×

Modeling periodic and non-periodic response of dynamical systems using an efficient Chebyshev-based time-spectral approach. (English) Zbl 1437.76038

Summary: A Chebyshev-based time-spectral method (C-TSM) is developed to model periodic and non-periodic nonlinear dynamical systems. It is shown that the proposed technique can accurately model such problems eliminating the need to use expensive classical dual-timestepping time-accurate integration. Furthermore, for autonomous dynamical systems subjected to single or multiple fundamental frequencies, the C-TSM analysis can be used without the prior knowledge of those frequencies. This offers an apparent advantage over the Fourier-based time-spectral methods. In addition, the current approach lends itself as a very useful tool for transient adjoint-based sensitivity analysis since it greatly reduces the memory requirements by solving and storing time-dependent response at a handful of collocation points instead of storing the entire time-history of the primal solution. The efficacy of the present technique is demonstrated by directly comparing the results with Fourier-based time-spectral, as well as time-accurate methods.

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
76N06 Compressible Navier-Stokes equations
76D17 Viscous vortex flows
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jameson, A., Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, (AIAA Paper 1991-1596 (1991))
[2] Eliasson, P.; Weinerfelt, P., High-order implicit time integration for unsteady turbulent flow simulations, Comput. Fluids, 112, 35-49 (2015) · Zbl 1390.76426
[3] Jameson, A., Evaluation of fully implicit Runge Kutta schemes for unsteady flow calculations, J. Sci. Comput., 73, 2-3, 819-852 (2017) · Zbl 1397.65104
[4] Li, H.; Ekici, K., Revisiting the one-shot method for modeling limit cycle oscillations: extension to two-degree-of-freedom systems, Aerosp. Sci. Technol., 69, 686-699 (2017)
[5] Gopinath, A.; Jameson, A., Application of the time spectral method to periodic unsteady vortex shedding, (AIAA Paper 2006-449 (2006))
[6] Ning, W.; He, L., Computation of unsteady flows around oscillating blades using linear and nonlinear harmonic Euler methods, J. Turbomach., 120, 3, 508-514 (1998)
[7] Hall, K. C.; Thomas, J. P.; Clark, W. S., Computation of unsteady nonlinear flows in cascades using a harmonic balance technique, AIAA J., 40, 5, 879-886 (2002)
[8] He, L.; Ning, W., Efficient approach for analysis of unsteady viscous flows in turbomachines, AIAA J., 36, 11, 2005-2012 (1998)
[9] McMullen, M.; Jameson, A.; Alonso, J., Demonstration of nonlinear frequency domain methods, AIAA J., 44, 7, 1428-1435 (2006)
[10] Jameson, A.; Alonso, J.; McMullen, M., Application of a non-linear frequency domain solver to the Euler and Navier-Stokes equations, (AIAA Paper 2002-120 (2002))
[11] Ekici, K.; Hall, K. C.; Dowell, E. H., Computationally fast harmonic balance methods for unsteady aerodynamic predictions of helicopter rotors, J. Comput. Phys., 227, 12, 6206-6225 (2008) · Zbl 1388.76245
[12] Howison, J.; Ekici, K., Dynamic stall analysis using harmonic balance and correlation-based γ-\( \text{Re}_{\theta t}\) transition models for wind turbine applications, Wind Energy, 18, 12, 2047-2063 (2015)
[13] Howison, J.; Thomas, J.; Ekici, K., Aeroelastic analysis of a wind turbine blade using the harmonic balance method, Wind Energy, 21, 4, 226-241 (2018)
[14] Li, H.; Ekici, K., Improved one-shot approach for modeling viscous transonic limit cycle oscillations, AIAA J., 56, 8, 3138-3152 (2018)
[15] Li, H.; Ekici, K., A novel approach for flutter prediction of pitch-plunge airfoils using an efficient one-shot method, J. Fluids Struct., 82, 651-671 (2018)
[16] Ekici, K.; Hall, K. C., Nonlinear analysis of unsteady flows in multistage turbomachines using harmonic balance, AIAA J., 45, 5, 1047-1057 (2007)
