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On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations. (English) Zbl 1165.65389
Summary: We present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions. This method belongs to the class of Arbitrary Lagrangian-Eulerian (ALE) methods for solving moving boundary problems. Its tools are either (a) separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or (b) domain decomposition and separate mappings of each subdomain using suitable coordinate systems. The latter is shown to be more versatile and general. In both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by Y. Dimakopoulos and J. A. Tsamopoulos [J. Comput. Phys. 192, No. 2, 494–522 (2003; Zbl 1047.76043)]. We shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasi-elliptic set of PDEs fail to produce an acceptable mesh and accurate solutions in such geometries. Furthermore, in contrast to other methods, appropriate boundary conditions and constraints such as, orthogonality of specific mesh lines and prespecified node distributions on them, can be easily implemented along a specific part of the domain or its boundary. Hence, no attractive terms at specific corners or singular points are needed. To increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time, a local mesh refinement technique has been incorporated as well. The method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations. In the first one, the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined, while in the second one the linear and non-linear dynamics of two bubbles in a viscous medium are determined in an acoustic field. The large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein. The governing equations are solved using the finite element/Galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid. These are coupled with an implicit Euler method for time integration or with Arnoldi’s algorithm for normal mode analysis.

MSC:
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Software:
ARPACK; PARDISO
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[1] Khamayseh, A.; Kuprat, A.; Mastin, C.W., Boundary orthogonality in elliptic grid generation, ()
[2] Saksono, P.H.; Peric, D., On finite element modeling of surface tension. variational formulation and applications – part II: dynamic problems, Comput. mech., 38, 251-263, (2006) · Zbl 1176.76073
[3] Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. comput. phys., 39, 201-225, (1981) · Zbl 0462.76020
[4] Unverdi, S.O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. comput. phys., 100, 25-37, (1992) · Zbl 0758.76047
[5] Zacharioudaki, M.; Kouris, C.; Dimakopoulos, Y.; Tsamopoulos, J., A direct comparison between volume and surface tracking methods with a boundary-fitted coordinate transformation and third-order upwinding, J. comput. phys., 227, 1428-1469, (2007) · Zbl 1388.76219
[6] Smith, R.E., Algebraic grid generation, Appl. math. comput., 10-11, 137-170, (1982) · Zbl 0493.65054
[7] ()
[8] Thompson, J.F.; Warsi, Z.U.A.; Mastin, W.C., Boundary-fitted coordinate systems for numerical solutions partial differential equations – a review, J. comput. phys., 47, 1-108, (1982) · Zbl 0492.65011
[9] Ryskin, G.; Leal, L.G., Orthogonal mapping, J. comput. phys., 50, 71-100, (1983) · Zbl 0579.65123
[10] Christodoulou, K.N.; Scriven, L.E., Discretization of free surface flows and another moving boundary problems, J. comput. phys., 99, 39-55, (1992) · Zbl 0743.76050
[11] Tsiveriotis, K.; Brown, R.A., Boundary-conforming mapping applied to computations of highly deformed solidification interface, Int. J. numer. meth. fluids, 14, 981-1003, (1992) · Zbl 0753.76106
[12] Tsiveriotis, K.; Brown, R.A., Solution of free-boundary problems using finite-element/Newton methods and locally refined grids: application to analysis of solidification microstructure, Int. J. numer. meth. fluids, 16, 827-843, (1993) · Zbl 0775.76104
[13] Dimakopoulos, Y.; Tsamopoulos, J.A., A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations, J. comput. phys., 192, 494-522, (2003) · Zbl 1047.76043
[14] Dimakopoulos, Y.; Tsamopoulos, J.A., Transient displacement of a viscoplastic material by air in straight and suddenly constricted tubes, J. non-Newton. fluid mech., 112, 43-75, (2003) · Zbl 1038.76505
[15] Dimakopoulos, Y.; Tsamopoulos, J.A., Transient displacement of a Newtonian fluid by air in straight and suddenly constricted tubes, Phys. fluids, 15, 7, 1973-1991, (2003) · Zbl 1186.76141
[16] Dimakopoulos, Y.; Tsamopoulos, J.A., On the gas-penetration in straight tubes completely filled with a viscoelastic fluid, J. non-Newton. fluid mech., 117, 117-139, (2004) · Zbl 1130.76315
[17] Dimakopoulos, Y.; Tsamopoulos, J.A., Transient displacement of a Newtonian and viscoplastic liquid by air in complex tubes, J. non-Newton. fluid mech., 142, 162-182, (2007) · Zbl 1143.76324
[18] Foteinopoulou, K.; Mavrantzas, V.; Dimakopoulos, Y.; Tsamopoulos, J.A., Numerical simulation of multiple bubbles growing in a Newtonian liquid filament undergoing stretching, Phys. fluids, 18, 4, (2006), 042106(24pp.)
