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Local-Maximum-and-Minimum-Preserving Solution Remapping Technique to Accelerate Flow Convergence for Discontinuous Galerkin Methods in Shape Optimization Design

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Abstract

In this work, a solution remapping technique is developed to accelerate the flow convergence for the intermediate shapes when a high-order discontinuous Galerkin (DG) method is employed as a compressible Euler flow solver in the airfoil design problems. Once the shape is updated, the proposed technique is applied to initialize the flow simulation for the new shape via a solution remapping formula and a maximum-and-minimum-preserving limiter. First, the solution remapping formula is used to remap the solution of the current shape into a piecewise polynomial on the mesh of the new shape. Then the piecewise polynomial is constrained with the maximum-and-minimum-preserving limiter. The modified piecewise polynomial is used as the initial value for the new shape. Numerical experiments show that the proposed technique can attractively accelerate flow convergence and significantly reduce up to 80% of the computational time in the airfoil design problems with a high-order DG solver.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

References

  1. Bhabra, M., Nadarajah, S.: Aerodynamic shape optimization for the NURBS-enhanced discontinuous Galerkin method. In: AIAA Aviation 2019 Forum (2019). https://doi.org/10.2514/6.2019-3197

  2. Blazek, J.: Computational Fluid Dynamics: Principles and Applications. Elsevier (2005). https://doi.org/10.1016/B978-0-08-044506-9.X5000-0

  3. de Boer, A., van der Schoot, M.S., Bijl, H.: Mesh deformation based on radial basis function interpolation. Comput. Struct. 85(11–14), 784–795 (2007). https://doi.org/10.1016/j.compstruc.2007.01.013

    Article  Google Scholar 

  4. Chan, C., Bai, H., He, D.: Blade shape optimization of the Savonius wind turbine using a genetic algorithm. Appl. Energy 213, 148–157 (2018) https://doi.org/10.1016/j.apenergy.2018.01.029. http://www.sciencedirect.com/science/article/pii/S0306261918300291

  5. Chen, G., Fidkowski, K.J.: Discretization error control for constrained aerodynamic shape optimization. J. Comput. Phys. 387, 163–185 (2019) https://doi.org/10.1016/j.jcp.2019.02.038. http://www.sciencedirect.com/science/article/pii/S002199911930155X

  6. Cockburn, B., Hou, S., Shu, C.W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54(190), 545–581 (1990). https://doi.org/10.1090/S0025-5718-1990-1010597-0

    Article  MathSciNet  MATH  Google Scholar 

  7. 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. Comput. Phys. 84(1), 90–113 (1989). https://doi.org/10.1016/0021-9991(89)90183-6

    Article  MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52(186), 411–435 (1989). https://doi.org/10.1090/S0025-5718-1989-0983311-4

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998) https://doi.org/10.1006/jcph.1998.5892. http://www.sciencedirect.com/science/article/pii/S0021999198958922

  10. Hartmann, R., Houston, P.: Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. J. Comput. Phys. 183(2), 508–532 (2002) https://doi.org/10.1006/jcph.2002.7206. http://www.sciencedirect.com/science/article/pii/S0021999102972062

  11. Hartmann, R., Houston, P.: Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24(3), 979–1004 (2003). https://doi.org/10.1137/S1064827501389084

    Article  MathSciNet  MATH  Google Scholar 

  12. Hicks, R.M., Henne, P.A.: Wing design by numerical optimization. J. Aircr. 15(7), 407–412 (1978). https://doi.org/10.2514/3.58379

    Article  Google Scholar 

  13. Holland, J.: Adaptation in Natural and Artificial Systems. University of Michigan Press (1975)

  14. Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. In: AIAA 14th Fluid and Plasma Dynamics Conference, p. 1259 (1981). https://doi.org/10.2514/6.1981-1259

  15. Kaland, L., Sonntag, M., Gauger, N.R.: Adaptive aerodynamic design optimization for Navier-Stokes using shape derivatives with discontinuous Galerkin methods. In: D. Greiner, B. Galván, J. Périaux, N. Gauger, K. Giannakoglou, G. Winter (eds.) Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences, pp. 143–158. Springer International Publishing, Cham (2015). https://doi.org/10.1007/978-3-319-11541-2_9

  16. LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2002)

  17. Li, D., Hartmann, R.: Adjoint-based airfoil optimization with discretization error control. Int. J. Numeri. Methods Fluids 77(1), 1–17 (2015). https://doi.org/10.1002/fld.3971

    Article  MathSciNet  Google Scholar 

  18. Lu, J.: An a Posteriori Error Control Framework for Adaptive Precision Optimization Using Discontinuous Galerkin Finite Element Method. Ph.D. thesis, Massachusetts Institute of Technology (2005)

