×

On a coupled discontinuous/continuous Galerkin framework and an adaptive penalty scheme for poroelasticity problems. (English) Zbl 1230.74189

Summary: In this paper a coupled discontinuous/continuous Galerkin framework and an adaptive penalty scheme are proposed to promote the applicability and efficiency of discontinuous Galerkin (DG) methods in modeling practical large-scale poroelasticity problems with popular equal-order linear elements. The idea of this coupled DG/continuous Galerkin (CG) framework is to apply DG elements locally, in which the inefficiency of DG methods due to an explosion in unknowns resulted from a full DG discretization for the whole domain is greatly improved. Moreover, the implementation of this coupled framework is based on a CG nodal-based program, in which additional effort for applying DG methods is simply focusing on breaking CG elements to form interfaces and computing surface stiffness using the same shape functions as CG elements. Such a CG nodal-based implementation for DG methods avoids establishing a new non-nodal based computer program, fits popular commercial finite element codes, and thus greatly promotes the applicability of DG methods for practical applications. More importantly, we observe and report that DG methods with constant penalties seriously slow down fluid diffusion at later times. The proposed adaptive penalty scheme targets to recover normal fluid diffusion rates for later time stages. Three numerical examples including the Mandel problem, a footing consolidation problem typical in civil engineering, and a horizontal oil production well problem under compaction popular in petroleum engineering are presented to demonstrate the good performance of the proposed coupled DG/CG framework and adaptive penalty scheme.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Terzaghi, K., Theoretical soil mechanics, (1942), Wiley New York
[2] Biot, M., General theory of 3-D consolidation, J. appl. phys., 12, 155-169, (1941) · JFM 67.0837.01
[3] Biot M, M.; Willis, D.G., The elastic coefficient of the theory of elasticity consolidation, J. appl. mech., 79, 594-601, (1957)
[4] Rice, J.R.; Cleary, M.P., Some basic stress-diffusion solutions for fluid saturated elastic porous media with compressible constituents, Rev. geophys. space phys., 14, 227-241, (1976)
[5] Mandel, J., Consolidation des sols, Geotechnique, 3, 287-299, (1953)
[6] Gibson, R.E.; Schiffman, R.L.; Pu, S.L., Plane strain and axially symmetric consolidation of a Clay layer on a smooth impervious base, Quart. J. mech. appl. math., 23, 505-520, (1970) · Zbl 0217.23705
[7] Cheng, A.H.D.; Detournay, E., A direct boundary element method for plane strain poroelasticity, Int. J. numer. anal. meth. geomech., 12, 551-572, (1988) · Zbl 0662.73056
[8] Sanhu, S.R.; Wilson, E.L., Finite element analysis of seepage in elastic media, J. engrg. mech. div. amer. soc. civil engrg., 95, 641-652, (1969)
[9] Yokoo, Y.; Yamagata, K.; Nagaoka, H., Variational principles for consolidation, Soils found., 11, 4, 25-36, (1971)
[10] Ghaboussi, J.; Wilson, E.L., Flow of compressible fluid in porous elastic media, Int. J. numer. meth. engrg., 5, 419-442, (1973) · Zbl 0248.76037
[11] Hwang, C.T.; Morgenstern, N.R.; Nurray, D.W., On solutions of plane strain consolidation problems by finite element method, Can. geotech. J., 8, 109-118, (1971)
[12] Vermeer, P.A.; Verruut, A., An accuracy condition for consolidation by finite elements, Int. J. numer. anal. meth. geomech., 5, 1-14, (1981) · Zbl 0456.73060
[13] Reed, M.B., An investigation of numerical errors in the analysis of consolidation by finite elements, Int. J. numer. anal. meth. geomech., 8, 243-2571, (1984) · Zbl 0536.73089
[14] Zienkiewicz, O.C.; Shiomi, T., Dynamic behavior of saturated porous media: the generalized Biot formulation and its numerical solution, Int. J. numer. anal. meth. geomech., 8, 71-96, (1984) · Zbl 0526.73099
[15] Booker, J.R.; Small, J.C., An investigation of the stability of numerical solutions of biot’s equations of consolidation, Int. J. solids struct., 11, 907-917, (1975) · Zbl 0311.73047
[16] Murad, M.A.; Loula, A.F.D., On stability and convergence of finite element approximations of biot’s consolidation problem, J. numer. meth. engrg., 37, 645-667, (1994) · Zbl 0791.76047