[19] Foteinopoulou, K.; Mavrantzas, V.; Tsamopoulos, J.A., Numerical simulation of bubble growth in Newtonian and viscoelastic filaments undergoing stretching, J. non-Newton. fluid mech., 122, 177-200, (2004) · Zbl 1143.76329
[20] Karapetsas, G.; Tsamopoulos, J.A., Transient squeeze flow of viscoplastic materials, J. non-Newton. fluid mech., 133, 35-56, (2006) · Zbl 1195.76030
[21] Tsamopoulos, J.; Dimakopoulos, Y.; Chatzidai, N.; Karapetsas, G.; Pavlidis, M., Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for their entrapment, J. fluid mech., 601, 123-164, (2008) · Zbl 1151.76602
[22] Karapetsas, G.; Tsamopoulos, J., Steady extrusion of viscoelastic materials from an annular die, J. non-Newton. fluid mech., 154, 136-152, (2008) · Zbl 1293.76031
[23] Carvalho, M.S.; Scriven, L.E., Three-dimensional stability analysis of free surface flows: application to forward deformable roll coating, J. comput. phys., 151, 534-562, (1999) · Zbl 0946.76028
[24] Kallinderis, Y.; Kontzialis, C., A priori mesh quality estimation via direct relation between truncation error and mesh distortion, J. comput. phys., 228, 3, 881-902, (2009) · Zbl 1158.65015
[25] Sackinger, P.A.; Schunk, P.R.; Rao, R.R., A newton – raphson pseudo-solid domain mapping technique for free and moving boundary problems: a finite element implementation, J. comput. phys., 125, 83-103, (1996) · Zbl 0853.65138
[26] Madasu, S.; Cairncross, R.A., Effect of substrate flexibility on dynamic wetting: a finite element model, Comput. meth. appl. mech. eng., 192, 2671-2702, (2003) · Zbl 1054.74734
[27] Thompson, J.F.; Warsi, Z.U.A.; Mastin, C.W., Numerical grid generation, (1985), North-Holland Amsterdam · Zbl 0598.65086
[28] Thompson, J.F.; Thames, F.C.; Mastin, C.W., Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary 2D bodies, J. comput. phys., 15, 299-319, (1974) · Zbl 0283.76011
[29] Villamizar, V.; Rojas, O.; Mabey, J., Generation of curvilinear coordinates on multiply connected regions with boundary singularities, J. comput. phys., 223, 571-588, (2007) · Zbl 1115.65124
[30] Thompson, J.F., A general 3D elliptic grid generation system on a composite block structure, Comput. meth. appl. mech. eng., 64, 377-411, (1987)
[31] Baker, T.J., Mesh generation: art or science?, Prog. aerosp. sci., 41, 29-63, (2005)
[32] Kim, C., Collapse of spherical bubbles in Maxwell fluids, J. non-Newton. fluid mech., 55, 37-58, (1994)
[33] Brujan, E.A., A first order bubble dynamics in a compressible viscoelastic liquid, J. non-Newton. fluid mech., 84, 83-103, (1999) · Zbl 0949.76083
[34] Papanastasiou, A.C.; Scriven, L.E.; Macosko, C.W., Bubble growth and collapse in viscoelastic liquids analyzed, J. non-Newton. fluid mech., 16, 53-75, (1984) · Zbl 0559.76012
[35] Bousfield, D.W.; Keunings, R.; Denn, M.M., Transient deformation of an inviscid inclusion in a viscoelastic extensional flow, J. non-Newton. fluid mech., 27, 205-221, (1988)
[36] Bjerknes, V.F.K., Fields of force, (1906), Columbia University Press
[37] V.F.K. Bjerknes, Die Craftfelder, Vieweg, 1909.
