Unsteady Continuous Adjoint Approach for Aerodynamic Design on Dynamic Meshes

AIAA Journal ◽  
2015 ◽  
Vol 53 (9) ◽  
pp. 2437-2453 ◽  
Author(s):  
Thomas D. Economon ◽  
Francisco Palacios ◽  
Juan J. Alonso
Keyword(s):  
Aerodynamic Design ◽  
Adjoint Approach ◽  
Continuous Adjoint

Systematic Continuous Adjoint Approach to Viscous Aerodynamic Design on Unstructured Grids

AIAA Journal ◽  
2007 ◽  
Vol 45 (9) ◽  
pp. 2125-2139 ◽  
Author(s):  
Carlos Castro ◽  
Carlos Lozano ◽  
Francisco Palacios ◽  
Enrique Zuazua
Keyword(s):  
Unstructured Grids ◽  
Aerodynamic Design ◽  
Adjoint Approach ◽  
Continuous Adjoint

S2 Surface Quasi-3D Aerodynamic Design Using the Continuous Adjoint Method for Multi-Stage Turbine

2014 ◽  
Author(s):  
Lei Chen ◽  
Jiang Chen
Keyword(s):  
Objective Function ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Design System ◽  
Shape Design ◽  
Aerodynamic Design ◽  
Aerodynamic Shape ◽  
Multi Stage ◽  
Gradient Based ◽  
Continuous Adjoint

The adjoint method eliminates the dependence of the gradient of the objective function with respect to design variables on the flow field making the obtainment of the gradient both accurate and fast. For this reason, the adjoint method has become the focus of attention in recent years. This paper develops a continuous adjoint formulation for through-flow aerodynamic shape design in a multi-stage gas turbine environment based on a S2 surface quasi-3D problem governed by the Euler equations with source terms. Given the general expression of the objective function calculated via a boundary integral, the adjoint equations and their boundary conditions are derived in detail by introducing adjoint variable vectors. As a result, the final expression of the objective function gradient only includes the terms pertinent to those physical shape variations that are calculated by metric variations. The adjoint system is solved numerically by a finite-difference method with explicit Euler time-marching scheme and a Jameson spatial scheme which employs first and third order dissipative flux. Integrating the blade stagger angles and passage perturbation parameterization with the simple steepest decent method, a gradient-based aerodynamic shape design system is constructed. Finally, the application of the adjoint method is validated through a 5-stage turbine blade and passage optimization with an objective function of entropy generation. The result demonstrates that the gradient-based system can be used for turbine aerodynamic design.

scholarly journals Derivation of the Adjoint Drift Flux Equations for Multiphase Flow

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 31 ◽  
Author(s):  
Shenan Grossberg ◽  
Daniel S. Jarman ◽  
Gavin R. Tabor
Keyword(s):  
Multiphase Flow ◽  
Objective Function ◽  
Settling Velocity ◽  
Flow Problem ◽  
Adjoint System ◽  
Drift Flux ◽  
Adjoint Approach ◽  
Continuous Adjoint ◽  
First Time ◽  
Mathematical Derivation

The continuous adjoint approach is a technique for calculating the sensitivity of a flow to changes in input parameters, most commonly changes of geometry. Here we present for the first time the mathematical derivation of the adjoint system for multiphase flow modeled by the commonly used drift flux equations, together with the adjoint boundary conditions necessary to solve a generic multiphase flow problem. The objective function is defined for such a system, and specific examples derived for commonly used settling velocity formulations such as the Takacs and Dahl models. We also discuss the use of these equations for a complete optimisation process.

scholarly journals Continuous Adjoint Approach for the Spalart-Allmaras Model in Aerodynamic Optimization

AIAA Journal ◽  
2012 ◽  
Vol 50 (3) ◽  
pp. 631-646 ◽  
Author(s):  
Alfonso Bueno-Orovio ◽  
Carlos Castro ◽  
Francisco Palacios ◽  
Enrique Zuazua
Keyword(s):  
Aerodynamic Optimization ◽  
Adjoint Approach ◽  
Continuous Adjoint

scholarly journals A complexity-driven framework for waveform tomography with discrete adjoints

2020 ◽  
Vol 223 (2) ◽  
pp. 1247-1264
Author(s):  
Alexandre Szenicer ◽  
Kuangdai Leng ◽  
Tarje Nissen-Meyer
Keyword(s):  
Adjoint Method ◽  
Waveform Inversion ◽  
Spectral Element ◽  
Frequency Scaling ◽  
Seismic Waveform ◽  
Discrete Adjoint ◽  
Continuous Adjoint Method ◽  
Adjoint Approach ◽  
Pseudo Spectral Method ◽  
Continuous Adjoint

Summary We develop a new approach for computing Fréchet sensitivity kernels in full waveform inversion by using the discrete adjoint approach in addition to the widely used continuous adjoint approach for seismic waveform inversion. This method is particularly well suited for the forward solver AxiSEM3D, a combination of the spectral-element method (SEM) and a Fourier pseudo-spectral method, which allows for a sparse azimuthal wavefield parametrization adaptive to wavefield complexity, leading to lower computational costs and better frequency scaling than conventional 3-D solvers. We implement the continuous adjoint method to serve as a benchmark, additionally allowing for simulating off-axis sources in axisymmetric or 3-D models. The kernels generated by both methods are compared to each other, and benchmarked against theoretical predictions based on linearized Born theory, providing an excellent fit to this independent reference solution. Our verification benchmarks show that the discrete adjoint method can produce exact kernels, largely identical to continuous kernels. While using the continuous adjoint method we lose the computational advantage and fall back on a full-3-D frequency scaling, using the discrete adjoint retains the speedup offered by AxiSEM3D. We also discuss the creation of a data-coverage based mesh to run the simulations on during the inversion process, which would allow to exploit the flexibility of the Fourier parametrization and thus the speedup offered by our method.

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