[17] Ekici, K.; Hall, K. C., Nonlinear frequency-domain analysis of unsteady flows in turbomachinery with multiple excitation frequencies, AIAA J., 46, 8, 1912-1920 (2008)
[18] Guédeney, T.; Gomar, A.; Gallard, F.; Sicot, F.; Dufour, G.; Puigt, G., Non-uniform time sampling for multiple-frequency harmonic balance computations, J. Comput. Phys., 236, 317-345 (2013)
[19] Mavriplis, D. J.; Yang, Z., Time spectral method for periodic and quasi-periodic unsteady computations on unstructured meshes, Math. Model. Nat. Phenom., 6, 3, 213-236 (2011) · Zbl 1387.76078
[20] Mundis, N.; Mavriplis, D., Quasi-periodic time spectral method for aeroelastic flutter analysis, (AIAA Paper 2013-0638 (2013))
[21] Attar, P. J., Using pseudotime solution framework in harmonic balance methods for aperiodic problems, AIAA J., 51, 12, 2982-2987 (2013)
[22] Fahroo, F.; Ross, I. M., Direct trajectory optimization by a Chebyshev pseudospectral method, J. Guid. Control Dyn., 25, 1, 160-166 (2002)
[23] Dinu, A. D.; Botez, R. M.; Cotoi, I., Aerodynamic forces approximations using the Chebyshev method for closed-loop aero-servoelasticity studies, Can. Aeronaut. Space J., 51, 4, 167-175 (2005)
[24] Dinu, A. D.; Botez, R.; Cotoi, I., Chebyshev polynomials for unsteady aerodynamic calculations in aeroservoelasticity, J. Aircr., 43, 1, 165-171 (2006)
[25] Khater, A.; Temsah, R.; Hassan, M., A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math., 222, 2, 333-350 (2008) · Zbl 1153.65102
[26] Butcher, E. A.; Bobrenkov, O. A., On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 16, 3, 1541-1554 (2011) · Zbl 1221.65158
[27] Niu, J.; Zheng, L.; Yang, Y.; Shu, C.-W., Chebyshev spectral method for unsteady axisymmetric mixed convection heat transfer of power law fluid over a cylinder with variable transport properties, Int. J. Numer. Anal. Model., 11, 3 (2014) · Zbl 1499.65574
[28] Bayliss, A.; Turkel, E., Mappings and accuracy for Chebyshev pseudo-spectral approximations, J. Comput. Phys., 101, 2, 349-359 (1992) · Zbl 0757.65009
[29] Kosloff, D.; Tal-Ezer, H., A modified Chebyshev pseudospectral method with an \(\mathcal{O}(N - 1)\) time step restriction, J. Comput. Phys., 104, 2, 457-469 (1993) · Zbl 0781.65082
[30] Alexandrescu, A.; Bueno-Orovio, A.; Salgueiro, J. R.; Pérez-García, V. M., Mapped Chebyshev pseudospectral method for the study of multiple scale phenomena, Comput. Phys. Commun., 180, 6, 912-919 (2009) · Zbl 1198.35251
[31] Guo, X.; Zhu, M., Direct trajectory optimization based on a mapped Chebyshev pseudospectral method, Chin. J. Aeronaut., 26, 2, 401-412 (2013)
[32] Im, D. K.; Choi, S.; McClure, J. E.; Skiles, F., Mapped Chebyshev pseudospectral method for unsteady flow analysis, AIAA J., 53, 12, 3805-3820 (2015)
[33] Frankel, J. I., Several symbolic augmented Chebyshev expansions for solving the equation of radiative transfer, J. Comput. Phys., 117, 2, 350-363 (1995) · Zbl 0832.65146
[34] Zhan, L.; Xiong, J.; Liu, F.; Xiao, Z., Fully implicit Chebyshev time-spectral method for general unsteady flows, AIAA J., 56, 11, 4474-4486 (2018)
[35] Im, D. K.; Choi, S., Helicopter rotor flow analysis using mapped Chebyshev pseudospectral method and overset mesh topology, Math. Probl. Eng., 2018 (2018)
[36] Choi, J.-Y.; Im, D. K.; Park, J.; Choi, S., Prediction of dynamic stability using mapped Chebyshev pseudospectral method, Int. J. Aerosp. Eng., 2018 (2018)
[37] Zhan, L., Time spectral and space-time LU-SGS implicit methods for unsteady flow computations (2015), UC Irvine, Ph.D. thesis
[38] Zhan, L.; Xiong, J.; Liu, F., A space-time lower-upper symmetric Gauss-Seidel scheme for the time-spectral method, Int. J. Comput. Fluid Dyn., 30, 5, 337-355 (2016) · Zbl 07516897
[39] Patankar, S., Numerical Heat Transfer and Fluid Flow (1980), Taylor & Francis · Zbl 0521.76003
[40] Spalart, P.; Allmaras, S., A one-equation turbulence model for aerodynamic flows, (AIAA Paper 1992-439 (1992))
