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Elemental enriched spaces for the treatment of weak and strong discontinuous fields. (English) Zbl 1439.76075
Summary: This paper presents a finite element that incorporates weak, strong and both weak plus strong discontinuities with linear interpolations of the unknown jumps for the modeling of internal interfaces. The new enriched space is built by subdividing each triangular or tetrahedral element in several standard linear sub-elements. The new degrees of freedom coming from the assembly of the sub-elements can be eliminated by static condensation at the element level, resulting in two main advantages: first, an elemental enrichment instead of a nodal one, which presents an important reduction of the computing time when the internal interface is moving all around the domain and second, an efficient implementation involving minor modifications allowing to reuse existing finite element codes. The equations for the internal interface are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. To improve the continuity of the unknowns on both sides of the elements on which a static condensation is done, a contour integral has been added. These contour integrals named inter-elemental forces can be interpreted as a “do nothing” boundary condition [H. Coppola-Owen and R. Codina, Int. J. Numer. Methods Fluids 66, No. 7, 833–851 (2011; Zbl 05910005)] published in another context, or as the usage of weighting functions that ensure convergence of the approach as proposed by J. C. Simo and M. S. Rifai [Int. J. Numer. Methods Eng. 29, No. 8, 1595–1638 (1990; Zbl 0724.73222)]. A series of numerical tests for scalar unknowns as a simple representation of more general numerical simulations are presented to illustrate the performance of the enriched elemental space.
MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74A50 Structured surfaces and interfaces, coexistent phases
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[2] Hughes, T.; Liu, W.; Zimmermann, T., LagrangianEulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29, 329-349 (1981)
[3] Donea, J.; Huerta, A.; Ponthot, J.; Rodrguez Ferran, A., (Stein, E.; de Borst, R.; Hughes, T., Encyclopedia of Computational Mechanics. Encyclopedia of Computational Mechanics, Arbitrary Lagrangian-Eulerian Methods, vol. 1 (2004), John Wiley & Sons: John Wiley & Sons New York)
[4] Hirt, C.; Amsden, A.; Cook, J., An arbitrary LagrangianEulerian computing method for all flow speeds, J. Comput. Phys., 14, 227-253 (1974)
[5] Cruchaga, M.; Celentano, D.; Tezduyar, T., A moving Lagrangian interface technique for flow computations over fixed meshes, Comput. Methods Appl. Mech. Eng., 11, 525-543 (2001)
[6] Dettmer, W.; Saksono, P.; Peric, D., On a finite element formulation for incompressible newtonian fluid flows on moving domains in the presence of surface tension, Commun. Numer. Methods. Eng., 19, 659-668 (2003)
[7] Baiges, J.; Codina, R.; Coppola-Owen, H., The fixed-mesh ale approach for the numerical simulation of floating solids, Internat. J. Numer. Methods Fluids, 67, 8, 1004-1023 (2011)
[8] Idelsohn, S.; Oñate, E.; Del Pin, F., The particle finite element method a powerful tool to solve incompressible flows with free-surfaces and breaking waves, Int. J. Numer. Methods, 61, 964-989 (2004)
[9] Idelsohn, S.; Mier-Torrecilla, M.; Oñate, E., Multi-fluid flows with the particle finite element method, Comput. Methods Appl. Mech. Engrg., 198, 2750-2767 (2009)
[10] Mier-Torrecilla, M.; Idelsohn, S.; Oñate, E., Advances in the simulation of multi-fluid flows with the particle finite element method. Application to bubble dynamics, Internat. J. Numer. Methods Fluids, 67, 11, 1516-1539 (2011)
[11] Idelsohn, S.; Mier-Torrecilla, M.; Nigro, N.; Oñate, E., On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field, Comput. Mech., 46, 1, 115-124 (2010)
[12] Unverdi, S.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100, 25-37 (1992)
[13] Gueyffier, D.; Lie, J.; Nadim, A.; Scardovelli, R.; Zaleski, S., Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. Comput. Phys., 152, 423-456 (1999)
[14] Popinet, S.; Zaleski, S., A front-tracking algorithm for accurate representation of surface tension, Internat. J. Numer. Methods Fluids, 30, 775-793 (1999)
[15] Hirt, C.; Nichols, B., Volume of fluid (vof) method for the dynamics of free boundaries, J. Comput. Phys., 39, 1, 201-225 (1981)
[16] Kothe, D.; Rider, W.; Mosso, S.; Brock, J.; Hochstein, J., Volume Tracking of interfaces Having Surface Tension in Two and Three dimensions. Technical Report AIAA 96-0859 (1996), AIAA
[17] Cummins, S.; Francois, M.; Kothe, D., Estimating curvature from volume fraction, Comput. Struct., 83, 425-434 (2005)
[18] Adalsteinsson, D.; Sethian, J., A fast level set method for propagating interfaces, J. Comput. Phys., 118, 269-277 (1995)
[19] Sethian, J., Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comput. Phys., 169, 503-555 (2001)
[20] Osher, S. J.; Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 463-502 (2001)
[21] Guermond, J.; Quartapelle, L., A projection fem for variable density incompressible flows, J. Comput. Phys., 165, 167-188 (2000)
