# zbMATH — the first resource for mathematics

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations. (English) Zbl 1391.76590
Summary: The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 65N99 Numerical methods for partial differential equations, boundary value problems
Full Text:
##### References:
 [1] Peskin, C.S., Flow patterns around heart valves: a numerical method, J. comput. phys., 10, 252-271, (1972) · Zbl 0244.92002 [2] Peskin, C.S., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309 [3] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. rev. fluid mech., 37, 239-261, (2005) · Zbl 1117.76049 [4] Goldstein, D.; Handler, R.; Sirovich, L., Modeling a no-slip flow boundary with an external force field, J. comput. phys., 105, 354-366, (1993) · Zbl 0768.76049 [5] Lai, M.-C.; Peskin, C.S., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. comput. phys., 160, 705-719, (2000) · Zbl 0954.76066 [6] J. Mohd-Yusof, Combined immersed-boundary/B-spline methods for simulations of flow in complex geometries, CTR Annual Research Briefs, NASA Ames/Stanford University, 1997, pp. 317-327. [7] 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, 35-60, (2000) · Zbl 0972.76073 [8] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. comput. phys., 171, 132-150, (2001) · Zbl 1057.76039 [9] Tseng, Y.-H.; Ferziger, J.H., A ghost-cell immersed boundary method for flow in complex geometry, J. comput. phys., 192, 593-623, (2000) · Zbl 1047.76575 [10] Balaras, E., Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Comput. fluids, 33, 375-404, (2004) · Zbl 1088.76018 [11] Yang, J.; Balaras, E., An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. comput. phys., 215, 12-40, (2006) · Zbl 1140.76355 [12] Uhlmann, M., An immersed boundary method with direct forcing for the simulation of particulate flows, J. comput. phys., 209, 448-476, (2005) · Zbl 1138.76398 [13] Su, S.-W.; Lai, M.-C.; Lin, C.-A., An immersed boundary technique for simulating complex flows with rigid boundary, Comput. fluids, 36, 313-324, (2007) · Zbl 1177.76299 [14] Taira, K.; Colonius, T., The immersed boundary method: a projection approach, J. comput. phys., 225, 2118-2137, (2007) · Zbl 1343.76027 [15] M. Uhlmann, First experiments with the simulation of particulate flows, Technical Report No. 1020, CIEMAT, Madrid, Spain, 2003, ISSN: 1135-9420. [16] Kim, D.; Choi, H., Immersed boundary method for flow around an arbitrarily moving body, J. comput. phys., 212, 662-680, (2006) · Zbl 1161.76520 [17] Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392 [18] Engquist, B.; Tornberg, A.-K.; Tsai, R., Discretization of Dirac delta functions in level set methods, J. comput. phys., 207, 28-51, (2005) · Zbl 1074.65025 [19] Smereka, P., The numerical approximation of a delta function with application to level set methods, J. comput. phys., 211, 77-90, (2006) · Zbl 1086.65503 [20] LeVeque, R.J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 1019-1044, (1994) · Zbl 0811.65083 [21] Roma, A.M.; Peskin, C.S.; Berger, M.J., An adaptive version of the immersed boundary method, J. comput. phys., 153, 509-534, (1999) · Zbl 0953.76069 [22] Beyer, R.P.; LeVeque, R.J., Analysis of a one-dimensional model for the immersed boundary method, SIAM J. numer. anal., 29, 332-364, (1992) · Zbl 0762.65052 [23] Liu, C.; Zheng, X.; Sung, C., Preconditioned multigrid methods for unsteady incompressible flows, J. comput. phys., 139, 35-57, (1998) · Zbl 0908.76064 [24] Guilmineau, E.; Queutey, P., A numerical simulation of vortex shedding from an oscillating circular cylinder, J. fluids struct., 16, 773-794, (2002) [25] Leontini, J.S.; Thompson, M.C.; Hourigan, K., The beginning of branching behaviour of vortex-induced vibration during two-dimensional flow, J. fluids struct., 22, 857-864, (2006) [26] Yang, J.; Preidikman, S.; Balaras, E., A strongly coupled embedded-boundary method for fluid-structure interactions of elastically mounted rigid bodies, J. fluids struct., 24, 167-182, (2008)
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.