Volume 13, Issue 3
An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient

Asma Farooqi, Riaz Ahmad & Zhenping Feng

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 689-718.

Published online: 2020-03

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  • Abstract

An improved hybrid adjoint method to the viscous, compressible Reynold-Averaged Navier-Stokes Equation (RANS) is developed for the computation of objective function gradient and demonstrated for external aerodynamic design optimization. In this paper, the main idea is to extend the previous coupling of the discrete and continuous adjoint method by the grid-node coordinates variation technique for the computation of the variation in the gradients of flow variables. This approach in combination with the Jacobian matrices of flow fluxes refrained the objective function from field integrals and coordinate transformation matrix. Thus, it opens up the possibility of employing the hybrid adjoint method to evaluate the subsequent objective function gradient analogous to many shape parameters, comprises of only boundary integrals. This avoids the grid regeneration in the geometry for every surface perturbation in a structured and unstructured grid. Hence, this viable technique reduces the overall CPU cost. Moreover, the new hybrid adjoint method has been successfully applied to the computation of accurate sensitivity derivatives. Finally, for the investigation of the presented numerical method, simulations are carried out on NACA0012 airfoil in a transonic regime and its accuracy and effectiveness related to the new gradient equation have been verified with the Finite Difference Method (FDM). The analysis reveals that the presented methodology for the optimization provides the designer with an indispensable CPU-cost effective tool to reshape the complex geometry airfoil surfaces, useful relative to the state-of-the-art, in a less computing time.

  • Keywords

Continuous adjoint method, discrete adjoint method, variation technique, automatic differentiation.

  • AMS Subject Headings

65K10, 49Q12, 76H05, 65N08, 65N55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zpfeng@mail.xjtu.edu.cn (Zhenping Feng)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-13-689, author = {Farooqi , Asma and Ahmad , Riaz and Feng , Zhenping}, title = {An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {3}, pages = {689--718}, abstract = {

An improved hybrid adjoint method to the viscous, compressible Reynold-Averaged Navier-Stokes Equation (RANS) is developed for the computation of objective function gradient and demonstrated for external aerodynamic design optimization. In this paper, the main idea is to extend the previous coupling of the discrete and continuous adjoint method by the grid-node coordinates variation technique for the computation of the variation in the gradients of flow variables. This approach in combination with the Jacobian matrices of flow fluxes refrained the objective function from field integrals and coordinate transformation matrix. Thus, it opens up the possibility of employing the hybrid adjoint method to evaluate the subsequent objective function gradient analogous to many shape parameters, comprises of only boundary integrals. This avoids the grid regeneration in the geometry for every surface perturbation in a structured and unstructured grid. Hence, this viable technique reduces the overall CPU cost. Moreover, the new hybrid adjoint method has been successfully applied to the computation of accurate sensitivity derivatives. Finally, for the investigation of the presented numerical method, simulations are carried out on NACA0012 airfoil in a transonic regime and its accuracy and effectiveness related to the new gradient equation have been verified with the Finite Difference Method (FDM). The analysis reveals that the presented methodology for the optimization provides the designer with an indispensable CPU-cost effective tool to reshape the complex geometry airfoil surfaces, useful relative to the state-of-the-art, in a less computing time.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0087}, url = {http://global-sci.org/intro/article_detail/nmtma/15781.html} }
TY - JOUR T1 - An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient AU - Farooqi , Asma AU - Ahmad , Riaz AU - Feng , Zhenping JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 689 EP - 718 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0087 UR - https://global-sci.org/intro/article_detail/nmtma/15781.html KW - Continuous adjoint method, discrete adjoint method, variation technique, automatic differentiation. AB -

An improved hybrid adjoint method to the viscous, compressible Reynold-Averaged Navier-Stokes Equation (RANS) is developed for the computation of objective function gradient and demonstrated for external aerodynamic design optimization. In this paper, the main idea is to extend the previous coupling of the discrete and continuous adjoint method by the grid-node coordinates variation technique for the computation of the variation in the gradients of flow variables. This approach in combination with the Jacobian matrices of flow fluxes refrained the objective function from field integrals and coordinate transformation matrix. Thus, it opens up the possibility of employing the hybrid adjoint method to evaluate the subsequent objective function gradient analogous to many shape parameters, comprises of only boundary integrals. This avoids the grid regeneration in the geometry for every surface perturbation in a structured and unstructured grid. Hence, this viable technique reduces the overall CPU cost. Moreover, the new hybrid adjoint method has been successfully applied to the computation of accurate sensitivity derivatives. Finally, for the investigation of the presented numerical method, simulations are carried out on NACA0012 airfoil in a transonic regime and its accuracy and effectiveness related to the new gradient equation have been verified with the Finite Difference Method (FDM). The analysis reveals that the presented methodology for the optimization provides the designer with an indispensable CPU-cost effective tool to reshape the complex geometry airfoil surfaces, useful relative to the state-of-the-art, in a less computing time.

Asma Farooqi, Riaz Ahmad & Zhenping Feng. (2020). An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 689-718. doi:10.4208/nmtma.OA-2019-0087
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