zbMATH — the first resource for mathematics

Gradient-consistent enrichment of finite element spaces for the DNS of fluid-particle interaction. (English) Zbl 1453.76096
Summary: We present gradient-consistent enriched finite element spaces for the simulation of free particles in a fluid. This involves forces being exchanged between the particles and the fluid at the interface. In an earlier work [ibid. 393, 186–213 (2019; Zbl 1452.76117)] we derived a monolithic scheme which includes the interaction forces into the Navier-Stokes equations by means of a fictitious domain like strategy. Due to an inexact approximation of the interface oscillations of the pressure along the interface were observed. In multiphase flows oscillations and spurious velocities are a common issue. The surface force term yields a jump in the pressure and therefore the oscillations are usually resolved by extending the spaces on cut elements in order to resolve the discontinuity. For the construction of the enriched spaces proposed in this paper we exploit the Petrov-Galerkin formulation of the vertex-centered finite volume method (PG-FVM), as already investigated in [loc. cit.]. From the perspective of the finite volume scheme we argue that wrong discrete normal directions at the interface are the origin of the oscillations. The new perspective of normal vectors suggests to look at gradients rather than values of the enriching shape functions. The crucial parameter of the enrichment functions therefore is the gradient of the shape functions and especially the one of the test space. The distinguishing feature of our construction therefore is an enrichment that is based on the choice of shape functions with consistent gradients. These derivations finally yield a fitted scheme for the immersed interface. We further propose a strategy ensuring a well-conditioned system independent of the location of the interface. The enriched spaces can be used within any existing finite element discretization for the Navier-Stokes equation. Our numerical tests were conducted using the PG-FVM. We demonstrate that the enriched spaces are able to eliminate the oscillations.

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
Full Text: DOI
[1] 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
[2] Ausas, R.; Buscaglia, G.; Idelsohn, S., A new enrichment space for the treatment of discontinuous pressures in multi-fluid flows, Int. J. Numer. Methods Fluids, 70, 829-850 (2012) · Zbl 1412.76059
[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] Becker, R.; Burman, E.; Hansbo, P., A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Eng., 198, 3352-3360 (2009) · Zbl 1230.74169
[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., Finite volume methods (2010), Private communication
[9] Chen, L.; Wei, H.; Wen, M., An interface-fitted mesh generator and virtual element methods for elliptic interface problems, J. Comput. Phys., 334, 327-348 (2017) · Zbl 1380.65400
[10] Chessa, J.; Belytschko, T., An extended finite element method for two-phase fluids, J. Appl. Mech., 70, 12-17 (2003) · Zbl 1110.74391
[11] Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152, 457-492 (1999) · Zbl 0957.76052
[12] Ginzburg, I.; Wittum, G., Two-phase flows on interface refined grids modeled with VOF, staggered finite volumes, and spline interpolants, J. Comput. Phys., 166, 302-335 (2001) · Zbl 1030.76035
[13] Glowinski, R., Finite element methods for incompressible viscous flow, (Handbook of Numerical Analysis IX (2003), North-Holland: North-Holland Amsterdam), 3-1176 · 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] Gross, S.; Reusken, A., An extended pressure finite element space for two-phase incompressible flows with surface tension, J. Comput. Phys., 224, 1, 40-58 (2007) · Zbl 1261.76015
[17] Gross, S.; Ludescher, T.; Olshanskii, M.; Reusken, A., Robust preconditioning for XFEM applied to time-dependent Stokes problems, SIAM J. Sci. Comput., 38, 6, A3492-A3514 (2016) · Zbl 1398.76106
[18] Hackbusch, W., On first and second order box schemes, Computing, 41, 4, 277-296 (1989) · Zbl 0649.65052
[19] Hansbo, A.; Hansbo, P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., 191, 47-48, 5537-5552 (2002) · Zbl 1035.65125
[20] Hansbo, P.; Hermansson, J.; Svedberg, T., Nitsche’s method combined with space-time finite elements for ALE fluid-structure interaction problems, Comput. Methods Appl. Mech. Eng., 193, 39-41 spec. iss., 4195-4206 (2004) · Zbl 1175.74082
[21] Hansbo, P.; Larson, M. G.; Zahedi, S., A cut finite element method for a Stokes interface problem, Appl. Numer. Math., 85, 90-114 (2014) · Zbl 1299.76136
[22] Höllbacher, S., Voll gekoppelte Modellierung zur direkten numerischen Simulation partikulärer Fluide (2016), Universität Frankfurt, PhD thesis
[23] Höllbacher, S.; Wittum, G., Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction, J. Comput. Phys., 393, 186-213 (2019)
[24] 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
[25] Kirchhart, M.; Gross, S.; Reusken, A., Analysis of an XFEM discretization for Stokes interface problems, SIAM J. Sci. Comput., 38, 2, A1019-A1043 (2017) · Zbl 1381.76182
[26] 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
[27] Lehrenfeld, C., On a Space-Time Extended Finite Element Method for the Solution of a Class of Two-Phase Mass Transport Problems (2015), Universität Aachen, PhD thesis
[28] Liu, C. R., On partitions of unity property of nodal shape functions: rigid-body-movement reproduction and mass conservation, Int. J. Comput. Methods, 13, 1-13 (2016) · Zbl 1359.65261
[29] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 1, 239-261 (2005) · Zbl 1117.76049
[30] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. Numer. Methods Biomed. Eng., 46, 131-150 (1999) · Zbl 0955.74066
[31] Nägele, S., Mehrgitterverfahren für die inkompressiblen Navier-Stokes Gleichungen im laminaren und turbulenten Regime unter Berücksichtigung verschiedener Stabilisierungsmethoden (2003), Universität: Universität Heidelberg, PhD thesis · Zbl 1136.65116
[32] Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hamb., 36, 9-15 (1971) · Zbl 0229.65079
[33] Oevermann, M.; Scharfenberg, C.; Klein, R., A sharp interface finite volume method for elliptic equations on Cartesian grids, J. Comput. Phys., 228, 5184-5206 (2009) · Zbl 1169.65343
[34] Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 220-252 (1977) · Zbl 0403.76100
[35] Reusken, A., Analysis of an extended pressure finite element space for two-phase incompressible flows, Comput. Vis. Sci., 11, 4-6, 293-305 (2008)
[36] Sauerland, H.; Fries, T., The stable XFEM for two-phase flows, Comput. Fluids, 87, 41-49 (2013) · Zbl 1290.76073
[37] 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, 547-566 (1996) · Zbl 0874.76070
[38] Schweitzer, M. A., Stable enrichment and local preconditioning in the particle-partition of unity method, Numer. Math., 118, 137-170 (2011) · Zbl 1217.65210
[39] ten Cate, A.; Nieuwstad, C. H.; Derksen, J. J.; Akker, H. E.A., Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity, Phys. Fluids, 14, 4012-4025 (2002) · Zbl 1185.76073
[40] Tornberg, T.-K.; Engquist, B., Regularization techniques for numerical approximation of PDES with singularities, J. Sci. Comput., 19, 527-552 (2003) · Zbl 1035.65085
[41] Tornberg, T.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. Comput. Phys., 200, 462-488 (2004) · Zbl 1115.76392
[42] 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
[43] 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
[44] 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
[45] 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
[46] 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
[47] 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
[48] 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
[49] Wang, F.; Xiao, Y.; Xu, J., High-order extended finite element methods for solving interface problems, Numer. Anal., 1-25 (2016)
[50] 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
[51] 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
[52] 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
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.