# zbMATH — the first resource for mathematics

Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction. (English) Zbl 1452.76117
Summary: The paper presents a fully-implicit and stable finite element and finite volume scheme for the simulation of freely moving particles in a fluid. The developed method is based on the Petrov-Galerkin formulation of a vertex-centered finite volume method (PG-FVM) on unstructured grids. Appropriate extension of the ansatz and test spaces lead to a formulation comparable to a fictitious domain formulation. The purpose of this work is to introduce a new concept of numerical modeling reducing the mathematical overhead which many other methods require. It exploits the identification of the PG-FVM with a corresponding finite element bilinear form. The surface integrals of the finite volume scheme enable a natural incorporation of the interface forces purely based on the original bilinear operator for the fluid. As a result, there is no need to expand the system of equations to a saddle-point problem. Like for fictitious domain methods the extended scheme treats the particles as rigid parts of the fluid. The distinguishing feature compared to most existing fictitious domain methods is that there is no need for an additional Lagrange multiplier or other artificial external forces for the fluid-solid coupling. Consequently, only one single solve for the derived linear system for the fluid together with the particles is necessary and the proposed method does not require any fractional time stepping scheme to balance the interaction forces between fluid and particles. For the linear Stokes problem we will prove the stability of both schemes. Moreover, for the stationary case the conservation of mass and momentum is not violated by the extended scheme, i.e. conservativity is accomplished within the range of the underlying, unconstrained discretization scheme. The scheme is applicable for problems in two and three dimensions.

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 76M10 Finite element methods applied to problems in fluid mechanics 76T20 Suspensions 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
##### Software:
libGrid; UG4; libDiscretization; pcl; libAlgebra
Full Text:
##### References:
 [1] Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffuse-Interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 1, 139-165 (1998) · Zbl 1398.76051 [2] Apte, S. V.; Martin, M.; Patankar, N. A., A numerical method for fully resolved simulation (FRS) of rigid particle-flow interactions in complex flows, J. Comput. Phys., 228, 8, 2712-2738 (2009) · Zbl 1282.76148 [3] Bank, R. E.; Rose, D. J., Some error estimates for the box method, SIAM J. Numer. Anal., 24, 777-787 (1987) · Zbl 0634.65105 [4] Bey, J., Finite-Volumen- und Mehrgitterverfahren für elliptische Randwertprobleme (1997), Universität: Universität Tübingen, Dissertation · Zbl 0888.65126 [5] Blasco, J.; Calzada, M. C.; Marín, M., A fictitious domain, parallel numerical method for rigid particulate flows, J. Comput. Phys., 228, 20, 7596-7613 (2009) · Zbl 1173.76037 [6] Breugem, W.-P., A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows, J. Comput. Phys., 231, 13, 4469-4498 (2012) · Zbl 1245.76064 [7] Burman, E.; Fernandez, M. A., Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility, Comput. Methods Appl. Mech. Eng., 198, 5-8, 766-784 (2009) · Zbl 1229.76045 [8] Chen, L., Fitine volume methods (2010), Private communication [9] Elliott, C. M.; Stinner, B.; Styles, V.; Welford, R., Numerical computation of advection and diffusion on evolving diffuse interfaces, IMA J. Numer. Anal., 31, 786-812 (2011) · Zbl 1241.65081 [10] Fadlun, E. A.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. Comput. Phys., 161, 1, 35-60 (2000) · Zbl 0972.76073 [11] Feng, J.; Hu, H. H.; Joseph, D. D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid, part 1: sedimentation, J. Fluid Mech., 261, 95-134 (1994) · Zbl 0800.76114 [12] Feng, J.; Hu, H. H.; Joseph, D. D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid, part 2: Couette and Poiseuille flows, J. Fluid Mech., 277, 271-301 (1994) · Zbl 0876.76040 [13] Glowinski, R., Finite element methods for incompressible viscous flow, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis IX (2003), North-Holland: North-Holland Amsterdam) · Zbl 1040.76001 [14] Glowinski, R.; Pan, T.-W.; Hesla, T. I.; Joseph, D. D., A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiph. Flow, 25, 755-794 (1999) · Zbl 1137.76592 [15] Glowinski, R.; Pan, T.-W.; Hesla, T. I.; Joseph, D. D.; Périaux, J., 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, 2, 363-426 (2001) · Zbl 1047.76097 [16] Griffith, M. D.; Leontini, J. S., Sharp interface immersed boundary methods and their application to vortex-induced vibration of a cylinder, J. Fluids Struct., 72, 38-58 (2017) [17] Hackbusch, W., On First and Second Order Box Schemes, Computing, 41, 4, 277-296 (1989) · Zbl 0649.65052 [18] Hu, H. H.; Patankar, N. A.; Zhu, M. Y., Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys., 169, 427-462 (2001) · Zbl 1047.76571 [19] John, V.; Matthies, G., Higher-order finite element discretizations in a benchmark problem for incompressible flows, Int. J. Numer. Methods Fluids, 37, 885-903 (2001) · Zbl 1007.76040 [20] Johnson, A. A.; Tezduyar, T. E., Simulation of multiple spheres falling in a liquid-filled tube, Comput. Methods Appl. Mech. Eng., 134, 3-4, 351-373 (1996) · Zbl 0895.76046 [21] Krause, D.; Kummer, F., An incompressible immersed boundary solver for moving body flows using a cut cell discontinuous Galerkin method, Comput. Fluids, 153, 118-129 (2017) · Zbl 1390.76330 [22] Marsdon, J. E.; Hughes, T. J.R., Mathematical foundations of elasticity, SIAM J. Numer. Anal. (1983) [23] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 1, 239-261 (2005) · Zbl 1117.76049 [24] Nägele, S., Mehrgitterverfahren für die inkompressiblen Navier-Stokes Gleichungen im laminaren und turbulenten Regime unter Berücksichtigung verschiedener Stabilisierungsmethoden (2003), Universität Heidelberg, Dissertation · Zbl 1103.76338 [25] Nägele, S.; Wittum, G., On the influence of different stabilisation methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 224, 100-116 (2007) · Zbl 1117.76040 [26] Pan, T.