scholarly journals On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization

Fluids ◽  
2020 ◽  
Vol 5 (2) ◽  
pp. 70
Author(s):  
Farzad Mohebbi ◽  
Ben Evans
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Shape Optimization ◽  
Difference Method ◽  
Optimization Problems ◽  
Step Length ◽  
Aerodynamic Shape Optimization ◽  
Aerodynamic Shape ◽  
Design Variables ◽  
Gradient Based

This study proposeda novel exact expression for step length (size) in gradient-based aerodynamic shape optimization for an airfoil in steady inviscid transonic flows. The airfoil surfaces were parameterized using Bezier curves. The Bezier curve control points were considered as design variables and the finite-difference method was used to compute the gradient of the objective function (drag-to-lift ratio) with respect to the design variables. An exact explicit expression was derived for the step length in gradient-based shape optimization problems. It was shown that the derived step length was independent of the method used for calculating the gradient (adjoint method, finite-difference method, etc.). The obtained results reveal the accuracy of the derived step length.

scholarly journals On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization—Part II: Viscous Flows

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 106
Author(s):  
Farzad Mohebbi ◽  
Ben Evans ◽  
Mathieu Sellier
Keyword(s):  
Shape Optimization ◽  
Viscous Flows ◽  
Optimization Method ◽  
Step Length ◽  
Aerodynamic Shape Optimization ◽  
Ansys Fluent ◽  
Transonic Flows ◽  
Aerodynamic Shape ◽  
Design Variables ◽  
Gradient Based

This study presents an extension of a previous study (On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization) to viscous transonic flows. In this work, we showed that the same procedure to derive an explicit expression for an exact step length βexact in a gradient-based optimization method for inviscid transonic flows can be employed for viscous transonic flows. The extended numerical method was evaluated for the viscous flows over the transonic RAE 2822 airfoil at two common flow conditions in the transonic regime. To do so, the RAE 2822 airfoil was reconstructed by a Bezier curve of degree 16. The numerical solution of the transonic turbulent flow over the airfoil was performed using the solver ANSYS Fluent (using the Spalart–Allmaras turbulence model). Using the proposed step length, a gradient-based optimization method was employed to minimize the drag-to-lift ratio of the airfoil. The gradient of the objective function with respect to design variables was calculated by the finite-difference method. Efficiency and accuracy of the proposed method were investigated through two test cases.

A gradient-based shape optimization scheme via isogeometric exact reanalysis

2018 ◽  
Vol 35 (8) ◽  
pp. 2696-2721 ◽  
Author(s):  
Chensen Ding ◽  
Xiangyang Cui ◽  
Guanxin Huang ◽  
Guangyao Li ◽  
K.K. Tamma ◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Shape Optimization ◽  
Difference Method ◽  
Computer Aided Design ◽  
Content Type ◽  
Optimization Framework ◽  
Computer Aided ◽  
Gradient Based ◽  
The Finite Difference Method

PurposeThis paper aims to propose a gradient-based shape optimization framework in which traditional time-consuming conversions between computer-aided design and computer-aided engineering and the mesh update procedure are avoided/eliminated. The scheme is general so that it can be used in all cases as a black box, no matter what the objective and/or design variables are, whilst the efficiency and accuracy are guaranteed.Design/methodology/approachThe authors integrated CAD and CAE by using isogeometric analysis (IGA), enabling the present methodology to be robust and accurate. To overcome the difficulty in evaluating the sensitivities of objective and/or constraint functions by analytic method in some cases, the authors adopt the finite difference method to calculate these sensitivities, thereby providing a universal approach. Moreover, to further eliminate the inefficiency caused by the finite difference method, the authors advance the exact reanalysis method, the indirect factorization updating (IFU), to exactly and efficiently calculate functions and their sensitivities, which guarantees its generality and efficiency at the same time.FindingsThe proposed isogeometric gradient-based shape optimization using our IFU approach is reliable and accurate, as well as general and efficient.Originality/valueThe authors proposed a gradient-based shape optimization framework in which they first integrate IGA and the proposed exact reanalysis method for applicability to structural response and sensitivity analysis.

scholarly journals A Sequential Approach for Aerodynamic Shape Optimization with Topology Optimization of Airfoils

2021 ◽  
Vol 26 (2) ◽  
pp. 34
Author(s):  
Isaac Gibert Martínez ◽  
Frederico Afonso ◽  
Simão Rodrigues ◽  
Fernando Lau
Keyword(s):  
Topology Optimization ◽  
Shape Optimization ◽  
Volume Fraction ◽  
Optimization Techniques ◽  
Free Form ◽  
Aerodynamic Shape Optimization ◽  
Sequential Approach ◽  
Airfoil Surface ◽  
Aerodynamic Shape ◽  
Design Variables

