discret. L’étude est étayée par une analyse des rapports entre les phénomènes complexes de l’écoulement, les limites thermodynamiques et la performance des cycles. Un modèle 1D généralisé est développé et utilisé pour concevoir des éjecteurs et prédire leur taux d’entraînement afin de déterminer les conditions optimales d’un cycle. La résolution des équations Navier-Stokes moyennées 2D-axisymétrique complémenté par le modèle de turbulence k-ω SST et une équation d’état cubique pour le réfrigérant R134a permet l’analyse de l’écoulement et l’optimisation de forme. La dernière repose sur la méthode de l’état adjoint discret qui évalue efficacement le gradient de la fonction objectif par rapport à un nombre arbitraire de variables de conception. Il est démontré que la méthode, appliquée pour la première fois à un écoulement transsonique d’un réfrigérant dans un éjecteur, peut générer une forme d’éjecteur performante à partir d’une conception défaillante, malgré la discontinuité apparente de la fonction objectif au point critique. Les efficacités prédites avec les formes optimisées dépassent d’environ 15% ceux des meilleurs éjecteurs sur le marchéSupersonic ejectors for heat recovery refrigeration cycles are modelled, studied and optimised based on numerical fluid mechanics and the discrete adjoint method. The study is supported by an analysis of the relations between the complex flow phenomena, the thermodynamic limits and the cycle performance. A generalised 1D model is developed and used to conceive ejectors and predict their entrainment ratio in order to determine the optimal cycle conditions. The resolution of the Reynolds averaged Navier-Stokes equations complemented by the k-ω SST turbulence model and a cubic equation of state for the refrigerant R134a enables the flow analysis and shape optimisation. The latter relies on the discrete adjoint method to efficiently evaluate the gradient of the objective function with respect to an arbitrary number of design variables. It is shown that the method, applied here for the first time to a transonic flow of a refrigerant in an ejector, is capable of generating a well performing ejector shape from a failed design, despite the apparent discontinuity of the objective function at the critical point. The predicted efficiencies with the optimised shapes exceed those of the best ejectors on the market by around 15%
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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.
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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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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://www.theses.fr/2008ENAM0039.
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L’objet de ce travail a consisté en l’étude et la mise en oeuvre d. Une méthodede calcul des dérivées des fonctions d’intérêt d’un problème d’optimisation de forme par rapport aux paramètres géométriques décrivant la forme solide, appelées de façon générique gradients. . , pour un système aéroélastique soumis à un écoulement lointain stationnaire. Une méthodologie de calcul de l’équilibre aéroélastique statique a tout d. Abord été développée dans le cadre de laquelle le comportement du fluide peut être modélisé par les équations d’Euler ou de Navier-Stokes moyennées (RANS), résolues numériquement par elsA, code de simulation numérique pour la mécanique des fluides développé à l’ONERA, et le comportement de la structure est prédit par la théorie des poutres (équations d’Euler-Bernoulli écrites en formulation matrice de lexibilité). Un cadre de calcul des gradients relatifs à ce système a ensuite pu être mis en place. Ceux-ci sont calculés de manière analytique en utilisant, soit la méthode de l’équation linéarisée discrète, soit la méthode du vecteur adjoint discret. Ces méthodes impliquent la résolution de systèmes linéaires couplés effectuée, dans le cadre de cette étude, par un processus itératif doublement retardé. Enfin, 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 de complexité croissante, et validés par comparaison aux valeurs des gradients calculés par différences finies. Une dernière partie de ce 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 turbomachineThis work is mainly dedicated to the sensitivity analysis of a static aeroelastic system with respect to design parameters governing its jig-shape. First, a framework able to predict the static aeroelastic equilibrium has been set up. The fluid behavior can be governed either by the nonlinear Euler equations or by the Navier-Stokes Reynolds averaged (RANS) equations. They are numerically solved by an ONERA CFD solver: elsA. The structural behavior is governed by the Euler-Bernoulli equations within the context of beam theory. The aerodynamic loads are transferred to the structure using the matrix of the influence coefficients, also called the flexibility matrix. Only the bending and the twisting