Optimum Shape Design for Multirow Turbomachinery Configurations Using a Discrete Adjoint Approach and an Efficient Radial Basis Function Deformation Scheme for Complex Multiblock Grids

2015 ◽  
Vol 137 (8) ◽  
Author(s):  
Benjamin Walther ◽  
Siva Nadarajah
Keyword(s):  
Pressure Ratio ◽  
Optimization Method ◽  
Adjoint Equations ◽  
Navier Stokes ◽  
Optimum Shape ◽  
Perturbation Scheme ◽  
Discrete Adjoint ◽  
Radial Basis ◽  
Gradient Calculation ◽  
Adjoint Approach

This paper proposes a framework for fully automatic gradient-based constrained aerodynamic shape optimization in a multirow turbomachinery environment. The concept of adjoint-based gradient calculation is discussed and the development of the discrete adjoint equations for a turbomachinery Reynolds-averaged Navier–Stokes (RANS) solver, particularly the derivation of flow-consistent adjoint boundary conditions as well as the implementation of a discrete adjoint mixing-plane formulation, are described in detail. A parallelized, automatic grid perturbation scheme utilizing radial basis functions (RBFs), which is accurate and robust as well as able to handle highly resolved complex multiblock turbomachinery grid configurations, is developed and employed to calculate the gradient from the adjoint solution. The adjoint solver is validated by comparing its sensitivities with finite-difference gradients obtained from the flow solver. A sequential quadratic programming (SQP) algorithm is then utilized to determine an improved blade shape based on the gradient information from the objective functional and the constraints. The developed optimization method is used to redesign a single-stage transonic flow compressor in both inviscid and viscous flow. The design objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.

Endwall Contour Optimization of a Turning Mid Turbine Frame Using an Adjoint Method

2014 ◽  
Author(s):  
M. Bugra Akin ◽  
Wolfgang Sanz ◽  
Paul Pieringer
Keyword(s):  
Adjoint Method ◽  
Pressure Ratio ◽  
Computation Time ◽  
Optimization Method ◽  
Secondary Flows ◽  
Adjoint Equations ◽  
Navier Stokes ◽  
Optimized Design ◽  
Flow Equations ◽  
Gradient Based

This paper presents the application of a viscous adjoint method in the optimization of the endwall contour of a turning mid turbine frame (TMTF). The adjoint method is a gradient based optimization method that allows for the computation of the complete gradient information by solving the governing flow equations and their corresponding adjoint equations only once per function of interest (objective and constraints), so that the computation time of the optimization is nearly independent of the number of parameters. With the use of a greater number of parameters a more detailed definition of endwall contours is possible, so that an optimum can be approached more precisely. A Navier-Stokes flow solver coupled with Menter’s SST k–ω turbulence model is utilized for the CFD simulations, whereas the adjoint formulation is based on the constant eddy viscosity approximation for turbulence. The total pressure ratio is used as the objective function in the optimization. The effect of contouring on the secondary flows is evaluated and the performance of the axisymmetric TMTF is calculated and compared with the optimized design.

An Adjoint-Based Optimization Method for Constrained Aerodynamic Shape Design of Three-Dimensional Blades in Multi-Row Turbomachinery Configurations

2014 ◽  
Author(s):  
Benjamin Walther ◽  
Siva Nadarajah
Keyword(s):  
Pressure Ratio ◽  
Three Dimensional ◽  
Optimization Method ◽  
Adjoint Equations ◽  
Programming Algorithm ◽  
Flow Solver ◽  
Aerodynamic Shape ◽  
Transonic Compressor ◽  
Discrete Adjoint ◽  
Gradient Based

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.

Constrained Adjoint-Based Aerodynamic Shape Optimization of a Single-Stage Transonic Compressor

2012 ◽  
Author(s):  
Benjamin Walther ◽  
Siva Nadarajah
Keyword(s):  
Shape Optimization ◽  
Pressure Ratio ◽  
Internal Flow ◽  
Optimization Method ◽  
Single Stage ◽  
Programming Algorithm ◽  
Aerodynamic Shape Optimization ◽  
Aerodynamic Shape ◽  
Transonic Compressor ◽  
Adjoint Approach

This paper develops a discrete adjoint formulation for the constrained aerodynamic shape optimization in a multistage turbomachinery environment. The adjoint approach for viscous, internal flow problems and the corresponding adjoint boundary conditions are discussed. To allow for a concurrent rotor/stator optimization a non-reflective adjoint mixing-plane formulation is proposed. A sequential-quadratic programming algorithm is utilized to determine an improved airfoil 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 midspan section 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.