[17] Sanhu, R.S.; Lee, S.C.; The, H.L., Special finite element for analysis of soil consolidation, Int. J. numer. anal. meth. geomech., 9, 125-147, (1985)
[18] Turska, I.; Schrefler, R.B.A., On convergence conditions of partitioned solution procedures for consolidation problems, Comput. meth. appl. mech. engrg., 106, 51-93, (1993) · Zbl 0783.76064
[19] Nitsche, J., Über ein variationsprinzip zur Lösung von Dirichlet bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abh. math. univ. Hamburg, 36, 9-15, (1970) · Zbl 0229.65079
[20] Douglas, J.; Dupont, T., Interior penalty procedures for elliptic and parabolic Galerkin methods, Lect. notes phys., 58, 207-216, (1976)
[21] Baker, G.A., Finite element methods for elliptic equations using nonconforming elements, Math. comput., 31, 45-59, (1977) · Zbl 0364.65085
[22] Wheeler, M.F., An elliptic collocation finite element method with interior penalties, SIAM J. numer. anal., 15, 152-161, (1978) · Zbl 0384.65058
[23] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 742-760, (1982) · Zbl 0482.65060
[24] Oden, J.T.; Bubuska, I.; Baumann, C.E., A discontinuous hp finite element method for diffusion problems, J. comput. phys., 146, 491-519, (1998) · Zbl 0926.65109
[25] Riviere, B.; Shaw, S.; Wheeler, M.F.; Whiteman, J.R., Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity, Numer. math., 95, 2, 347-376, (2003) · Zbl 1253.74114
[26] Dawson, C.N.; Sun, S.; Wheeler, M.F., Compatible algorithms for coupled flow and transport, Comput. meth. appl. meth. engrg., 193, 2565-2580, (2004) · Zbl 1067.76565
[27] Liu, R.; Wheeler, M.F.; Dawson, C., A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity, Comput. struct., 87, 141-150, (2009)
[28] Liu, R.; Wheeler, M.F.; Dawson, C.; Dean, R., Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method, Comput. meth. appl. mech. engrg., 198, 912-919, (2009) · Zbl 1229.76053
[29] Cockburn, B.; Shu, C., The local discontinuous Galerkin finite element method for convection – diffusion systems, SIAM J. numer. anal., 35, 2440-2463, (1998) · Zbl 0927.65118
[30] Cockburn, B.; Shu, C., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[31] Baumann, C.E.; Oden, J.T., A discontinuous hp finite element method for the Euler and navier – stokes equations, Int J. numer. meth. fluid., 31, 79-95, (1999) · Zbl 0985.76048
[32] Hansbo, P.; Larson, M.G., Discontinuous Galerkin method for incompressible and nearly incompressible elasticity by nitche’s methods, Computer meth. appl. mech. engrg., 191, 1895-1908, (2002) · Zbl 1098.74693
[33] Wihler, T.P., Locking-free DGFEM for elasticity problems in polygons, IMA J. numer. anal., 24, 45-75, (2004) · Zbl 1057.74046
[34] Noels, L.; Radovitzky, R., A general discontinuous Galerkin method for finite hyperelasticity. formulation numerical applications, J. numer. meth. engrg., 68, 64-97, (2006) · Zbl 1145.74039
[35] Eyck, A.T.; Lew, A., Discontinuous Galerkin methods for nonlinear elasticity, J. numer. meth. engrg., 67, 1204-1243, (2006) · Zbl 1113.74068
[36] Eyck, A.T.; Celiker, F.; Lew, A., Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity, analytical estimates, Comput. meth. appl. mech. engrg., 197, 2989-3000, (2008) · Zbl 1194.74390
[37] A.T. Eyck, A. Lew, Adaptive stabilization strategy for enhanced strain methods in nonlinear elasticity, J. Numer. Meth. Engrg., in press. · Zbl 1183.74270
[38] Wihler, T.P.; FrauenFelder, P.; Schwab, C., Exponential convergence of hp-DGFEM for diffusion problems, Int J. comp. math. appl., 46, 183-205, (2003) · Zbl 1059.65095
[39] Richter, G., The discontinuous Galerkin method with diffusion, Math. comput., 58, 631-643, (1992) · Zbl 0783.65078
[40] RiViere, B.; Wheeler, M.F.; Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. part I, Comput. geosci., 8, 337-360, (1999) · Zbl 0951.65108
[41] Cockburn, B.; Karniadakis, G.E.; Shu, C., Discontinuous Galerkin methods: theory computation and application, (2000), Springer-Verlag
[42] Lewis, R.L.; Schrefler, B.A., The finite element method in the static and dynamic deformation and consolidation of porous media, (1998), Wiley Chichester · Zbl 0935.74004
[43] Ainsworth, M.; Tinsley Oden, J., A posteriori error estimation in finite element analysis, (2000), Wiley New York · Zbl 1008.65076
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.