Diagnositcs With Adjoint Modelling
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Abstract We consider the problem of optimizing a non‐linear aeroelastic system in steady‐state conditions, where the structure is represented by a detailed finite element model, and the aerodynamic loads are predicted by the discretization of the non‐linear Euler equations. We present a solution method for this problem that is based on the three‐field formulation of fluid–structure interaction problems, and the adjoint approach for coupled sensitivity analysis. We discuss the computational complexity of the proposed solution method, describe its implementation on parallel processors, and illustrate its computational efficiency with the aeroelastic optimization of various wings. Copyright © 2002 John Wiley & Sons, Ltd.
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An approach for the aerodynamic optimization design of elastic configurations is implemented and tested. Aeroelastic analysis was carried out by combining an Euler equations solver and finite-element structural solver. Two important techniques used in high-fidelity aerostructural design optimization are discussed: the grid deformation method and finite-element mesh (FEM) update. An improved grid deformation methodology based on transfinite interpolation (TFI) and the radial basis function (RBF) method is presented. It adapts to complex configurations very well. The technique to update the FEM is based on a bilinear interpolation method and RBF method, and it adapts to any grid of finite-element model. The discrete adjoint method is used to get the gradient of the objective function with respect to design variables. Optimizations of a wing and a more realistic wing-body configuration are done to demonstrate the effectiveness of the proposed approach. Results show that the lift-to-drag ratio can be improved with constraints through optimization, which indicates that the present methodology can be successfully applied to design optimization of jig shapes of aircraft.
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This thesis presents a shape optimization framework for problems that are encountered routinely in Aerodynamic Design. The nature of the framework is numerical. Its focus is wide as different aspects of the shape optimization practice are treated, e.g., the solution of the flow equations, the sensitivity analysis and the parameterization of the shape. The framework components are not taken as black-boxes but are conceived and implemented within the present work. A considerable part of the thesis describes the characteristics and the implementation of those components. Additional work on unsteady flows, which may find applications in aeroelastic analysis, is presented in the appendix.
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An adjoint system of the Euler equations of gas dynamics is derived in order to solve aerodynamic shape optimization problems with gradient-based methods. The derivation is based on the fully discr ...
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This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a heat transport benchmark problem that simulates the radial heat conduction in a heated fuel rod together with the axial heat transport along the coolant channel surrounding the rod. This paradigm benchmark is sufficiently simple to admit an exact solution, thereby making transparent the mathematical derivations underlying the 2nd-ASAM, and also for performing solution verification of computational fluid dynamics simulation tools such as the FLUENT Adjoint Solver, a code that has been used for computing thermal-hydraulic processes within the G4M Reactor. The G4M Reactor is an innovative small modular fast reactor cooled by lead-bismuth eutectic.The benchmark problem has six thermal-hydraulic parameters representative of the G4M Reactor, as well as of a test section that is under design for analyzing thermal-hydraulic phenomena expected within the reactor's core. Thus, the benchmark exhibits 6 first-order and 21 second-order sensitivities for the temperature distribution at any location within the heated rod (and/or on its surface) and just as many sensitivities for the temperature distribution at any location within the coolant. Locations of particular importance are those where the rod temperature attains its maximum along the rod's axis or on the rod's surface, as well as the coolant's exit, where the coolant temperature attains its maximum.The general theory underlying the 2nd-ASAM indicates that for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires Nα large-scale computations involving correspondingly constructed adjoint systems, which we called second-level adjoint sensitivity systems (2nd-LASS). In practice, however, the actual number of large-scale adjoint computations can be significantly smaller; in particular, all 54 first- and second-order rod and coolant temperature sensitivities for the benchmark presented in this work were obtained using only seven independent adjoint computations. Furthermore, the construction and solution of the 2nd-LASS require very little additional effort beyond the construction of the first-level adjoint sensitivity system (1st-LASS) needed for computing the first-order sensitivities. Very significantly, only the sources on the right side of the differential operator needed to be modified; the left side of the respective differential equations (and hence the solver in large-scale practical applications) remained unchanged from the 1st-LASS and from the original equations when these are linear in the unknown state function.This work also shows that the second-order sensitivities have the following impacts on the computed moments of the response distribution: (a) they cause the expected value of the response to differ from the computed nominal value of the response, (b) they contribute to the response variances and covariances, and (c) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations stemming from normally distributed parameters and hence would nullify the skewness of the response; consequently, events occurring in a response's long and/or short tails could be missed.
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