Research article

A Gaussian Process Regression approach within a data-driven POD framework for engineering problems in fluid dynamics

  • Received: 22 November 2020 Accepted: 20 July 2021 Published: 06 August 2021
  • This work describes the implementation of a data-driven approach for the reduction of the complexity of parametrical partial differential equations (PDEs) employing Proper Orthogonal Decomposition (POD) and Gaussian Process Regression (GPR). This approach is applied initially to a literature case, the simulation of the Stokes problem, and in the following to a real-world industrial problem, within a shape optimization pipeline for a naval engineering problem.

    Citation: Giulio Ortali, Nicola Demo, Gianluigi Rozza. A Gaussian Process Regression approach within a data-driven POD framework for engineering problems in fluid dynamics[J]. Mathematics in Engineering, 2022, 4(3): 1-16. doi: 10.3934/mine.2022021

    Related Papers:

  • This work describes the implementation of a data-driven approach for the reduction of the complexity of parametrical partial differential equations (PDEs) employing Proper Orthogonal Decomposition (POD) and Gaussian Process Regression (GPR). This approach is applied initially to a literature case, the simulation of the Stokes problem, and in the following to a real-world industrial problem, within a shape optimization pipeline for a naval engineering problem.



    加载中


    [1] M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, et al., The FEniCS project version 1.5, Archive of Numerical Software, 3 (2015), 9-23.
    [2] F. Ballarin, A. Manzoni, A. Quarteroni, G. Rozza, Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Int. J. Numer. Meth. Eng., 102 (2015), 1136-1161. doi: 10.1002/nme.4772
    [3] D. Boffi, F. Brezzi, M. Fortin, Finite elements for the stokes problem, In: Mixed finite elements, compatibility conditions, and applications, Springer, 2008, 45-100.
    [4] M. D. Buhmann, Radial basis functions: theory and implementations, Cambridge University Press, 2003.
    [5] P. Davidson, Turbulence: an introduction for scientists and engineers, Oxford University Press, 2015.
    [6] N. Demo, G. Ortali, G. Gustin, G. Rozza, G. Lavini, An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniques. Boll. Unione Mat. Ital., 14 (2021), 211-230.
    [7] N. Demo, M. Tezzele, G. Gustin, G. Lavini, G. Rozza, Shape optimization by means of proper orthogonal decomposition and dynamic mode decomposition, In: Technology and science for the ships of the future: proceedings of NAV 2018: 19th international conference on ship & maritime research, IOS Press, 2018,212-219.
    [8] N. Demo, M. Tezzele, G. Rozza, EZyRB: Easy reduced basis method, JOSS, 3 (2018), 661. doi: 10.21105/joss.00661
    [9] N. Demo, M. Tezzele, G. Rozza, A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces, CR. Mécanique, 347 (2019), 873-881. doi: 10.1016/j.crme.2019.11.012
    [10] F. A. Fortin, F. M. De Rainville, M. A. Gardner, M. Parizeau, C. Gagné, DEAP: Evolutionary algorithms made easy, J. Mach. Learn. Res., 13 (2012), 2171-2175.
    [11] GPy, GPy: A Gaussian process framework in Python, 2012. Available from: http://github.com/SheffieldML/GPy.
    [12] M. Guo, J. S. Hesthaven, Reduced order modeling for nonlinear structural analysis using Gaussian process regression, Comput. Method. Appl. M., 341 (2018), 807-826. doi: 10.1016/j.cma.2018.07.017
    [13] J. S. Hesthaven, G. Rozza, B. Stamm, Certified reduced basis methods for parametrized partial differential equations, 1 Eds., Switzerland: Springer, 2016.
    [14] J. S. Hesthaven, S. Ubbiali, Non-intrusive reduced order modeling of nonlinear problems using neural networks, J. Comput. Phys., 363 (2018), 55-78. doi: 10.1016/j.jcp.2018.02.037
