Adjoint Aerodynamic Design Optimization for Blades in Multistage Turbomachines—Part I: Methodology and Verification

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
D. X. Wang ◽  
L. He
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Design Optimization ◽  
Difference Method ◽  
Method Development ◽  
Adjoint Method ◽  
Flow Characteristics ◽  
Compressor Stage ◽  
Flow Domain ◽  
Domain Conservation

The adjoint method for blade design optimization will be described in this two-part paper. The main objective is to develop the capability of carrying out aerodynamic blading shape design optimization in a multistage turbomachinery environment. To this end, an adjoint mixing-plane treatment has been proposed. In the first part, the numerical elements pertinent to the present approach will be described. Attention is paid to the exactly opposite propagation of the adjoint characteristics against the physical flow characteristics, providing a simple and consistent guidance in the adjoint method development and applications. The adjoint mixing-plane treatment is formulated to have the two fundamental features of its counterpart in the physical flow domain: conservation and nonreflectiveness across the interface. The adjoint solver is verified by comparing gradient results with a direct finite difference method and through a 2D inverse design. The adjoint mixing-plane treatment is verified by comparing gradient results against those by the finite difference method for a 2D compressor stage. The redesign of the 2D compressor stage further demonstrates the validity of the adjoint mixing-plane treatment and the benefit of using it in a multi-bladerow environment.

Adjoint Aerodynamic Design Optimization for Blades in Multi-Stage Turbomachines: Part I—Methodology and Verification

2008 ◽  
Author(s):  
D. X. Wang ◽  
L. He
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Design Optimization ◽  
Difference Method ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Steepest Descent Method ◽  
Compressor Stage ◽  
Multi Stage ◽  
Continuous Adjoint

The adjoint method for blade design optimization will be described in this two-part paper. The main objective is to develop the capability of carrying out aerodynamic blading shape design optimization in a multi-stage turbomachinery environment. To this end, an adjoint mixing-plane treatment has been proposed. In the first part, the numerical elements pertinent to the present approach will be described. The gradients of a single objective function of a weighted sum of objectives and constraints with respect to detailed blade shape perturbations are obtained very efficiently by the continuous adjoint method. The steepest descent method is used to drive the design to an optimum. The adjoint mixing-plane treatment enables the adjoint equations to be solved in a multi-stage environment. The adjoint solver is verified by comparing gradient results with a direct finite difference method and through a 2D inverse design. The adjoint mixing-plane treatment is verified by comparing gradient results against those by the finite difference method for a 2D compressor stage. The redesign of the 2D compressor stage further demonstrates the validity of the adjoint mixing-plane treatment and the benefit of using it in a multi-bladerow environment.

An implementation of adjoint-based topology optimization in magnetostatics

2019 ◽  
Vol 38 (3) ◽  
pp. 1023-1035 ◽  
Author(s):  
Rtimi Youness ◽  
Frederic Messine
Keyword(s):  
Topology Optimization ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Design Problem ◽  
Large Scale ◽  
Adjoint Method ◽  
Processing Unit ◽  
Original Design ◽  
Content Type

Purpose In magnetostatics, topology optimization (TO) addresses the problem of finding the distributions of both current densities and ferromagnetic materials to comply with fixed magnetic specifications. The purpose of this paper is to develop TO in order to design Hall-effect Thrusters (HETs). Design/methodology/approach In fact, TO problems are known to be large-scale optimization problems. The authors therefore adopt the adjoint method to reduce the computation time required to obtain the gradient information. In this paper, they illustrate the continuous variant of the adjoint method in the context of magnetostatics TO. Herein, the authors propose an implementation of the adjoint method then use it within a gradient-based optimization solver fmincon-MATLAB to solve a HET TO design problem. Findings By comparison with finite difference method, the authors validate the accuracy of the suggested implementation of the adjoint method. Then, they solve a large-scale HET TO design problem. The resultant design of TO is distinctly original and not intuitive. Research limitations/implications In this paper, the authors introduce TO as a tool that has allowed them to explore new and innovative design of a HET. However, although the design presented is original, its manufacture is not feasible. Thus, a discussion section has been included at the end of paper to suggest a possible way to concretize topological solutions. Practical implications TO helps to explore more original design possibilities. In this paper, the authors present an implementation of the adjoint method that makes it possible to solve efficiently and in less central processing unit time large-scale TO design problem. Originality/value An easy implementation of the adjoint method is presented in magnetostatics TO. This implementation was first validated by comparison with the finite difference method and then used to solve a large-scale design problem. The result of the TO design problem is distinctly original and non-intuitive.

A Quasi-First Order Optimization Method and its Demonstration on the Optimization of a U-Bend

2015 ◽  
Author(s):  
M. Bugra Akin ◽  
Wolfgang Sanz
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Adjoint Method ◽  
Optimization Method ◽  
Second Order ◽  
Computational Time ◽  
First Order ◽  
Finite Differences Method ◽  
The Finite Difference Method

Optimal shape design is widely used today to improve a variety of designs. It is a challenging task and several methods have been developed. These methods are generally classified by the order of derivatives used. They are zero, first and second order methods, which, as their names imply, use only the function values, first and second order derivatives, respectively. There are two common approaches to first order methods. These are the finite difference method and the adjoint method. The finite difference method requires an additional CFD calculation for each parameter, which quickly becomes computationally very expensive as the number of parameters rise. The adjoint method provides a computationally efficient alternative in such cases. But the computational cost of the adjoint method also becomes expensive if additional constraints are introduced or when multi-objective optimizations are considered. This paper presents a novel optimization strategy which can be classified as a quasi-gradient based optimization method. As with the finite differences method an additional CFD calculation is performed for each parameter. But in order to save computational time the simulations are not performed to full convergence so that the derivatives are not calculated accurately. The only information that can be obtained in this way is whether the chosen contour manipulation leads to an improvement. A line search method is introduced that can find an optimum using this incomplete gradient information. The optimization method is demonstrated by the quasi-3d optimization of a U-bend.

HEAT TRANSFER IN LARGE RUBBER ELEMENTS THROUGH OF MODELING BY FINITE DIFFERENCE METHOD

2019 ◽  
Author(s):  
Lucas Peixoto ◽  
Ane Lis Marocki ◽  
Celso Vieira Junior ◽  
Viviana Mariani
Keyword(s):  
Heat Transfer ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method

Weighted Finite Difference and Boundary Element Methods Applied to Groundwater Pollution Problems

1991 ◽  
Vol 23 (1-3) ◽  
pp. 517-524
Author(s):  
M. Kanoh ◽  
T. Kuroki ◽  
K. Fujino ◽  
T. Ueda
Keyword(s):  
Boundary Element Method ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Element ◽  
Difference Method ◽  
Groundwater Pollution ◽  
Small Region ◽  
Boundary Element Methods ◽  
Computational Time ◽  
Element Method

The purpose of the paper is to apply two methods to groundwater pollution in porous media. The methods are the weighted finite difference method and the boundary element method, which were proposed or developed by Kanoh et al. (1986,1988) for advective diffusion problems. Numerical modeling of groundwater pollution is also investigated in this paper. By subdividing the domain into subdomains, the nonlinearity is localized to a small region. Computational time for groundwater pollution problems can be saved by the boundary element method; accurate numerical results can be obtained by the weighted finite difference method. The computational solutions to the problem of seawater intrusion into coastal aquifers are compared with experimental results.

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