zbMATH — the first resource for mathematics

Time-integration for ALE simulations of fluid-structure interaction problems: stepsize and order selection based on the BDF. (English) Zbl 1423.76238
Summary: We present an adaptive algorithm for time integration of fluid-structure integration problems. The method relies on a fully coupled procedure to solve FSI problems in which a naturally GCL-compliant ALE formulation for the finite-element spatial discretization is used. The main originality of the proposed solution procedure is that time integration is performed using automatic order and stepsize selections (hp-adaptivity) based on the Backward Differentiation Formulas (BDF). The stepsize selection is based on a local error estimate, an error controller and a step rejection mechanism. It guarantees that the solution precision is within the user targeted tolerance. The order selection is based on a stability test and a quarantine mechanism. The selection is performed to ensure that no other methods within the family of 0-stable BDF methods would produce a solution of the targeted precision for a larger stepsize (and thus a lower computational time). To improve efficiency, the time integration procedure also relies on a modified Newton method and a predictor. The time adaptive algorithm behaviors and performances are assessed on the vortex-induced translational and rotational vibrations of a square cylinder and on the wake-induced vibrations of 3 cylinders in an in-line arrangement. The algorithm yields substantial CPU time savings (compared to constant stepsize and order integration) while delivering solutions of prescribed accuracies.

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76D05 Navier-Stokes equations for incompressible viscous fluids
74H45 Vibrations in dynamical problems in solid mechanics
Full Text: DOI
[1] Yu, K. R.; Hay, A.; Pelletier, D.; Corbeil-Létourneau, S.; Ghasemi, S.; Etienne, S., Two degrees of freedom vortex-induced vibration responses with zero mass and damping at low Reynolds number, (32nd International Conference on Ocean, (2013), Offshore and Arctic Engineering Nantes, France)
[2] Joly, A.; Etienne, S.; Pelletier, D., Galloping of square cylinders in cross-flow at low Reynolds numbers, J. Fluid Struct., 28, 232-243, (2012)
[3] Hay, A.; Yu, K.; Etienne, S.; Garon, A.; Pelletier, D., High-order temporal accuracy for 3D finite-element ALE flow simulations, Comput. Fluids, 100, 204-217, (2014) · Zbl 1391.76335
[4] Mavriplis, D.; Yang, Z., Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes, J. Comput. Phys., 213, 557-573, (2006) · Zbl 1136.76402
[5] Hairer, E.; Nørsett, S.; Wanner, G., Solving ordinary differential equations, (Nonstiff Problems, second revised ed., vol. 1, (2002), Springer Germany)
[6] Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical solution of initial-value problems in differential-algebraic equations, (1989), Elsevier New-York, New-York · Zbl 0699.65057
[7] Malidi, A.; Dufour, S.; Ndri, D., A study of time integration schemes for the numerical modelling of free surface flows, Internat. J. Numer. Methods Fluids, 48, 1123-1147, (2005) · Zbl 1071.76030
[8] Kay, D.; Gresho, P.; Griffiths, D.; Silverster, D., Adaptive time-stepping for incompressible flow part II: Navier-Stokes equations, SIAM J. Sci. Comput., 32, 1, 111-128, (2010) · Zbl 1410.76293
[9] John, V.; Rang, J., Adaptive time step control for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 199, 514-524, (2010) · Zbl 1227.76048
[10] Nigro, A.; Ghidoni, A.; Rebay, S.; Bassi, F., Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows, Internat. J. Numer. Methods Fluids, 76, 549-574, (2014)
[11] Hay, A.; Etienne, S.; Pelletier, D.; Garon, A., Hp-adaptive time integration based on the BDF for viscous flows, J. Comput. Phys., 291, 151-176, (2015) · Zbl 1349.76217
[12] Hartog, D., Mechanical vibrations, fourth revised ed., (1956), McGraw-Hill New York, USA
[13] Païdoussis, M.; Price, S.; de Langre, E., Fluid-structure interactions: cross-flow induced instabilities, (2011), Cambridge University Press New York, USA · Zbl 1220.74007
[14] Asher, U. M.; Petzold, L. R., Computer methods for ordinary differential equations and differential-algebraic equations, (1998), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, Pennsylvania
[15] Hairer, E.; Wanner, G., Solving ordinary differential equations, volume 2: stiff and differential-algebraic problems, second revised ed., (2002), Springer Germany
[16] Sackinger, P.; Schunk, P.; Rao, R., A Newton-raphson pseudo-solid domain mapping technique for free and moving boundary problems : a finite element implementation, J. Comput. Phys., 125, 83-103, (1996) · Zbl 0853.65138
[17] Petzold, L. R., Differential/algebraic equations are not odes, SIAM J. Sci. Stat. Comp., 3, 367-384, (1982) · Zbl 0482.65041
[18] Lötstedt, P.; Petzold, L., Numerical solution of nonlinear differential equations with algebraic constraints I: convergence results for backward differentiation formulas., Math. Comp., 46, 491-516, (1986) · Zbl 0601.65060
[19] Gear, C. W.; Petzold, L. R., ODE methods for the solution of differential/algebraic systems, SIAM J. Numer. Anal., 21, 716-728, (1984) · Zbl 0557.65053
[20] Gear, C. W., Numerical initial value problems in ordinary differential equations, (1971), Prentice-Hall · Zbl 1145.65316
[21] Gresho, P. M.; Sani, R. L., Incompressible flow and the finite-element method: volume 1, advection-diffusion, (1998), Wiley England, United Kingdom · Zbl 0941.76002
[22] Shampine, L. F.; Gordon, M. K., Computer solution of ordinary differential equations, (1975), W. H. Freeman San Fransisco, California · Zbl 0347.65001
[23] Petzold, L. R., (A Description of DASSL: A Differential/Algebraic System Solver, Tech. Rep., Applied Mathematics Division, (1982), Sandia National Laboratories Livermore, California), SAND82-8637
[24] Skelboe, S., The control of order and steplength for backward differentiation methods, BIT, 17, 91-107, (1977) · Zbl 0361.65062
[25] Gear, C. W.; Tu, K., The effect of variable mesh size on the stability of multistep methods, SIAM J. Numer. Anal., 11, 1025-1043, (1974) · Zbl 0292.65041
[26] Grigorieff, R. D., Stability of multistep-methods on variable grids, Numer. Math., 42, 3, 359-377, (1983) · Zbl 0554.65051
[27] Blevins, R., Flow-induced vibrations, second revised ed., (1990), Van Nostrand Reinhold New York, USA
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.