    [15] S. Hijazi, G. Stabile, A. Mola, G. Rozza, Data-driven pod-galerkin reduced order model for turbulent flows, J. Comput. Phys., 416 (2020), 109513. doi: 10.1016/j.jcp.2020.109513
    [16] C. Hirt, B. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39 (1981), 201-225. doi: 10.1016/0021-9991(81)90145-5
    [17] A. Koshakji, A. Quarteroni, G. Rozza, Free form deformation techniques applied to 3D shape optimization problems, CAIM, 4 (2013), 1-26.
    [18] F. Moukalled, L. Mangani, M. Darwish, The finite volume method in computational fluid dynamics: an advanced introduction with OpenFOAM and Matlab, Cham: Springer, 2015.
    [19] OpenCFD, OpenFOAM - The Open Source CFD Toolbox - User's Guide, 2018. Available from: https://www.openfoam.com/documentation/user-guide.
    [20] G. Ortali, A data-driven reduced order optimization approach for Cruise ship design, Master's thesis, Politecnico di Torino, 2019.
    [21] F. Pichi, F. Ballarin, G. Rozza, J. S. Hesthaven, Artificial neural network for bifurcating phenomena modelled by nonlinear parametrized PDEs, PAMM, 20 (2021), e202000350.
    [22] F. Pichi, G. Rozza, Reduced basis approaches for parametrized bifurcation problems held by non-linear von kármán equations, J. Sci. Comput., 81 (2019), 112-135. doi: 10.1007/s10915-019-01003-3
    [23] PyGeM, PyGeM: Python geometrical morphing. Available from: https://github.com/mathLab/PyGeM.
    [24] J. Quiñonero-Candela, C. E. Rasmussen, A unifying view of sparse approximate gaussian process regression, J. Mach. Learn. Res., 6 (2005), 1939-1959.
    [25] G. Rozza, M. H. Malik, N. Demo, M. Tezzele, M. Girfoglio, G. Stabile, et al., Advances in reduced order methods for parametric industrial problems in computational fluid dynamics, In: Proceedings of the ECCOMAS Congress 2018, 2018.
    [26] F. Salmoiraghi, F. Ballarin, L. Heltai, G. Rozza, Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes, Adv. Model. and Simul. in Eng. Sci., 3 (2016), 21. doi: 10.1186/s40323-016-0076-6
    [27] F. Salmoiraghi, A. Scardigli, H. Telib, G. Rozza, Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation, Int. J. Comput. Fluid Dyn., 32 (2018), 233-247. doi: 10.1080/10618562.2018.1514115
    [28] T. W. Sederberg, S. R. Parry, Free-form deformation of solid geometric models, In: ACM SIGGRAPH Computer Graphics, 1986,151-160.
    [29] G. Stabile, G. Rozza, Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations, Comput. Fluids, 173 (2018), 273-284. doi: 10.1016/j.compfluid.2018.01.035
    [30] G. Stabile, M. Zancanaro, G. Rozza, Efficient Geometrical parametrization for finite-volume based reduced order methods. Int. J. Numer. Meth. Eng., 121 (2020), 2655-2682.
    [31] C. Taylor, P. Hood, A numerical solution of the navier-stokes equations using the finite element technique, Comput. Fluids, 1 (1973), 73-100. doi: 10.1016/0045-7930(73)90027-3
    [32] M. Tezzele, N. Demo, G. Rozza, Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces, In: Proceedings of MARINE 2019: VIII International Conference on Computational Methods in Marine Engineering, 2019,122-133.
    [33] M. Tezzele, N. Demo, G. Stabile, A. Mola, G. Rozza, Enhancing cfd predictions in shape design problems by model and parameter space reduction, Adv. Model. and Simul. in Eng. Sci., 7 (2020), 40. doi: 10.1186/s40323-020-00177-y
    [34] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods, 17 (2020), 261-272. doi: 10.1038/s41592-019-0686-2
    [35] S. Volkwein, Proper orthogonal decomposition: theory and reduced-order modelling, 2012.
    [36] Q. Wang, J. S. Hesthaven, D. Ray, Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem, J. Comput. Phys., 384 (2019), 289-307. doi: 10.1016/j.jcp.2019.01.031
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3356) PDF downloads(235) Cited by(10)

Article outline

Figures and Tables

Figures(6)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog