zbMATH — the first resource for mathematics

A systematic approach for constructing higher-order immersed boundary and ghost fluid methods for fluid-structure interaction problems. (English) Zbl 1426.76443
Summary: A systematic approach is presented for constructing higher-order immersed boundary and ghost fluid methods for CFD in general, and fluid – structure interaction problems in particular. Such methods are gaining popularity because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid – structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, immersed boundary and ghost fluid methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid – structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the immersed interfaces. In some cases, they are also provably inconsistent at these locations. A methodology is presented in this paper for addressing this issue. It is developed for inviscid flows and prescribed structural motions. For the sake of clarity, but without any loss of generality, this methodology is described in one and two dimensions. However, its extensions to flow-induced structural motions and three dimensions are straightforward. The proposed methodology leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. Instead, it accounts for the pattern and properties of a preferred higher-order discretization scheme, and attributes ghost values as to preserve the formal order of spatial accuracy of this scheme. It is illustrated in this paper by its application to various finite difference and finite volume methods. Its impact is also demonstrated by one- and two-dimensional numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of the immersed interfaces.

76M12 Finite volume methods applied to problems in fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Full Text: DOI
[1] Fedkiw, R.P., Coupling an eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method, J. comput. phys., 175, 200-224, (2002) · Zbl 1039.76050
[2] M. Brenk, H.-J. Bungartz, M. Mehl, T. Neckel, Fluid-Structure Interaction on Cartesian Grids: Flow Simulation and Coupling Environment, Lecture Notes in Computational Science and Engineering, vol. 53, first ed., Springer, Berlin, 2006, pp. 233-269. · Zbl 1323.76047
[3] van Loon, R.; Anderson, P.D.; van de Vosse, F.N.; Sherwin, S.J., Comparison of various fluid-structure interaction methods for deformable bodies, Comput. struct., 85, 833-843, (2007)
[4] Cirak, F.; Deiterding, R.; Mauch, S.P., Large-scale fluid-structure interaction simulation of viscoplastic and fracturing thin-shells subjected to shocks and detonations, Comput. struct., 85, 1049-1065, (2007)
[5] Farhat, C.; Lesoinne, M.; Maman, N., Mixed explicit/implicit time integration of coupled aeroelastic problems: three-field formulation, geometric conservation and distributed solution, Int. J. numer. methods fluids, 21, 807-835, (1995) · Zbl 0865.76038
[6] Farhat, C.; Geuzaine, P.; Grandmont, C., The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids, J. comput. phys., 174, 669-694, (2001) · Zbl 1157.76372
[7] Tseng, Y.-H.; Ferziger, J.H., A ghost-cell immersed boundary method for flow in complex geometry, J. comput. phys., 192, 593-623, (2003) · Zbl 1047.76575
[8] Berger, M.J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64-84, (1989) · Zbl 0665.76070
[9] Ghias, R.; Mittal, R.; Dong, H., A sharp interface immersed boundary method for compressible viscous flows, J. comput. phys., 225, 528-553, (2007) · Zbl 1343.76043
[10] 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
[11] Wang, C.W.; Liu, T.G.; Khoo, B.C., A real ghost fluid method for the simulation of multimedium compressible flow, SIAM J. sci. comput., 28, 278-302, (2006) · Zbl 1114.35119
[12] Farhat, C.; Rallu, A.; Shankaran, S., A higher-order generalized ghost fluid method for the poor for the three dimensional two-phase flow computation of underwater implosions, J. comput. phys., 227, 7674-7700, (2008) · Zbl 1269.76073
[13] 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
[14] Lax, P.; Wendroff, B., Systems of conservation laws, Commun. pure appl. math., 13, 217-237, (1960) · Zbl 0152.44802
[15] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[16] Fedkiw, R.P.; Marquina, A.; Merriman, B., An isobaric fix for the overheating problem in multimaterial compressible flows, J. comput. phys., 148, 545-578, (1999) · Zbl 0933.76075
[17] van Leer, B., Towards the ultimate conservative difference scheme V. A second-order sequel to godunov’s method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223
[18] Boris, J.P.; Book, D.L., Flux-corrected transport. I. shasta, a fluid transport algorithm that works, J. comput. phys., 11, 38-69, (1973) · Zbl 0251.76004
[19] Colella, P.; Woodward, P.R., The piecewise parabolic method (ppm) for gas-dynamical simulations, J. comput. phys., 54, 174-201, (1984) · Zbl 0531.76082
[20] Shu, C.-W., High order finite difference and finite volume weno schemes and discontinuous Galerkin methods for CFD, Int. J. comput. fluid dyn., 17, 107-118, (2003) · Zbl 1034.76044
[21] Wang, K.; Rallu, A.; Gerbeau, J.-F.; Farhat, C., Algorithms for interface treatment and load computation in embedded boundary methods for fluid and fluid-structure interaction problems, Int. J. numer. methods fluids, 67, 1175-1206, (2011) · Zbl 1426.76436
[22] Gustafsson, B., High order difference methods for time dependent PDE, Springer series in computational mathematics, vol. 38, (2008), Springer
[23] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM rev., 43, 89-112, (2001) · Zbl 0967.65098
[24] Henshaw, W.D.; Schwendeman, D.W., Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow, J. comput. phys., 216, 744-779, (2006) · Zbl 1220.76052
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.