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Combining Machine Learning with Computational Fluid Dynamics using OpenFOAM and SmartSim (2402.16196v2)

Published 25 Feb 2024 in cs.LG and physics.flu-dyn

Abstract: Combining ML with computational fluid dynamics (CFD) opens many possibilities for improving simulations of technical and natural systems. However, CFD+ML algorithms require exchange of data, synchronization, and calculation on heterogeneous hardware, making their implementation for large-scale problems exceptionally challenging. We provide an effective and scalable solution to developing CFD+ML algorithms using open source software OpenFOAM and SmartSim. SmartSim provides an Orchestrator that significantly simplifies the programming of CFD+ML algorithms and a Redis database that ensures highly scalable data exchange between ML and CFD clients. We show how to leverage SmartSim to effectively couple different segments of OpenFOAM with ML, including pre/post-processing applications, solvers, function objects, and mesh motion solvers. We additionally provide an OpenFOAM sub-module with examples that can be used as starting points for real-world applications in CFD+ML.

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References (10)
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Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Marić, T., Höpken, J., Mooney, K.G.: The OpenFOAM Technology Primer, v2112 edn. Zenodo, zenodo.org (2021). https://doi.org/10.5281/zenodo.4630596 Maric et al. [2024] Maric, T., Fadeli, M.E., Shao, A., Rigazzi, A., Weiner, A.: openfoam-smartsim (1.0). Zenodo (2024). https://doi.org/10.5281/zenodo.10702885 [5] Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Maric, T., Fadeli, M.E., Shao, A., Rigazzi, A., Weiner, A.: openfoam-smartsim (1.0). Zenodo (2024). https://doi.org/10.5281/zenodo.10702885 [5] Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  2. Marić, T., Höpken, J., Mooney, K.G.: The OpenFOAM Technology Primer, v2112 edn. Zenodo, zenodo.org (2021). https://doi.org/10.5281/zenodo.4630596 Maric et al. [2024] Maric, T., Fadeli, M.E., Shao, A., Rigazzi, A., Weiner, A.: openfoam-smartsim (1.0). Zenodo (2024). https://doi.org/10.5281/zenodo.10702885 [5] Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Maric, T., Fadeli, M.E., Shao, A., Rigazzi, A., Weiner, A.: openfoam-smartsim (1.0). Zenodo (2024). https://doi.org/10.5281/zenodo.10702885 [5] Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. 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  3. Maric, T., Fadeli, M.E., Shao, A., Rigazzi, A., Weiner, A.: openfoam-smartsim (1.0). Zenodo (2024). https://doi.org/10.5281/zenodo.10702885 [5] Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. 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[1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  4. Pitz, R., Daily, J.: Experimental study of combustion in a turbulent free shear layer formed at a rearward facing step. https://doi.org/10.2514/6.1981-106 Shirzadi et al. [2017] Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  5. Shirzadi, M., Mirzaei, P.A., Naghashzadegan, M.: Improvement of k-epsilon turbulence model for cfd simulation of atmospheric boundary layer around a high-rise building using stochastic optimization and monte carlo sampling technique. Journal of Wind Engineering and Industrial Aerodynamics 171, 366–379 (2017) https://doi.org/10.1016/j.jweia.2017.10.005 Edeling et al. [2014] Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  6. Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k-ε𝜀\varepsilonitalic_ε turbulence model. J. Comput. Phys. 258(C), 73–94 (2014) Brunton and Kutz [2019] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  7. Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, ??? (2019). https://doi.org/10.1017/9781108380690 Weiner and Semaan [2023] Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  8. Weiner, A., Semaan, R.: Robust dynamic mode decomposition methodology for an airfoil undergoing transonic shock buffet. AIAA Journal 61(10), 4456–4467 (2023) https://doi.org/10.2514/1.J062546 Liang et al. [2016] Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  9. Liang, F., Shi, R., Mo, Q.: A split-and-merge approach for singular value decomposition of large-scale matrices. Statistics and Its Interface 9(4), 453–459 (2016) https://doi.org/10.4310/SII.2016.v9.n4.a5 Schäfer et al. [1996] Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39 Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
  10. Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: In: Hirschel, E.H. (ed.) Benchmark Computations of Laminar Flow Around a Cylinder, pp. 547–566. Vieweg+Teubner Verlag, Wiesbaden (1996). https://doi.org/10.1007/978-3-322-89849-4_39
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