[22] Shu, C.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471 (1988)
[23] Jiang, G.; Peng, D., Weighted eno schemes for hamilton-jacobi quations, SIAM J. Sci. Comput., 21, 2126-2144 (2000)
[24] Sweby, P., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21, 995-1011 (1984)
[25] Enright, D.; Fedkiw, R.; Ferziger, J.; Mitchell, I., A hybrid particle level set method for improved interface capturing, J. Comput. Phys., 183, 1, 83-116 (2002)
[26] Marchandise, E.; Remacle, J.; Chevaugeon, N., A quadrature-free discontinuous galerkin method for the level set equation, J. Comput. Phys., 212, 338-357 (2006)
[27] Di Pietro, D.; Lo Forte, S.; Parolini, N., Mass preserving finite element implementations of the level set method, Appl. Numer. Math., 56, 1179-1195 (2006)
[28] Enright, D.; Losasso, F.; Fedkiw, R., A fast and accurate semi-Lagrangian particle level set method, Comput. Struct., 83, 479-490 (2005)
[29] Strain, J., Semi Lagrangian methods for level set equations, J. Comput. Phys., 151, 498-533 (1999)
[30] Strain, J., Tree methods for moving interfaces, J. Comput. Phys., 151, 616-648 (1999)
[31] Brackbill, J.; Kothe, D.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354 (1992)
[32] Löhner, R.; Yang, C.; Oñate, E., On the simulation of flows with violent free surface motion, Comput. Methods Appl. Mech. Engrg., 195, 5597-5620 (2006)
[33] Carrica, P.; Wilson, R.; Stern, F., An unsteady single-phase level set method for viscous free surface flows, Internat. J. Numer. Methods Fluids, 53, 229-256 (2007)
[34] Ganesan, S.; Matthies, G.; Tobiska, L., On spurious velocities in incompressible flow problems with interfaces, Comput. Methods Appl. Mech. Engrg., 196, 1193-1202 (2007)
[35] Ausas, R.; Sousa, F.; Buscaglia, G., An improved finite element space for discontinuous pressures, Comput. Methods Appl. Mech. Engrg., 199, 1019-1031 (2010)
[36] Buscaglia, G.; Ausas, R., Variational formulations for surface tension, capillarity and wetting, Comput. Methods Appl. Mech. Engrg., 200, 45-46, 3011-3025 (2011)
[37] Minev, P. D.; Chen, T.; Nandakumar, K., A finite element technique for multifluid incompressible flow using eulerian grids, J. Comput. Phys., 187, 255-273 (2003)
[38] Chessa, J.; Belytschko, T., An extended finite element method for two-phase fluids, J. Appl. Mech., 70, 10-17 (2003)
[39] Belytschko, T.; Moës, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg., 50, 993-1013 (2001)
[40] Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the generalized finite element method, Comput. Methods Appl. Mech. Engrg., 181, 43-69 (2000)
[41] Duarte, C.; Babuška, I.; Oden, J., Generalized finite element methods for three-dimensional structural mechanics problems, Comput. Struct., 77, 2, 215-232 (2000)
[42] Gross, S.; Reusken, A., An extended pressure finite element space for two-phase incompressible flows with surface tension, J. Comput. Phys., 224, 40-58 (2007)
[43] Sauerland, Henning; Fries, Thomas-Peter, The stable XFEM for two-phase flows, Comput. & Fluids, 87, 41-49 (2013)
[44] Barrett, John W.; Garcke, Harald; Nürnberg, Robert, A stable parametric finite element discretization of two-phase navier-stokes flow, J. Sci. Comput., 63, 1, 78-117 (2015)
[45] Fries, T.; Belytschko, T., The extended/generalized finite element method: An overview of the method and its applications, Internat. J. Numer. Methods Engrg., 84, 253-304 (2010)
[46] Coppola-Owen, H.; Codina, R., Improving eulerian two-phase on finite element approximation with discontinuous gradient pressure shape functions, Internat. J. Numer. Methods Fluids, 49, 1287-1304 (2005)
[47] Gimenez, J.; González, L., An extended validation of the last generation of particle finite element method for free surface flows, J. Comput. Phys., 284, 0, 186-205 (2015)
[48] Ausas, R.; Buscaglia, G.; Idelsohn, S., A new enrichment space for the treatment of discontinuous pressures in multi-fluid flows, Internat. J. Numer. Methods Fluids, 70, 7, 829-850 (2012)
[49] Oliver, J.; Huespe, A.; Samaniego, E., A study on finite elements for capturing strong discontinuities, Internat. J. Numer. Methods Engrg., 56, 2135-2161 (2003)
[50] Oliver, J.; Huespe, A.; Sánchez, P., A comparative study on finite elements for capturing strong discontinuities: e-fem vs x-fem, Comput. Methods Appl. Mech. Engrg., 195, 4732-4752 (2006)
[51] Linder, C.; Armero, F., Finite elements with embedded strong-discontinuities for the modeling of failure of solids, Internat. J. Numer. Methods Engrg., 72, 1391-1433 (2007)
[52] Papanastasiou, T.; Malamataris, N.; Ellwood, K., A new outflow boundary condition, Internat. J. Numer. Methods Fluids, 14, 587-608 (1992)
[53] Behr, M., On the application of slip boundary conditions on curved boundaries, Internat. J. Numer. Methods Fluids, 45, 43-51 (2004)
[54] Coppola-Owen, H.; Codina, R., A free surface finite element model for low froude number mould filling problems on fixed meshes, Internat. J. Numer. Methods Fluids, 66, 833-851 (2011)
[55] Simo, J.; Rifai, M., A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 29, 1595-1638 (1990)
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