Optimization of Endwall Contours of a Turbine Blade Row Using an Adjoint Method

2011 ◽  
Author(s):  
Jiaqi Luo ◽  
Ivan McBean ◽  
Feng Liu
Keyword(s):  
Turbine Blade ◽  
Adjoint Method ◽  
Optimization Method ◽  
Adjoint Equations ◽  
Navier Stokes ◽  
Design Parameters ◽  
Flow Equations ◽  
Blade Row ◽  
Flow Loss ◽  
Design Case

This paper presents the application of a viscous adjoint method in the optimization of a low-aspect-ratio turbine blade through spanwise restaggering and endwall contouring. A generalized wall-function method is implemented in a Navier-Stokes flow solver coupled with Menter’s SST k-ω turbulence model to simulate secondary flow with reduced requirements on grid density. Entropy production through the blade row combined with a flow turning constraint is used as the objective function in the optimization. With the viscous adjoint method, the complete gradient information needed for optimization can be obtained by solving the governing flow equations and their corresponding adjoint equations only once, regardless of the number of design parameters. The endwall profiles are contoured alone in the first design case, while it is combined with spanwise restaggering in the second design case. The results demonstrate that it is feasible to reduce flow loss through the blade redesign while maintaining the same mass-averaged flow turning by using the viscous adjoint optimization method. The performance of the redesigned blade is calculated and compared at off-design conditions.

Constrained Adjoint-Based Aerodynamic Shape Optimization of a Single-Stage Transonic Compressor

2012 ◽  
Vol 135 (2) ◽  
Author(s):  
Benjamin Walther ◽  
Siva Nadarajah
Keyword(s):  
Shape Optimization ◽  
Pressure Ratio ◽  
Internal Flow ◽  
Optimization Method ◽  
Single Stage ◽  
Programming Algorithm ◽  
Aerodynamic Shape Optimization ◽  
Aerodynamic Shape ◽  
Transonic Compressor ◽  
Adjoint Approach

This paper develops a discrete adjoint formulation for the constrained aerodynamic shape optimization in a multistage turbomachinery environment. The adjoint approach for viscous internal flow problems and the corresponding adjoint boundary conditions are discussed. To allow for a concurrent rotor/stator optimization, a nonreflective adjoint mixing-plane formulation is proposed. A sequential-quadratic programming algorithm is utilized to determine an improved airfoil 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 midspan section 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.

scholarly journals Development of Discrete Adjoint Approach Based on the Lattice Boltzmann Method

2014 ◽  
Vol 6 ◽  
pp. 230854 ◽  
Author(s):  
Mohamad Hamed Hekmat ◽  
Masoud Mirzaei
Keyword(s):  
Cost Function ◽  
Optimization Problems ◽  
General Procedure ◽  
Adjoint Equation ◽  
Adjoint Equations ◽  
Test Case ◽  
Inverse Optimization Problem ◽  
Discrete Adjoint ◽  
Adjoint Approach ◽  
The Cost

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.

Discrete Adjoint Solution of Unsteady Turbulent Flow in Compressor

2015 ◽  
Author(s):  
Can Ma ◽  
Xinrong Su ◽  
Xin Yuan
Keyword(s):  
Unsteady Flow ◽  
Stokes Equations ◽  
Flow Simulation ◽  
Adjoint Equations ◽  
Navier Stokes ◽  
Aerodynamic Optimization ◽  
Flow Equations ◽  
Discrete Adjoint ◽  
Continuous Adjoint ◽  
Unsteady Flow Simulation

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.

scholarly journals A New Hyper-Parameter Optimization Method for Power Load Forecast Based on Recurrent Neural Networks

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 163
Author(s):  
Yaru Li ◽  
Yulai Zhang ◽  
Yongping Cai
Keyword(s):  
Neural Networks ◽  
Parameter Optimization ◽  
Recurrent Neural Networks ◽  
Critical Role ◽  
Optimization Method ◽  
Bayesian Optimization ◽  
Power Load ◽  
Gradient Calculation ◽  
Truncated Newton ◽  
Selection Of

The selection of the hyper-parameters plays a critical role in the task of prediction based on the recurrent neural networks (RNN). Traditionally, the hyper-parameters of the machine learning models are selected by simulations as well as human experiences. In recent years, multiple algorithms based on Bayesian optimization (BO) are developed to determine the optimal values of the hyper-parameters. In most of these methods, gradients are required to be calculated. In this work, the particle swarm optimization (PSO) is used under the BO framework to develop a new method for hyper-parameter optimization. The proposed algorithm (BO-PSO) is free of gradient calculation and the particles can be optimized in parallel naturally. So the computational complexity can be effectively reduced which means better hyper-parameters can be obtained under the same amount of calculation. Experiments are done on real world power load data,where the proposed method outperforms the existing state-of-the-art algorithms,BO with limit-BFGS-bound (BO-L-BFGS-B) and BO with truncated-newton (BO-TNC),in terms of the prediction accuracy. The errors of the prediction result in different models show that BO-PSO is an effective hyper-parameter optimization method.

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