-W.; Glowinsky, R.; Joseph, D. D.; Bai, R., Direct simulation of the motion of settling ellipsoids in Newtonian fluid, (Fourteenth International Conference on Domain Decomposition Methods (2003)), 119-129 [27] Patankar, N. A., A formulation for fast computations of rigid particulate flows, Annu. Res. Briefs, 185-196 (2001) [28] Patankar, N. A.; Singh, P.; Joseph, D. D.; Glowinski, R.; Pan, T.-W., A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiph. Flow, 26, 9, 1509-1524 (2000) · Zbl 1137.76712 [29] Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 220-252 (1977) · Zbl 0403.76100 [30] Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517 (2002) · Zbl 1123.74309 [31] Prignitz, R.; Bänsch, E., Particulate flows with the subspace projection method, J. Comput. Phys., 260, 249-272 (2014) · Zbl 1349.76254 [32] Reichert, H.; Wittum, G., Solving the Navier-Stokes equations on unstructured grids, (Flow Simulation with High-Performance Computers I (1993)), 38 [33] Reuther, S.; Voigt, A., Incompressible two-phase flows with an inextensible Newtonian fluid interface, J. Comput. Phys., 322, 850-858 (2016) · Zbl 1351.76326 [34] Schäfer, M.; Turek, S., Benchmark computations of laminar flow around a cylinder, (Flow Simulation with High-Performance Computers II. Flow Simulation with High-Performance Computers II, Notes on Numerical Fluid Mechanics, vol. 2 (1996)), 547-566 · Zbl 0874.76070 [35] Sharma, N.; Patankar, N. A., A fast computation technique for the direct numerical simulation of rigid particulate flows, J. Comput. Phys., 205, 2, 439-457 (2005) · Zbl 1087.76533 [36] Teigen, K. E.; Li, X.; Lowengrub, J.; Wang, F.; Voigt, A., A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci., 4, 7, 1009-1037 (2009) · Zbl 1186.35168 [37] Teigen, K. E.; Song, P.; Lowengrub, J.; Voigt, A., A diffuse-interface method for two-phase flows with soluble surfactants, J. Comput. Phys., 230, 2, 375-393 (2011) · Zbl 1428.76210 [38] Tschisgale, S.; Kempe, T.; Fröhlich, J., A non-iterative immersed boundary method for spherical particles of arbitrary density ratio, J. Comput. Phys., 339, 432-452 (2017) · Zbl 1375.76202 [39] Uhlmann, M., An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys., 209, 2, 448-476 (2005) · Zbl 1138.76398 [40] Uhlmann, M.; Dušek, J., The motion of a single heavy sphere in ambient fluid: A benchmark for interface-resolved particulate flow simulations with significant relative velocities, Int. J. Multiph. Flow, 59, 221-243 (2014) [41] Unverdi, S. O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Fluids, 100, 25-37 (1992) · Zbl 0758.76047 [42] Veeramani, C.; Minev, P. D.; Nandakumar, K., A fictitious domain formulation for flows with rigid particles: A non-Lagrange multiplier version, J. Comput. Phys., 224, 2, 867-879 (2007) · Zbl 1123.76069 [43] Vogel, A.; Reiter, S.; Rupp, M.; Nägel, A.; Wittum, G., UG 4: A novel flexible software system for simulating PDE based models on high performance computers, Comput. Vis. Sci., 16, 165-179 (2013) · Zbl 1375.35003 [44] Wachs, A., A DEM-DLM/FD method for direct numerical simulation of particulate flows: Sedimentation of polygonal isometric particles in a Newtonian fluid with collisions, Comput. Fluids, 38, 8, 1608-1628 (2009) · Zbl 1242.76142 [45] Wachs, A.; Hammouti, A.; Vinay, G.; Rahmani, M., Accuracy of finite volume/staggered grid distributed Lagrange multiplier/fictitious domain simulations of particulate flows, Comput. Fluids, 115, 154-172 (2015) · Zbl 1390.76530 [46] Wagner, G. J.; Moës, N.; Liu, W. K.; Belytschko, T., The extended finite element method for rigid particles in Stokes flow, Int. J. Numer. Methods Eng., 51, 293-313 (2001) · Zbl 0998.76054 [47] Wan, D.; Turek, S., Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method, Int. J. Numer. Methods Fluids, 51, 5, 531-566 (2006) · Zbl 1145.76406 [48] Wan, D.; Turek, S., Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows, J. Comput. Phys., 222, 1, 28-56 (2007) · Zbl 1216.76036 [49] Xu, J.; Zou, Q., Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numer. Math., 111, 3, 469-492 (2009) · Zbl 1169.65110 [50] Yang, J.; Stern, F., A non-iterative direct forcing immersed boundary method for strongly-coupled fluid-solid interactions, J. Comput. Phys., 295, 779-804 (2015) · Zbl 1349.76556 [51] Ye, X., On the relationship between finite volume and finite element methods applied to the Stokes equations, Numer. Methods Partial Differ. Equ., 17, 440-453 (2001) · Zbl 1017.76057 [52] Yu, Z.; Shao, X., A direct-forcing fictitious domain method for particulate flows, J. Comput. Phys., 227, 1, 292-314 (2007) · Zbl 1280.76052
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.