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Luo, P.; Rodrigo, Carmen; Gaspar, F. J.; Oosterlee, C. W.

Multigrid method for nonlinear poroelasticity equations. (English) Zbl 1388.74013

Comput. Vis. Sci. 17, No. 5, 255-265 (2015).
Summary: In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies.

Cited in 8 Documents

MSC:

74B20 Nonlinear elasticity
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S10 Finite volume methods applied to problems in solid mechanics
65N08 Finite volume methods for boundary value problems involving PDEs
35Q74 PDEs in connection with mechanics of deformable solids

Keywords:

multigrid; poroelasticity; coupled system; collocated grid; nonlinearity; heterogeneity; Weibull function

Software:

Wesseling
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References:

[1] Curtis, J.B.: Fractured shale-gas systems. AAPG Bull. 86(11), 1921-1938 (2002)
[2] King, G.E.: Thirty years of gas shale fracturing: what have we learned? In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2010)
[3] Hubbert, M.K., Willis, D.G.: Mechanics of hydraulic fracturing (1972) · Zbl 0979.74018
[4] Coussy, O.: Poromechanics. Wiley, Chichester (2004) · Zbl 1120.74447
[5] Cowin, S.C.: Bone poroelasticity. J. Biomech. 32(3), 217-238 (1999) · doi:10.1016/S0021-9290(98)00161-4
[6] Detournay, E., Cheng, A.H.D.: Fundamentals of poroelasticity. Anal. Des. Methods: Compr. Rock Eng.: Princ. Pract. Proj. 2, 113-171 (1993)
[7] Terzaghi, K.: Theoretical Soil Mechanics. Wiley, New York (1943) · doi:10.1002/9780470172766
[8] Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155-164 (1941) · JFM 67.0837.01 · doi:10.1063/1.1712886
[9] Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182-185 (1955) · Zbl 0067.23603 · doi:10.1063/1.1721956
[10] Showalter, R.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251(1), 310-340 (2000) · Zbl 0979.74018 · doi:10.1006/jmaa.2000.7048
[11] Barucq, H., Madaune-Tort, M., Saint-Macary, P.: On nonlinear biot’s consolidation models. Nonlinear Anal,: Theory Methods Appl. 63(5), e985-e995 (2005) · Zbl 1224.76137 · doi:10.1016/j.na.2004.12.010
[12] Gaspar, F.J., Lisbona, F.J., Matus, P., Tuyen, V.T.K.: Numerical methods for a one-dimensional non-linear Biot’s model. J. Comput. Appl. Math. 293, 62-72 (2016) · Zbl 1329.76231
[13] Gaspar, F.J., Lisbona, F.J., Oosterlee, C.W.: A stabilized difference scheme for deformable porous media and its numerical resolution by multigrid methods. Comput. Vis. Sci. 11(2), 67-76 (2008) · doi:10.1007/s00791-007-0061-1
[14] Gaspar, F.J., Lisbona, F.J., Vabishchevich, P.N.: Staggered grid discretizations for the quasi-static biot’s consolidation problem. Appl. Numer. Math. 56(6), 888-898 (2006) · Zbl 1091.76047 · doi:10.1016/j.apnum.2005.07.002
[15] Brandt, A., Livne, O.E.: Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, vol. 67. SIAM, Philadelphia (2011) · Zbl 1227.65121 · doi:10.1137/1.9781611970753
[16] Briggs, W.L., McCormick, S.F., Henson, V.E.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000) · Zbl 0958.65128
[17] Hackbusch, W.: Multi-Grid Methods and Applications, vol. 4. Springer-Verlag, Berlin (1985) · Zbl 0595.65106
[18] Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic Press, New York (2000) · Zbl 0976.65106
[19] Wesseling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992) · Zbl 0760.65092
[20] Hubbert, M.K.: Darcy’s law and the field equations of the flow of underground fluids. Hydrol. Sci. J. 2(1), 23-59 (1957)
[21] Oosterlee, C.W., Gaspar, F.J.: Multigrid relaxation methods for systems of saddle point type. Appl. Numer. Math. 58(12), 1933-1950 (2008) · Zbl 1148.76040 · doi:10.1016/j.apnum.2007.11.014
[22] Brandt, A., Dinar, N.: Multigrid solutions to elliptic flow problems. Numer. Methods PDE’s 42, 53-147 (1979) · Zbl 0447.76020
[23] Vanka, S.P.: Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comput. Phys. 65(1), 138-158 (1986) · Zbl 0606.76035 · doi:10.1016/0021-9991(86)90008-2
[24] Henson, V.E.: Multigrid methods for nonlinear problems: an overview. In: Electronic Imaging 2003, pp. 36-48. International Society for Optics and Photonics (2003)
[25] Tang, C.: Numerical simulation of progressive rock failure and associated seismicity. Int. J. Rock Mech. Min. Sci. 34(2), 249-261 (1997) · doi:10.1016/S0148-9062(96)00039-3
[26] De Sterck, H., Miller, K., Sanders, G., Winlaw, M.: Recursively accelerated multilevel aggregation for markov chains. SIAM J. Sci. Comput. 32(3), 1652-1671 (2010) · Zbl 1213.65018 · doi:10.1137/090770114
[27] Washio, T., Oosterlee, C.W.: Krylov subspace acceleration for nonlinear multigrid schemes. Electron. Trans. Numer. Anal. 6(271-290), 3 (1997) · Zbl 0903.65096
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