[38] Parlitz, U.; Mettin, R.; Luther, S.; Akhatov, I.; Voss, M.; Lauterborn, W., Spatio-temporal dynamics of acoustic cavitation bubble clouds, Phil. trans. R. soc. lond., 357, 313-334, (1999)
[39] Plesset, M.S.; Prosperetti, A., Bubble dynamics and cavitation, Annu. rev. fluid mech., 9, 145-185, (1977) · Zbl 0418.76074
[40] Doinikov, A.A.; Zavtrak, S.T., On the mutual interaction of two gas bubbles in a sound field, Phys. fluids, 7, 8, 1923-1930, (1995) · Zbl 1032.76674
[41] Pelekasis, N.A.; Tsamopoulos, J.A., Bjerknes forces between two bubbles. part 1. response to a step change in pressure, J. fluid mech., 254, 467-499, (1993) · Zbl 0780.76014
[42] Pelekasis, N.A.; Tsamopoulos, J.A., Bjerknes forces between two bubbles. part 2. response to an oscillatory pressure field, J. fluid mech., 254, 501-527, (1993) · Zbl 0780.76014
[43] Phan-Thien, N.; Tanner, R.I., A new constitutive equation derived from network theory, J. non-Newton. fluid mech., 2, 353-365, (1977) · Zbl 0361.76011
[44] Phan-Thien, N.; Tanner, R.I., A nonlinear network viscoelastic model, J. rheol., 22, 259-283, (1978) · Zbl 0391.76010
[45] Brown, R.A.; Szady, M.J.; Northey, P.J.; Armstrong, R.C., On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations, Theor. comput. fluid dyn., 5, 77-106, (1993) · Zbl 0785.76039
[46] Papanastasiou, T.C.; Malamataris, N.; Ellwood, K., A new outflow boundary condition, Int. J. numer. meth. fluids, 14, 587-608, (1992) · Zbl 0747.76039
[47] Saltzman, J.; Brackbill, J., Applications and generalizations of variational methods for generating adaptive meshes, (), 865-884 · Zbl 0493.65066
[48] Dvinsky, A.S., Adaptive grid generation from harmonic maps on Riemannian manifold, J. comput. phys., 95, 450-476, (1991) · Zbl 0733.65074
[49] Chung, T.J., Computational fluid dynamics, (2002), Cambridge University Press, pp. 533-580
[50] Szabo, B.; Babuska, I., Finite element analysis, (1991), Wiley New York
[51] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/petrov – galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier – stokes equations, Comput. appl. mech. eng., 32, 199-259, (1982) · Zbl 0497.76041
[52] Zienkiewicz, O.C.; Taylor, R.L., The finite element method, (2000), Butterworth-Heinemann Oxford · Zbl 0991.74002
[53] Codina, R., Comparison of some finite element methods for solving the diffusion – convection – reaction equation, Comput. meth. appl. mech. eng., 156, 185-210, (1998) · Zbl 0959.76040
[54] Gresho, P.M.; Lee, R.L.; Sani, R.L., On the time-dependent solution of the incompressible navier – stokes equations in two and three dimensions, (), 27-81 · Zbl 0446.76034
[55] Fortin, M.; Fortin, A., A new approach for the FEM simulation of viscoelastic flows, J. non-Newton. fluid mech., 32, 295-310, (1989) · Zbl 0672.76010
[56] Cockburn, B., Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws, J. comput. appl. math., 128, 187-204, (2001) · Zbl 0974.65092
[57] Schenk, O.; Gärtner, K., Solving unsymmetric sparse systems of linear equations with PARDISO, J. future generat. comput. syst., 20, 3, 475-487, (2004) · Zbl 1062.65035
[58] Schenk, O.; Gärtner, K., On fast factorization pivoting methods for symmetric indefinite systems, Electron. trans. numer. anal., 23, 158-179, (2006) · Zbl 1112.65022
[59] Jester, W.; Kallinderis, Y., Numerical study of incompressible flow around fixed cylinder pairs, J. fluids struct., 17, 561-577, (2003)
[60] Kallinderis, Y.; Nakajima, K., Finite element method for incompressible viscous flows with adaptive hybrid grids, Aiaa j., 32, 1617-1625, (1994) · Zbl 0815.76043
[61] Bird, R.B.; Armstrong, R.C.; Hassager, O., Dynamics of polymeric liquids, (1987), John Wiley and Sons New York
[62] Miller, C.A.; Scriven, L.E., The oscillations of a fluid droplet immersed in another fluid, J. fluid mech., 32, 417-435, (1968) · Zbl 0169.56203
[63] Lehoucq, R.B.; Sorensen, D.C.; Young, C., ARPACK user’s guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods, (1998), SIAM Philadelphia, PA · Zbl 0901.65021
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