The objective of this work is to study the coupling of two efficient optimization techniques, Aerodynamic Shape Optimization (ASO) and Topology Optimization (TO), in 2D airfoils. To achieve such goal two open-source codes, SU2 and Calculix, are employed for ASO and TO, respectively, using the Sequential Least SQuares Programming (SLSQP) and the Bi-directional Evolutionary Structural Optimization (BESO) algorithms; the latter is well-known for allowing the addition of material in the TO which constitutes, as far as our knowledge, a novelty for this kind of application. These codes are linked by means of a script capable of reading the geometry and pressure distribution obtained from the ASO and defining the boundary conditions to be applied in the TO. The Free-Form Deformation technique is chosen for the definition of the design variables to be used in the ASO, while the densities of the inner elements are defined as design variables of the TO. As a test case, a widely used benchmark transonic airfoil, the RAE2822, is chosen here with an internal geometric constraint to simulate the wing-box of a transonic wing. First, the two optimization procedures are tested separately to gain insight and then are run in a sequential way for two test cases with available experimental data: (i) Mach 0.729 at α=2.31°; and (ii) Mach 0.730 at α=2.79°. In the ASO problem, the lift is fixed and the drag is minimized; while in the TO problem, compliance minimization is set as the objective for a prescribed volume fraction. Improvements in both aerodynamic and structural performance are found, as expected: the ASO reduced the total pressure on the airfoil surface in order to minimize drag, which resulted in lower stress values experienced by the structure.

scholarly journals the numerical solution of two point boundary value problem for the Helmholtz type equation by finite difference method with non regular step length between nodes

2021 ◽  
Vol 1 (1) ◽  
pp. 18-23
Author(s):  
Pramod Pandey
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
Boundary Value ◽  
Type Equation ◽  
Step Length ◽  
Computational Results ◽  
Order Of Accuracy ◽  
Variable Step

In this article, we have presented a variable step finite difference method for solving second order boundary value problems in ordinary differential equations. We have discussed the convergence and established that proposed has at least cubic order of accuracy. The proposed method tested on several model problems for the numerical solution. The numerical results obtained for these model problems with known / constructed exact solution confirm the theoretical conclusions of the proposed method. The computational results obtained for these model problems suggest that method is efficient and accurate.

Parameter assessment for virtual Stackelberg game in aerodynamic shape optimization

2018 ◽  
Vol 32 (12n13) ◽  
pp. 1840044
Author(s):  
Jing Wang ◽  
Fangfang Xie ◽  
Yao Zheng ◽  
Jifa Zhang
Keyword(s):  
Shape Optimization ◽  
Stackelberg Game ◽  
Critical Parameters ◽  
Aerodynamic Shape Optimization ◽  
Aerodynamic Optimization ◽  
Aerodynamic Shape ◽  
Design Variables ◽  
Design Cycle ◽  
Parametric Studies ◽  
The Impact

In this paper, parametric studies of virtual Stackelberg game (VSG) are conducted to assess the impact of critical parameters on aerodynamic shape optimization, including design cycle, split of design variables and role assignment. Typical numerical cases, including the inverse design and drag reduction design of airfoil, have been carried out. The numerical results confirm the effectiveness and efficiency of VSG. Furthermore, the most significant parameters are identified, e.g. the increase of design cycle can improve the optimization results but it will also add computational burden. These studies will maximize the productivity of the effort in aerodynamic optimization for more complicated engineering problems, such as the multi-element airfoil and wing-body configurations.

Calculation on the Internal Force of Deeply Buried Anti-Slide Pile by Using Finite Difference Method Based on the m-k Type Method

2011 ◽  
Vol 130-134 ◽  
pp. 128-134 ◽  
Author(s):  
Qiu Yan Fan ◽  
Mei Qian Wang ◽  
Sheng Cai Xu
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Solution Process ◽  
Step Length ◽  
Internal Force ◽  
Power Series Method ◽  
Type Method ◽  
New Type

In the past, when used the foundation coefficient to calculate the internal force of anti-slide pile, power series method is usually adopted. The deformation compatibility conditions and continuity conditions of sliding surface between non-anchoring section and anchoring section are exploited to determine the final result, causing the lengthy solution process and that there is no guarantee for the calculation accuracy. This paper uses the foundation coefficient method in the calculation of internal force of anti-slide pile and employs the “m-k” method with a more complicated up-down foundation structure to get the finite difference equation to determine the new-type deeply buried anti-slide pile displacement and internal force. Then the calculation on the internal force and displacement of the whole pile can be realized easily through the procedure method. Finally, this paper makes a contrastive analysis on the result of the finite difference method and finite element calculation through the case study. As long as the equal differential step length is small enough, the calculation accuracy can meet the demand of engineer design and the program graph processing result can optimize the design of anti-slide pile.

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