aerodynamic load components are consistently transmitted to the structure, and only the bending and the torsional displacements of the structure are calculated under the small displacement hypothesis. The deformation induced on the fluid domain mesh is analytically prescribed using an analogy to solid mechanics. Finally, the resulting coupled aeroelastic system of equations is solved by an iterative process inspired from the fixed-point algorithm. Second, a framework aiming at computing the gradients of the functions of interest (objective and constraints) with respect to a vector of shape parameters related to the jig-shape of the aeroelastic system previously depicted, has been raised. These gradients can be computed either by the discrete direct differentiation method or by the discrete adjoint vector method. In both cases, a coupled linear system of equations has to be solved, which is carried out using a doubly lagged iterative process. Finally, this framework has been applied to the computation of the gradients of the drag and lift aerodynamic coefficients with respect to different shape parameters for three aerodynamic configurations of growing complexity: Euler equations solved on a multiblock mesh with matching boundaries, RANS equations on a monoblock mesh, and, at last, RANS equations solved on a multiblock mesh with non-matching boundaries. The analytical gradients have been validated through the comparison with the finite difference gradients. A last part of this work has been dedicated to the evaluation of the performances of four surrogate models within the shape optimization of a bidimensional turbomachinery configuration
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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/.
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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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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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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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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.
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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.
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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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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.
Full textBooks on the topic "Adjoint discret"
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Edmunds, D. E., and W. D. Evans. Capacity and Compactness Criteria. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198812050.003.0008.
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In this chapter, necessary and sufficient conditions are derived for the Poincaré inequality to hold, for the embedding of W01,p(Ω) in Lp(Ω) to be compact, and for a self-adjoint realization of − aijDiDj + q to have a wholly discrete spectrum when q is real and bounded below. The results are proved using a method of Maz’ya.
Book chapters on the topic "Adjoint discret"
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Wong, M. W. "Self-Adjoint Operators." In Discrete Fourier Analysis, 113–16. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0116-4_16.
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Lotz, Johannes, Uwe Naumann, Max Sagebaum, and Michel Schanen. "Discrete Adjoints of PETSc through dco/c++ and Adjoint MPI." In Euro-Par 2013 Parallel Processing, 497–507. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-40047-6_51.
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Giles, M. B. "Discrete Adjoint Approximations with Shocks." In Hyperbolic Problems: Theory, Numerics, Applications, 185–94. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/978-3-642-55711-8_16.
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Catlin, Donald E. "Adjoints, Projections, Pseudoinverses." In Estimation, Control, and the Discrete Kalman Filter, 92–113. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-4528-5_4.
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Schmüdgen, Konrad. "Discrete Spectra of Self-adjoint Operators." In Graduate Texts in Mathematics, 265–80. Dordrecht: Springer Netherlands, 2012. http://dx.doi.org/10.1007/978-94-007-4753-1_12.
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Fichtner, Andreas. "The Frequency-Domain Discrete Adjoint Method." In Full Seismic Waveform Modelling and Inversion, 189–92. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-15807-0_10.
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Shu, Hanlin, Liangzhi Cao, Qingming He, Tao Dai, Zhangpeng Huang, and Hongchun Wu. "Study on Unstructured-Mesh-Based Importance Sampling Method of Monte Carlo Simulation." In Springer Proceedings in Physics, 431–44. Singapore: Springer Nature Singapore, 2023. http://dx.doi.org/10.1007/978-981-99-1023-6_38.
