Abstract
We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity-reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem.
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Afanasiev, K., Hinze, M.: Adaptive control of a wake flow using proper orthogonal decomposition. Lecture Notes in Pure and Applied Mathematics pp. 317–332 (2001)
Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis, vol. 37. Wiley, New York (2011)
Ali, M., Steih, K., Urban, K.: Reduced basis methods with adaptive snapshot computations. Adv. Comput. Math. 43(2), 257–294 (2017)
Arian, E., Fahl, M., Sachs, E.W.: Trust-region proper orthogonal decomposition for flow control. Tech rep (2000)
Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20(3), 179–192 (1973)
Ballarin, F., Manzoni, A., Quarteroni, A., Rozza, G.: Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations. Int. J. Numer. Methods Eng. 102(5), 1136–1161 (2015)
Bergmann, M., Bruneau, C.H., Iollo, A.: Enablers for robust POD models. J. Comput. Phys. 228(2), 516–538 (2009)
Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25(1), 539–575 (1993)
Besier, M., Wollner, W.: On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes. Internat. J. Numer. Methods Fluids 69(6), 1045–1064 (2012)
Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. Revue franċaise d’automatique, informatique, recherche opérationnelle. Analyse Numérique 8(R2), 129–151 (1974)
Caiazzo, A., Iliescu, T., John, V., Schyschlowa, S.: A numerical investigation of velocity–pressure reduced order models for incompressible flows. J. Comput. Phys. 259, 598–616 (2014)
Carlberg, K.: Adaptive h-refinement for reduced-order models. Int. J. Numer. Methods Eng. 102(5), 1192–1210 (2015)
Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (5), 2524–2550 (2008)
Constantin, P., Foias, C.: Navier-Stokes Equations. The University of Chicago Press (1988)
Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)
Frutos, J.D., John, V., Novo, J.: Projection methods for incompressible flow problems with WENO finite difference schemes. J. Comput. Phys. 309, 368–386 (2016)
Graham, W.R., Peraire, J., Tang, K.Y.: Optimal control of vortex shedding using low-order models. I. Open-loop model development. Int. J. Numer. Meth. Eng. 44(7), 945–972 (1999)
Gräßle, C., Hinze, M.: POD reduced order modeling for evolution equations utilizing arbitrary finite element discretizations. Adv. Comput. Math. 44 (6), 1941–1978 (2018)
Gunzburger, M.D., Peterson, J.S., Shadid, J.N.: Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Engrg. 196(4–6), 1030–1047 (2007)
Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constraints, vol. 23. Springer Science & Business Media (2008)
Ito, K., Ravindran, S.S.: A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143(2), 403–425 (1998)
John, V.: Finite element methods for incompressible flow problems, Springer Series in Computational Mathematics, vol. 51. Springer, Cham (2016)
Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40(2), 492–515 (2002)
Ladyženskaja, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Mathematics and Its Applications, vol. 12. Gordon and Breach Science Publishers, New York (1969)
Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: Model order reduction in fluid dynamics: challenges and perspectives. In: Reduced Order Methods for Modeling and Computational Reduction, pp. 235–273. Springer (2014)
Mitchell, W.F.: Adaptive refinement for arbitrary finite-element spaces with hierarchical bases. J. Comput. Appl. Math. 36(1), 65–78 (1991)
Noack, B.R., Papas, P., Monkewitz, P.A.: The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365 (2005)
Peterson, J.S.: The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10(4), 777–786 (1989)
Quarteroni, A., Rozza, G.: Numerical solution of parametrized Navier–Stokes equations by reduced basis methods. Numerical Methods for Partial Differential Equations: An International Journal 23(4), 923–948 (2007)
Ravindran, S.S.: Proper orthogonal decomposition in optimal control of fluids. Int. J. Numer. Methods Fluids 34(5), 425–448 (2000)
Raymond, J.P.: Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Annales de l’Institut Henri Poincare – Non Linear Analysis 24 (6), 921–951 (2007)
Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196(7), 1244–1260 (2007)
Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I, II and III. Quart. Appl. Math. 45, 561–590 (1987)
Stabile, G., Rozza, G.: Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations. Comput. Fluids 173, 273–284 (2018)
Steih, K.: Reduced basis methods for time-periodic parametric partial differential equations. Ph.D. thesis, Universität Ulm (2014)
Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007)
Temam, R.: Navier-Stokes equations, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York (1979)
Ullmann, S.: Triangular Taylor Hood finite elements, version 1.5. http://www.mathworks.com/matlabcentral/fileexchange/49169. Retrieved: 28 September 2018
Ullmann, S.: POD-Galerkin modeling for incompressible flows with stochastic boundary conditions. Ph.D. thesis, Technische Universität Darmstadt (2014)
Ullmann, S., Rotkvic, M., Lang, J.: POD-Galerkin reduced-order modeling with adaptive finite element snapshots. J. Comput. Phys. 325, 244–258 (2016)
Verfürth, R.: Computational fluid dynamics, lecture notes summer term 2018. http://www.rub.de/num1/files/lectures/CFD.pdf. Retrieved: 27 September 2018
Verfürth, R.: A posteriori error estimation techniques for finite element methods. Numerical mathematics and scientific computation. OUP Oxford (2013)
Veroy, K., Patera, A.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47(8-9), 773–788 (2005)
Weller, J., Lombardi, E., Bergmann, M., Iollo, A.: Numerical methods for low-order modeling of fluid flows based on POD. Int. J. Numer. Methods Fluids 63(2), 249–268 (2010)
Yano, M.: A minimum-residual mixed reduced basis method: exact residual certification and simultaneous finite-element reduced-basis refinement. ESAIM: Mathematical Modelling and Numerical Analysis 50(1), 163–185 (2016)
Yano, M.: A reduced basis method with an exact solution certificate and spatio-parameter adaptivity: application to linear elasticity, pp. 55–76. Springer International Publishing (2017)
Yano, M.: A reduced basis method for coercive equations with an exact solution certificate and spatio-parameter adaptivity: energy-norm and output error bounds. SIAM J. Sci. Comput. 40(1), A388–A420 (2018)
Funding
Carmen Gräßle and Michael Hinze were financially supported by the DFG through the priority program SPP 1962. Jens Lang was supported by the German Research Foundation within the collaborative research center TRR154 “Mathematical Modeling, Simulation and Optimization Using the Example of Gas Networks” (DFG-SFB TRR154/2-2018, TP B01). The work of Sebastian Ullmann was supported by the Excellence Initiative of the German federal and state governments and the Graduate School of Computational Engineering at the Technische Universität Darmstadt.
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Communicated by: Anthony Nouy
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Gräßle, C., Hinze, M., Lang, J. et al. POD model order reduction with space-adapted snapshots for incompressible flows. Adv Comput Math 45, 2401–2428 (2019). https://doi.org/10.1007/s10444-019-09716-7
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DOI: https://doi.org/10.1007/s10444-019-09716-7
Keywords
- Model order reduction
- Proper orthogonal decomposition
- Adaptive finite element discretization
- Navier-Stokes equations
- Incompressible flow
- Inhomogeneous Dirichlet conditions