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A space-time adaptive algorithm to illustrate the lack of collision of a rigid disk falling in an incompressible fluid. (English) Zbl 1476.65225

Summary: A space-time adaptive algorithm to solve the motion of a rigid disk in an incompressible Newtonian fluid is presented, which allows collision or quasi-collision processes to be computed with high accuracy. In particular, we recover the theoretical result proven in [M. Hillairet, Commun. Partial Differ. Equations 32, No. 9, 1345–1371 (2007; Zbl 1221.35279)], that the disk will never touch the boundary of the domain in finite time. Anisotropic, continuous piecewise linear finite elements are used for the space discretization, the Euler scheme for the time discretization. The adaptive criteria are based on a posteriori error estimates for simpler problems.

MSC:

65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)

Citations:

Zbl 1221.35279

Software:

BL2D-V2
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Full Text: DOI

References:

[1] M. Ainsworth, J. Z. Zhu, A. W. Craig and O. C. Zienkiewicz, Analysis of the Zienkiewicz-Zhu a posteriori error estimator in the finite element method, Internat. J. Numer. Methods Engrg. 28 (1989), no. 9, 2161-2174. · Zbl 0716.73082
[2] F. Alauzet and A. Loseille, A decade of progress on anisotropic mesh adaptation for computational fluid dynamics, Comput. Aided Design 72 (2016), 13-39.
[3] F. Alauzet and M. Mehrenberger, \mathbf{P}^1-conservative solution interpolation on unstructured triangular meshes, Internat. J. Numer. Methods Engrg. 84 (2010), no. 13, 1552-1588. · Zbl 1202.76096
[4] Y. C. Chang, T. Y. Hou, B. Merriman and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys. 124 (1996), no. 2, 449-464. · Zbl 0847.76048
[5] T. Coupez, G. Jannoun, N. Nassif, H. C. Nguyen, H. Digonnet and E. Hachem, Adaptive time-step with anisotropic meshing for incompressible flows, J. Comput. Phys. 241 (2013), 195-211. · Zbl 1349.76597
[6] S. Dubuis, Adaptive algorithms for two fluids flows with anisotropic finite elements and order two time discretizations, PhD thesis, Ecole Polytechnique fédérale de Lausanne, 2019.
[7] S. Dubuis and M. Picasso, An adaptive algorithm for the time dependent transport equation with anisotropic finite elements and the Crank-Nicolson scheme, J. Sci. Comput. 75 (2018), no. 1, 350-375. · Zbl 1398.65249
[8] L. P. Franca and S. L. Frey, Stabilized finite element methods. II. The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 99 (1992), no. 2-3, 209-233. · Zbl 0765.76048
[9] R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph and J. Périaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. Comput. Phys. 169 (2001), no. 2, 363-426. · Zbl 1047.76097
[10] C. Grandmont and Y. Maday, Existence de solutions d’un problème de couplage fluide-structure bidimensionnel instationnarie, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 4, 525-530. · Zbl 0924.76022
[11] D. Guignard, F. Nobile and M. Picasso, A posteriori error estimation for the steady Navier-Stokes equations in random domains, Comput. Methods Appl. Mech. Engrg. 313 (2017), 483-511. · Zbl 1439.76022
[12] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1345-1371. · Zbl 1221.35279
[13] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl. 9 (1999), no. 2, 633-648. · Zbl 0966.76016
[14] H. H. Hu, N. A. Patankar and M. Y. Zhu, Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys. 169 (2001), no. 2, 427-462. · Zbl 1047.76571
[15] J. A. Janela, A. Lefebvre and B. Maury, A penalty method for the simulation of fluid-rigid body interaction, CEMRACS 2004—Mathematics and Applications to Biology and Medicine, ESAIM Proc. 14, EDP Science, Les Ulis (2005), 115-123. · Zbl 1079.76043
[16] V. John and J. Rang, Adaptive time step control for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 9-12, 514-524. · Zbl 1227.76048
[17] P. Laug and H. Borouchaki, The BL2D mesh generator: Beginner’s guide, user’s and programmer’s manual, Technical report RT-0194, Institut National de Recherche en Informatique et Automatique (INRIA), Le Chesnay-Rocquencourt, 1996.
[18] A. Lozinski, M. Picasso and V. Prachittham, An anisotropic error estimator for the Crank-Nicolson method: Application to a parabolic problem, SIAM J. Sci. Comput. 31 (2009), no. 4, 2757-2783. · Zbl 1215.65154
[19] B. Maury, Numerical analysis of a finite element/volume penalty method, SIAM J. Numer. Anal. 47 (2009), no. 2, 1126-1148. · Zbl 1191.65157
[20] S. Micheletti and S. Perotto, Space-time adaptation for purely diffusive problems in an anisotropic framework, Int. J. Numer. Anal. Model. 7 (2010), no. 1, 125-155. · Zbl 1499.65515
[21] S. Micheletti, S. Perotto and M. Picasso, Stabilized finite elements on anisotropic meshes: A priori error estimates for the advection-diffusion and the Stokes problems, SIAM J. Numer. Anal. 41 (2003), no. 3, 1131-1162. · Zbl 1053.65089
[22] M. Picasso, Adaptive finite elements for a linear parabolic problem, Comput. Methods Appl. Mech. Engrg. 167 (1998), no. 3-4, 223-237. · Zbl 0935.65105
[23] M. Picasso, An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: Application to elliptic and parabolic problems, SIAM J. Sci. Comput. 24 (2003), no. 4, 1328-1355. · Zbl 1061.65116
[24] M. Picasso, An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio, Appl. Numer. Math. 54 (2005), no. 3-4, 470-490. · Zbl 1114.76045
[25] J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 (2002), no. 2, 113-147. · Zbl 1018.76012
[26] N. Sauerwald, An adaptive method for solving stokes’ flow around a falling sphere, Master’s thesis, Institute of Mathematics, 2014.
[27] V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near a boundary, Free Boundary Problems (Trento 2002), Internat. Ser. Numer. Math. 147, Birkhäuser, Basel (2004), 313-327. · Zbl 1060.76038
[28] T. Takahashi, Existence of strong solutions for the problem of a rigid-fluid system, C. R. Math. Acad. Sci. Paris 336 (2003), no. 5, 453-458. · Zbl 1044.35062
[29] N. Verdon, A. Lefebvre-Lepot, L. Lobry and P. Laure, Contact problems for particles in a shear flow, Eur. J. Comput. Mech. 19 (2010), no. 5-7, 513-531.
[30] O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337-357. · Zbl 0602.73063
[31] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331-1364. · Zbl 0769.73084
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