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AbstractMonte Carlo (MC) method is widely adopted in radiation transport calculation due to its high accuracy, but suffers from high variance in deep-penetration problems. To obtain reasonable results, variance reduction techniques are necessary and thus be widely studied worldwide. The Consistent Adjoint Driven Importance Sampling (CADIS) method is proved to be an effective variance reduction technique, which generally employs finite-difference discrete ordinate (SN) code to obtain the adjoint flux, and generates parameters of source biasing and weight window for MC code. However, the finite-difference method, which models through structural meshes, will introduce considerable geometric approximations in complex geometry. The finite element method (FEM) performs calculations with lower truncation error and can employ unstructured meshes, which are capable of modeling complex geometry with relatively lower geometric approximations. Therefore, the adjoint flux calculated by unstructured-mesh FEM is able to generate more appropriate parameters of source biasing and weight window which will further reduce the variance of forward MC calculation. A fully automatic unstructured-mesh CADIS method is studied and implemented in this paper, parallel three-dimensional unstructured-mesh neutron-photon coupled transport calculation code NECP-SUN based on the SN method and discontinuous FEM is developed and embedded into the MC code NECP-MCX to calculate the adjoint flux with unstructured meshes. The updated code is applied to the HBR-2 benchmark, the numerical results show that the relative statistic error is reduced by up to 22% compared to the traditional CADIS method while the calculation results are closer to the measurements and the figure of merit (FOM) is increased by 3–4 orders comparing to direct MC simulation.
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Anil, N., N. K. S. Rajan, Omesh Reshi, and S. M. Deshpande. "A Low Dissipative Discrete Adjoint m-KFVS Method." In Computational Fluid Dynamics 2008, 619–24. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01273-0_82.
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Towara, Markus, Johannes Lotz, and Uwe Naumann. "Discrete Adjoint Approaches for CHT Applications in OpenFOAM." In Computational Methods in Applied Sciences, 163–78. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-57422-2_11.
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Brezillon, Joël, and Mohammad Abu-Zurayk. "Aerodynamic Inverse Design Framework Using Discrete Adjoint Method." In Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 489–96. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-35680-3_58.
Full textConference papers on the topic "Adjoint discret"
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Biava, Massimo, Mark Woodgate, and George N. Barakos. "Fully Implicit Discrete Adjoint Methods." In 53rd AIAA Aerospace Sciences Meeting. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2015. http://dx.doi.org/10.2514/6.2015-1491.
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Frey, Christian, Hans-Peter Kersken, and Dirk Nu¨rnberger. "The Discrete Adjoint of a Turbomachinery RANS Solver." In ASME Turbo Expo 2009: Power for Land, Sea, and Air. ASMEDC, 2009. http://dx.doi.org/10.1115/gt2009-59062.
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Since adjoint flow solvers allow for the computation of sensitivities of global flow parameters under geometric variations in an amount of time which is nearly independent of the number of geometric parameters, automatic shape optimization can be accelerated considerably by the use of an adjoint solver. In this article, a systematic approach for the development of an exact discrete adjoint of a turbomachinery flow solver is described. By using finite differences to differentiate the numerical fluxes, the problems associated with automatic and hand differentiation are circumvented. Moreover, a general treatment of the adjoint numerical boundary conditions is presented. As a result, an exact adjoint boundary condition for the conservative mixing planes is obtained. In combination with nonreflecting boundary conditions the latter are crucial for accurate flow simulations in turbomachinery. The adjoint is validated on the basis of a transonic compressor stage.
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Schäfer, Fellcitas, Luca Magri, and Wolfgang Polifke. "A Hybrid Adjoint Network Model for Thermoacoustic Optimization." In ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/gt2021-59866.
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Abstract A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.
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Martins, Joaquim R. R. A., Charles Mader, and Juan Alonso. "ADjoint: An Approach for Rapid Development of Discrete Adjoint Solvers." In 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2006. http://dx.doi.org/10.2514/6.2006-7121.
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Zhang, Chaolei, and Zhenping Feng. "Aerodynamic Shape Design Optimization for Turbomachinery Cascade Based on Discrete Adjoint Method." In ASME 2011 Turbo Expo: Turbine Technical Conference and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/gt2011-45805.