  19. Lyu, Z., Kenway, G.K.W., Martins, J.R.R.A.: Aerodynamic shape optimization investigations of the common research model wing benchmark. AIAA J. 53(4), 968–985 (2015). https://doi.org/10.2514/1.J053318

    Article  Google Scholar 

  20. Naumann, D., Evans, B., Walton, S., Hassan, O.: A novel implementation of computational aerodynamic shape optimisation using Modified Cuckoo Search. Appl. Math. Modell. 40(7), 4543–4559 (2016) https://doi.org/10.1016/j.apm.2015.11.023. http://www.sciencedirect.com/science/article/pii/S0307904X15007374

  21. Persson, P.O., Peraire, J.: Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations. SIAM J. Sci. Comput. 30(6), 2709–2733 (2008). https://doi.org/10.1137/070692108

    Article  MathSciNet  MATH  Google Scholar 

  22. Salmoiraghi, F., Scardigli, A., Telib, H., Rozza, G.: Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation. Int. J. Comput. Fluid Dyn. 32(4–5), 233–247 (2018). https://doi.org/10.1080/10618562.2018.1514115

    Article  MathSciNet  Google Scholar 

  23. Slotnick, J., Khodadoust, A., Alonso, J., Darmofal, D., Gropp, W., Lurie, E., Mavriplis, D.: CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences. NASA/CR-2014-218178, NF1676L-18332 (2014)

  24. Spall, J.C.: An overview of the simultaneous perturbation method for efficient optimization. Johns Hopkins APL Tech. Digest 19(4), 482–492 (1998)

    Google Scholar 

  25. Spall, J.C.: Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. John Wiley & Sons, New Jersey (2005)

  26. Toman, U.T., Hassan, A.K.S., Owis, F.M., Mohamed, A.S.: Blade shape optimization of an aircraft propeller using space mapping surrogates. Adv. Mech. Eng. 11(7) (2019). https://doi.org/10.1177/1687814019865071

  27. Wang, J., Wang, Z., Liu, T.: Solution remapping technique to accelerate flow convergence for finite volume methods applied to shape optimization design. Numeri. Math. Theory Methods Appl. 13(4), 863–880 (2020) https://doi.org/10.4208/nmtma.OA-2019-0164. http://global-sci.org/intro/article_detail/nmtma/16957.html

  28. Wang, K., Yu, S., Wang, Z., Feng, R., Liu, T.: Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 344, 602–625 (2019). https://doi.org/10.1016/j.cma.2018.10.033

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, Z.: A perspective on high-order methods in computational fluid dynamics. Sci. China Phys. Mech. Astron. 59(1), 614701 (2016). https://doi.org/10.1007/s11433-015-5706-3

    Article  Google Scholar 

  30. Wang, Z.J.: High-order computational fluid dynamics tools for aircraft design. Philos. Trans. Roy. Soc. A Math. Phys. Eng. Sci. 372(2022), 20130318 (2014). https://doi.org/10.1098/rsta.2013.0318

    Article  Google Scholar 

  31. 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. Int. J. Numer. Methods Fluids 72(8), 811–845 (2013). https://doi.org/10.1002/fld.3767

    Article  MathSciNet  MATH  Google Scholar 

  32. Xing, X.Q., Damodaran, M.: Application of simultaneous perturbation stochastic approximation method for aerodynamic shape design optimization. AIAA J. 43(2), 284–294 (2005). https://doi.org/10.2514/1.9484

    Article  Google Scholar 

  33. Zahr, M.J., Persson, P.O.: High-order, time-dependent aerodynamic optimization using a discontinuous Galerkin discretization of the Navier-Stokes equations. In: 54th AIAA Aerospace Sciences Meeting (2016). https://doi.org/10.2514/6.2016-0064

  34. Zahr, M.J., Persson, P.O.: Energetically optimal flapping wing motions via adjoint-based optimization and high-order discretizations. In: H. Antil, D.P. Kouri, M.D. Lacasse, D. Ridzal (eds.) Frontiers in PDE-Constrained Optimization, pp. 259–289. Springer, New York, NY (2018). https://doi.org/10.1007/978-1-4939-8636-1_7

  35. Zhang, X., Shu, C.W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091–3120 (2010) https://doi.org/10.1016/j.jcp.2009.12.030. http://www.sciencedirect.com/science/article/pii/S0021999109007165

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Acknowledgements

The authors would like to acknowledge the support of National Numerical Wind Tunnel Project and the National Natural Science Foundation of China (No.U1730118 and No.91530325).

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

    Jufang Wang & Tiegang Liu

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  1. Jufang Wang
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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 Sci Comput 87, 79 (2021). https://doi.org/10.1007/s10915-021-01499-8

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