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GCRO with dynamic deflated restarting for solving adjoint systems of equations for aerodynamic shape optimization

Chih-Hao Chen (Department of Mechanical Engineering, McGill University, Montreal, Canada)
Siva Nadarajah (Department of Mechanical Engineering, McGill University, Montreal, Canada)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 16 July 2019

Issue publication date: 30 August 2019

Abstract

Purpose

This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.

Design/methodology/approach

The proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗.

Findings

The authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver.

Originality/value

This paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix.

Keywords

Acknowledgements

The authors gratefully acknowledge the comments and feedback raised by the reviewers as it greatly improved the article. They also acknowledge the generous support from the Natural Sciences and Engineering Research Council (NSERC). The authors also thank McGill HPC, Calcul Quebec for providing the computational facilities.

Citation

Chen, C.-H. and Nadarajah, S. (2019), "GCRO with dynamic deflated restarting for solving adjoint systems of equations for aerodynamic shape optimization", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 7, pp. 2179-2205. https://doi.org/10.1108/HFF-10-2018-0586

Publisher

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Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited