Papers
Topics
Authors
Recent
Detailed Answer
Quick Answer
Concise responses based on abstracts only
Detailed Answer
Well-researched responses based on abstracts and relevant paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses
Gemini 2.5 Flash
Gemini 2.5 Flash 48 tok/s
Gemini 2.5 Pro 48 tok/s Pro
GPT-5 Medium 26 tok/s Pro
GPT-5 High 19 tok/s Pro
GPT-4o 107 tok/s Pro
Kimi K2 205 tok/s Pro
GPT OSS 120B 473 tok/s Pro
Claude Sonnet 4 37 tok/s Pro
2000 character limit reached

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.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (10)
  1. Zanna, L.: A New Generation of Climate Models. https://medium.com/@lz1955/a-new-generation-of-climate-models-aefd851d47bd. Accessed: 02-19-2024 (2023) Marić et al. [2021] 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 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. [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
  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. 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
  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
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
Lightbulb On Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now