[41] Jameson, A.; Schmidt, W.; Turkel, E., Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes, (AIAA Paper 1981-1259 (1981))
[42] Huang, H.; Ekici, K., Stabilization of high-dimensional harmonic balance solvers using time spectral viscosity, AIAA J., 52, 8, 1784-1794 (2014)
[43] Djeddi, R.; Howison, J.; Ekici, K., A fully coupled turbulent low-speed preconditioner for harmonic balance applications, Aerosp. Sci. Technol., 53, 22-37 (2016)
[44] Van Der Weide, E.; Gopinath, A.; Jameson, A., Turbomachinery applications with the time spectral method, (AIAA Paper 2005-4905 (2005))
[45] Thomas, J.; Custer, C.; Dowell, E.; Hall, K., Unsteady flow computation using a harmonic balance approach implemented about the OVERFLOW 2 flow solver, (AIAA Paper 2009-4270 (2009))
[46] Campobasso, M. S.; Baba-Ahmadi, M. H., Analysis of unsteady flows past horizontal axis wind turbine airfoils based on harmonic balance compressible Navier-Stokes equations with low-speed preconditioning, J. Turbomach., 134, 6 (2012)
[47] Zhan, L.; Liu, F.; Papamoschou, D., Fourier time spectral method for subsonic and transonic flows, Acta Mech. Sin., 32, 3, 380-396 (2016) · Zbl 1348.76091
[48] Frankel, J. I., A Galerkin solution to a regularized Cauchy singular integro-differential equation, Q. Appl. Math., 53, 2, 245-258 (1995) · Zbl 0823.65145
[49] Moin, P., Fundamentals of Engineering Numerical Analysis (2010), Cambridge University Press · Zbl 1228.65003
[50] Landon, R., NACA 0012 oscillatory and transient pitching (2000), Aircraft Research Association Ltd: Aircraft Research Association Ltd Bedford (United Kingdom), Tech. Rep.
[51] Da Ronch, A.; McCracken, A. J.; Badcock, K. J.; Widhalm, M.; Campobasso, M., Linear frequency domain and harmonic balance predictions of dynamic derivatives, J. Aircr., 50, 3, 694-707 (2013)
[52] Economon, T. D.; Palacios, F.; Copeland, S. R.; Lukaczyk, T. W.; Alonso, J. J., SU2: an open-source suite for multiphysics simulation and design, AIAA J., 54, 3, 828-846 (2016)
[53] Sekar, W. K., Viscous-inviscid interaction methods for flutter calculations (2006), Technische Universität München, Ph.D. thesis
[54] Howlett, J. T.; Bland, S. R., Calculation of viscous effects on transonic flow for oscillating airfoils and comparisons with experiment (1987), NASA Technical Report 1987-2731
[55] Ekici, K.; Hall, K. C., Harmonic balance analysis of limit cycle oscillations in turbomachinery, AIAA J., 49, 7, 1478-1487 (2011)
[56] Choi, S.; Choi, H.; Kang, S., Characteristics of flow over a rotationally oscillating cylinder at low Reynolds number, Phys. Fluids, 14, 8, 2767-2777 (2002) · Zbl 1185.76086
[57] Yamaleev, N.; Diskin, B.; Nielsen, E., Adjoint-based methodology for time-dependent optimization, (AIAA Paper 2008-5857 (2008))
[58] Ntanakas, G.; Meyer, M., Towards unsteady adjoint analysis for turbomachinery applications, (6th European Conference on Computational Fluid Dynamics (ECFD VI). 6th European Conference on Computational Fluid Dynamics (ECFD VI), Barcelona, Spain (2014)), 1-11
[59] Wang, Q.; Moin, P.; Iaccarino, G., Minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation, SIAM J. Sci. Comput., 31, 4, 2549-2567 (2009) · Zbl 1196.65050
[60] Sen, A.; Towara, M.; Naumann, U., A discrete adjoint version of an unsteady incompressible solver for OpenFOAM using algorithmic differentiation, (6th European Conference on Computational Fluid Dynamics (ECFD VI). 6th European Conference on Computational Fluid Dynamics (ECFD VI), Barcelona, Spain (2014)), 1-10
[61] Hückelheim, J. C.; Müller, J.-D., Checkpointing with time gaps for unsteady adjoint CFD, (Minisci, E.; Vasile, M.; Periaux, J.; Gauger, N. R.; Giannakoglou, K. C.; Quagliarella, D., Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences (2019), Springer International Publishing: Springer International Publishing Cham, Switzerland), 117-130 · Zbl 1493.76076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.