非结构网格上针对间断有限元方法的初值重映流场收敛加速技术

doi:10.12288/szjs.s2020-0725

数值计算与计算机应用 ›› 2021, Vol. 42 ›› Issue (2): 169-182.DOI: 10.12288/szjs.s2020-0725

• 论文 • 上一篇

非结构网格上针对间断有限元方法的初值重映流场收敛加速技术

王居方, 刘铁钢

LMIB, 北京航空航天大学数学科学学院, 北京 100191

收稿日期:2020-12-07 出版日期:2021-06-15 发布日期:2021-06-03
基金资助:
国家自然科学基金（U1730118，91530325）资助.

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王居方, 刘铁钢. 非结构网格上针对间断有限元方法的初值重映流场收敛加速技术[J]. 数值计算与计算机应用, 2021, 42(2): 169-182.

Wang Jufang, Liu Tiegang. SOLUTION REMAPPING TECHNIQUE TO ACCELERATE FLOW CONVERGENCE FOR DISCONTINUOUS GALERKIN METHODS ON UNSTRUCTURED MESHES[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(2): 169-182.

SOLUTION REMAPPING TECHNIQUE TO ACCELERATE FLOW CONVERGENCE FOR DISCONTINUOUS GALERKIN METHODS ON UNSTRUCTURED MESHES

Wang Jufang, Liu Tiegang

LMIB, School of Mathematical Sciences, Beihang University, Beijing 100191, China

Received:2020-12-07 Online:2021-06-15 Published:2021-06-03

给出一种非结构网格上针对间断有限元方法的初值重映流场收敛加速技术框架, 以提高气动优化过程中间断有限元方法的收敛速度, 减少优化过程所需时间. 在保持网格拓扑结构不变的前提下, 通过从给定的参考单元到物理域上每个网格单元的一一映射, 在不同外形的网格上相应单元之间建立局部的一一对应关系; 每次更新外形时, 将当前外形的数值解重映到新外形的网格上作为初值, 以加快间断有限元方法的收敛速度. 将该技术应用于三角形网格上的翼型优化设计问题, 取得了很好的效果, 对于三阶间断有限元方法能够减少超过70%的计算时间.

关键词： 初值重映技术, 间断有限元, 非结构网格, 翼型优化设计

A framework of the solution remapping technique on unstructured meshes is proposed to accelerate flow convergence for the discontinuous Galerkin (DG) methods in aerodynamic shape optimization. A local one-to-one correspondence between the meshes of different shapes is established by the one-to-one mappings from a given reference cell to each grid cell in the physical domain if the mesh topology is preserved in the optimization process. Then when the shape is updated, the solution of the current shape is remapped to the mesh of the new shape as the initial value to accelerate the convergence of the DG solver. The proposed framework is implemented in the airfoil design problem on triangular meshes. Numerical experiments show that more than 70% of the computational time can be saved with a third-order DG solver.

Key words: solution remapping technique, discontinuous Galerkin method, unstructured meshes, airfoil optimization design

MR(2010)主题分类:

65M60

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[1] Slotnick J, Khodadoust A, Alonso J, Darmofal D, Gropp W, Lurie E, Mavriplis D. CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences[J]. NASA/CR-2014-218178, NF1676L-18332, 2014.
[2] Wang Z J, Fidkowski K, Abgrall R, Bassi F, Caraeni D, Cary A, Deconinck H, Hartmann R, Hillewaert K, Huynh H T, Kroll N, May G, Persson P O, van Leer B, Visbal M. High-order CFD methods: current status and perspective[J]. International Journal for Numerical Methods in Fluids, 2013, 72(8): 811–845.
[3] Cockburn B, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework[J]. Mathematics of Computation, 1989, 52(186): 411–435.
[4] Cockburn B, Lin S Y, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems[J]. Journal of Computational Physics, 1989, 84(1): 90–113.
[5] Cockburn B, Hou S, Shu C W. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case[J]. Mathematics of Computation, 1990, 54(190): 545–581.
[6] Cockburn B, Shu C W. The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems[J]. Journal of Computational Physics, 1998, 141(2): 199–224.
[7] Bhabra M, Nadarajah S. Aerodynamic Shape Optimization for the NURBS-Enhanced Discontinuous Galerkin Method [C]. In AIAA Aviation 2019 Forum, 2019.
[8] Kaland L, Sonntag M, Gauger N R. Adaptive Aerodynamic Design Optimization for Navier-Stokes Using Shape Derivatives with Discontinuous Galerkin Methods [G]. In Greiner D, Galván B, Périaux J, Gauger N, Giannakoglou K, Winter G, editors, Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences, pages 143–158. Springer International Publishing, Cham, 2015.
[9] Li D, Hartmann R. Adjoint-based airfoil optimization with discretization error control[J]. International Journal for Numerical Methods in Fluids, 2015, 77(1): 1–17.
[10] Lu J. An a posteriori Error Control Framework for Adaptive Precision Optimization using Discontinuous Galerkin Finite Element Method[D]. PhD thesis, Massachusetts Institute of Technology, 2005.
[11] Wang K, Yu S, Wang Z, Feng R, Liu T. Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 344: 602–625.
[12] Zahr M J, Persson P O. High-order, time-dependent aerodynamic optimization using a discontinuous Galerkin discretization of the Navier-Stokes equations [C]. In 54th AIAA Aerospace Sciences Meeting, 2016.
[13] Zahr M J, Persson P O. Energetically Optimal Flapping Wing Motions via Adjoint-Based Optimization and High-Order Discretizations [G]. In Antil H, Kouri D P, Lacasse M D, Ridzal D, editors, Frontiers in PDE-Constrained Optimization, pages 259–289. Springer New York, New York, NY, 2018.
[14] Wang Z. A perspective on high-order methods in computational fluid dynamics[J]. Science China Physics, Mechanics & Astronomy, 2016, 59(1): 614701.
[15] Wang J, Liu T. Local-Maximum-and-Minimum-Preserving Solution Remapping Technique to Accelerate Flow Convergence for Discontinuous Galerkin Methods in Shape Optimization Design[J]. Journal of Scientific Computing, 2021, 87: 79. https://doi.org/10.1007/s10915-021-01499-8
[16] Wang J, Wang Z, Liu T. Solution Remapping Technique to Accelerate Flow Convergence for Finite Volume Methods Applied to Shape Optimization Design[J]. Numerical Mathematics: Theory, Methods and Applications, 2020, 13(4): 863–880.
[17] de Boer A, van der Schoot M S, Bijl H. Mesh deformation based on radial basis function interpolation[J]. Computers & Structure, 2007, 85(11-14): 784–795.
[18] Hicks R M, Henne P A. Wing design by numerical optimization[J]. Journal of Aircraft, 1978, 15(7): 407–412.
[19] Spall J C. An overview of the simultaneous perturbation method for efficient optimization[J]. Johns Hopkins apl technical digest, 1998, 19(4): 482–492.
[20] Spall J C. Introduction to stochastic search and optimization: estimation, simulation, and control[M]. John Wiley & Sons, 2005.
[21] Xing X Q, Damodaran M. Application of simultaneous perturbation stochastic approximation method for aerodynamic shape design optimization[J]. AIAA Journal, 2005, 43(2): 284–294.

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非结构网格上针对间断有限元方法的初值重映流场收敛加速技术

SOLUTION REMAPPING TECHNIQUE TO ACCELERATE FLOW CONVERGENCE FOR DISCONTINUOUS GALERKIN METHODS ON UNSTRUCTURED MESHES

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