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Achieving higher aerodynamic performance in terms of efficiency, pressure ratio or stable operation range has been of interest to both researchers and engineers in the field of turbomachinery. The design of optimal shaped aerodynamic configurations based on Computational Fluid Dynamics (CFD) and predefined targets can be obtained by using deterministic search algorithms, which need to calculate the first and second order sensitivities of the objective function with respect to the design variables. With the characteristics of quick and exact sensitivity analysis, as well as less computational resource requirement, the adjoint method has become a research focus in aerodynamic shape design optimization over the past decades. In this paper, a discrete adjoint solver was developed and validated based on an in-house flow solver code. Moreover, a turbomachinery cascade optimization design system was established by coupling the flow solver, the discrete adjoint solver, the parameterization technology, the grid generation technology and the gradient-based optimization algorithms. During the development process of the discrete adjoint solver, the automatic differentiation tool was used in order to ease the construction of the discrete adjoint system based on the flow solver code. However, in order to save the memory requirement and to reduce the computational cost, the automatic differentiation tool was used selectively to build the fundamental subroutines. The top-most module of the discrete adjoint solver was established based on the discrete adjoint theory and the automatic differentiation technology manually. The treatments of the discontinuity in the flow field, such as strong shocks, and the imposition of strong boundary conditions which were implemented in the adjoint solver were discussed in detail. At the same time, several technologies were used to accelerate convergence. Based on the optimization system, a typical 2D transonic turbomachinery cascade was optimized under the viscous flow environment. The optimization results were analyzed in detail. The validity and efficiency of the present optimization design system were proved.
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Ma, Can, Xinrong Su, and Xin Yuan. "Discrete Adjoint Solution of Unsteady Turbulent Flow in Compressor." In ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/gt2015-42948.
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Unsteady blade row interactions play an important role in the performance of the compressor stages. However, due to the large cost of the unsteady flow simulation, most aerodynamic optimizations of the compressor are based on the steady flow simulation. This paper adopts the time spectral method to reduce the cost of the unsteady flow simulation and a discrete adjoint solver based on the unsteady flow solver has been developed. The unsteady flow equations and the adjoint equations are solved using an in-house code. The in-house code is based on the finite volume method and solves the URANS (Unsteady Reynolds Averaged Navier-Stokes) equations on the multi-block structured mesh. For spatial discretization the 3rd order WENO (Weighted Essentially Nonoscillatory) upwind scheme is used for reconstruction and the convective flux is computed with Roe’s approximate Riemann solver. The widely used one-equation Spalart-Allmaras turbulence model is adopted for the flow simulation. For the adjoint solution, the constant-eddy viscosity assumption is adopted and only the main flow adjoint equations are solved. The adjoint equations are formed in a discrete manner, which leads to more accurate discrete objective function sensitivity than the continuous adjoint method. The present work serves as an essential part of the system for efficient unsteady aerodynamic optimization of turbomachinery.
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Walther, Benjamin, and Siva Nadarajah. "An Adjoint-Based Optimization Method for Constrained Aerodynamic Shape Design of Three-Dimensional Blades in Multi-Row Turbomachinery Configurations." In ASME Turbo Expo 2014: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/gt2014-26604.
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This paper develops the discrete adjoint equations for a turbomachinery RANS solver and proposes a framework for fully-automatic gradient-based constrained aerodynamic shape optimization in a multistage turbomachinery environment. The systematic approach for the development of the discrete adjoint solver is discussed. Special emphasis is put on the development of the turbomachinery specific features of the adjoint solver, i.e. on the derivation of flow-consistent adjoint inlet/outlet boundary conditions and, to allow for a concurrent rotor/stator optimization and stage coupling, on the development of an exact adjoint counterpart to the non-reflective, conservative mixing-plane formulation used in the flow solver. The adjoint solver is validated by comparing its sensitivities with finite difference gradients obtained from the flow solver. A sequential quadratic programming algorithm is utilized to determine an improved blade shape based on the objective function gradient provided by the adjoint solution. The functionality of the proposed optimization method is demonstrated by the redesign of a single-stage transonic compressor. The objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.
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Wu, Hangkong, Shenren Xu, Xiuquan Huang, and Dingxi Wang. "The Development and Verification of a Fully Turbulent Discrete Adjoint Solver Using Algorithmic Differentiation." In ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/gt2021-59610.
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Abstract This paper presents the development and verification of a discrete adjoint solver using algorithmic differentiation (AD). The computational cost of sensitivity evaluation using the adjoint method is largely independent of the number of design variables, making it attractive for optimization applications where the design variables are far more than objectives and constraints. To obtain the gradients of a single objective function or constraint with respect to many design variables, the nonlinear flow and the adjoint equations need to be solved once at every design cycle. This paper makes a detailed presentation of how AD is used to develop a discrete adjoint solver. The data flow diagrams of the nonlinear flow, linear and adjoint solvers are compared. Moreover, a comparison of convergence history of sensitivity, asymptotic rate of residual convergence and computational cost between the linear and adjoint solvers is also made. Two cases — the subsonic Durham turbine and transonic NASA Rotor 67 are studied in this paper. The results show that the adjoint solver has the same asymptotic rate of residual convergence and produces consistent convergence history of sensitivity as the linear solver, but the adjoint solver consumes more time and memory.
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Lu, Juan, Chaolei Zhang, and Zhenping Feng. "Aerodynamic Optimization and Inverse Design of 2D and 3D Turbine Cascades Using the Discrete Adjoint Method." In ASME Turbo Expo 2013: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/gt2013-95284.
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The adjoint method has significant advantage in sensitivity analysis because its computation cost is independent of the number of the design variables. In recent years it has been applied greatly in aerodynamic design optimization of turbomachinery. This paper developed the discrete adjoint method based on the authors’ previous work and demonstrated the applications of the method in the aerodynamic design optimization for turbine cascades. The Non-uniform Rational B-Spline (NURBS) technology was introduced in the current design optimization system and a flexible parameterization method for 3D cascade was proposed. Based on the parameterization method, the stack line and the blade profile are parameterized together by using NURBS curves. During the design process, the control points of the profile, the stack point and the stagger angle of the blade on each section can be taken as the design variables. Moreover, the flow solver and the discrete adjoint solver were extended towards the turbulent flow environment by adopting the k – ω turbulence model. Based on the optimization design system, several applications including two optimization design cases and two inverse design cases for 2D and 3D turbine cascades were implemented with the mass flow ratio constraint. The gradient verification and the numerical cases showed the correctness and accuracy of the discrete adjoint solver. The numerical results demonstrated the validity and efficiency of the design optimization system based on discrete adjoint method.
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Chung, June, Jeonghwan Shim, and Ki D. Lee. "Inverse Design of 3D Compressor Blades With Adjoint Method." In ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/fedsm2003-45419.
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A three-dimensional (3D) CFD-based design method for high-speed axial compressor blades is being developed based on the discrete adjoint method. An adjoint code is built corresponding to RVC3D, a 3D turbomachinery Navier-Stokes analysis code developed at NASA Glenn. A validation study with the Euler equations indicates that the adjoint sensitivities are sensitive to the choice of boundary conditions for the adjoint variables in internal flow problems and constraints may be needed on internal boundaries to capture proper physics of the adjoint system. The design method is demonstrated with inverse design based on Euler physics, and the results indicate that the adjoint design method produces efficient 3D designs by drastically reducing the computational cost.
Reports on the topic "Adjoint discret"
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Slater, C. O. DRC2: A code with specialized applications for coupling localized Monte Carlo adjoint calculations with fluences from two-dimensional R-Z discrete ordinates air-over-ground calculations. Office of Scientific and Technical Information (OSTI), January 1992. http://dx.doi.org/10.2172/5973682.
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Slater, C. O. DRC2: A code with specialized applications for coupling localized Monte Carlo adjoint calculations with fluences from two-dimensional R-Z discrete ordinates air-over-ground calculations. Office of Scientific and Technical Information (OSTI), January 1992. http://dx.doi.org/10.2172/10110196.
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