Numerical solvers for a poromechanic problem with a moving boundary

Daniele Cerroni; Florin Adrian Radu; Paolo Zunino; Daniele Cerroni; Florin Adrian Radu; Paolo Zunino

doi:10.3934/mine.2019.4.824

Mathematics in Engineering

2019, Volume 1, Issue 4: 824-848. doi: 10.3934/mine.2019.4.824

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Research article

Numerical solvers for a poromechanic problem with a moving boundary

1.
MOX, Department of Mathematics, Politecnico di Milano, Italy
2.
PMG, Department of Mathematics, University of Bergen, Norway

Received: 05 March 2019 Accepted: 01 August 2019 Published: 09 October 2019

We study a poromechanic problem in presence of a moving boundary. The poroelastic material is described by means of the Biot model while the moving boundary accounts for the effect of surface erosion of the material. We focus on the numerical approximation of the problem, in the framework of the finite element method. To avoid re-meshing along with the evolution of the boundary, we adopt the cut finite element approach. The main issue of this strategy consists of the ill-conditioning of the finite element matrices in presence of cut elements of small size. We show, by means of numerical experiments and theory, that this issue significantly decreases the performance of the numerical solver. For this reason, we propose a strategy that allows to overcome the illconditioned behavior of the discrete problem. The resulting solver is based on the fixed stress approach, used to iteratively decompose the Biot equations, combined with the ghost penalty stabilization and preconditioning applied to the pressure and displacement sub-problems respectively.
- poromechanics,
- moving boundary,
- fixed-stress approach,
- cut finite elements,
- preconditioning
Citation: Daniele Cerroni, Florin Adrian Radu, Paolo Zunino. Numerical solvers for a poromechanic problem with a moving boundary[J]. Mathematics in Engineering, 2019, 1(4): 824-848. doi: 10.3934/mine.2019.4.824

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Abstract

We study a poromechanic problem in presence of a moving boundary. The poroelastic material is described by means of the Biot model while the moving boundary accounts for the effect of surface erosion of the material. We focus on the numerical approximation of the problem, in the framework of the finite element method. To avoid re-meshing along with the evolution of the boundary, we adopt the cut finite element approach. The main issue of this strategy consists of the ill-conditioning of the finite element matrices in presence of cut elements of small size. We show, by means of numerical experiments and theory, that this issue significantly decreases the performance of the numerical solver. For this reason, we propose a strategy that allows to overcome the illconditioned behavior of the discrete problem. The resulting solver is based on the fixed stress approach, used to iteratively decompose the Biot equations, combined with the ghost penalty stabilization and preconditioning applied to the pressure and displacement sub-problems respectively.

References

[1]	Bause M, Radu FA and Koecher U (2017) Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput Method Appl M 320: 745–768. doi: 10.1016/j.cma.2017.03.017
[2]	Bense VF and Person MA (2008) Transient hydrodynamics within intercratonic sedimentary basins during glacial cycles. during glacial cycles. J Geophys Res-Earth 113.
[3]	Borregales M, Kumar K, Radu FA, et al. (2019) A partially parallel-in-time fixed-stress splitting method for Biot's consolidation model. Comput Math Appl 77: 1466–1478. doi: 10.1016/j.camwa.2018.09.005
[4]	Both JW, Borregales M, Nordbotten JM, et al. (2017) Robust fixed stress splitting for Biot's equations in heterogeneous media. Appl Math Lett 68: 101–108. doi: 10.1016/j.aml.2016.12.019
[5]	Both JW, Kumar K, Nordbotten JM, et al. (2018) Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media. Comput Math Appl 77: 1479–1502.
[6]	Bukač M, Yotov I, Zakerzadeh R, et al. (2015) Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach. Comput Method Appl M 292: 138–170. doi: 10.1016/j.cma.2014.10.047
[7]	Burman E (2010) Ghost penalty. CR Math 348: 1217–1220.
[8]	Burman E, Claus S, Hansbo P, et al. (2015) Cutfem: discretizing geometry and partial differential equations. Int J Numer Meth Eng 104: 472–501. doi: 10.1002/nme.4823
[9]	Burman E and Hansbo P (2012) Fictitious domain finite element methods using cut elements: II. a stabilized Nitsche method. Appl Numer Math 62: 328–341.
[10]	Burman E and Hansbo P (2014) Fictitious domain methods using cut elements: III. a stabilized Nitsche method for Stokes' problem. ESAIM-Math Model Num 48: 859–874.
[11]	Burman E, Hansbo P and Larson M (2018) A cut finite element method with boundary value correction. Math Comput 87: 633–657.
[12]	Burman E and Zunino P (2012) Numerical Approximation of Large Contrast Problems with the Unfitted Nitsche Method. In: Blowey J and Jensen M (Eds.) Frontiers in Numerical Analysis - Durham 2010. (pp. 227-282). Springer Berlin Heidelberg: Berlin, Germany.
[13]	Cheng AHD (2016) Poroelasticity. volume 27, Springer.
[14]	Coussy O (2004) Poromechanics. John Wiley & Sons.
[15]	Dolbow J and Harari I (2009) An efficient finite element method for embedded interface problems. Int J Numer Meth Eng 78: 229–252. doi: 10.1002/nme.2486
[16]	Ern A and Guermond JL (2013) Theory and practice of finite elements. Springer.
[17]	Gaspar FJ and Rodrigo C (2017) On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics. Comput Method Appl M 326: 526–540. doi: 10.1016/j.cma.2017.08.025
[18]	Griebel M, Oeltz D and Schweitzer MA (2003) An algebraic multigrid method for linear elasticity. SIAM J Sci Comput 25: 385–407. doi: 10.1137/S1064827502407810
[19]	Gross S and Reusken A (2011) Numerical methods for two-phase incompressible flows. volume 40 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin.
[20]	Hansbo A and Hansbo P (2002) An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput Method Appl M 191: 5537–5552. doi: 10.1016/S0045-7825(02)00524-8
[21]	Hansbo A and Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Method Appl M 193: 3523–3540. doi: 10.1016/j.cma.2003.12.041
[22]	Hansbo P, Larson MG and Zahedi S (2014) A cut finite element method for a Stokes interface problem. Appl Numer Math 85: 90–114. doi: 10.1016/j.apnum.2014.06.009
[23]	Kim J, Tchelepi HA and Juanes R (2011) Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits. Comput Method Appl M 200: 1591–1606. doi: 10.1016/j.cma.2010.12.022
[24]	Lehrenfeld C and Reusken A (2017) Optimal preconditioners for Nitsche-xfem discretizations of interface problems. Numer Math 135: 313–332. doi: 10.1007/s00211-016-0801-6
[25]	Massing A, Larson MG, Logg A, et al. (2014) A stabilized Nitsche fictitious domain method for the Stokes problem. J Sci Comput 61: 604–628. doi: 10.1007/s10915-014-9838-9
[26]	Mikeli´ c A and Wheeler MF (2013) Convergence of iterative coupling for coupled flow and geomechanics. Computat Geosci 17: 455–461. doi: 10.1007/s10596-012-9318-y
[27]	Nasir O, Fall M, Nguyen ST, et al. (2013) Modeling of the thermo-hydro-mechanical–chemical response of sedimentary rocks to past glaciations. Int J Rock Mech Min 64: 160–174. doi: 10.1016/j.ijrmms.2013.08.002
[28]	Reusken A (2008) Analysis of an extended pressure finite element space for two-phase incompressible flows. Computing and Visualization in Science 11: 293–305. doi: 10.1007/s00791-008-0099-8
[29]	Schott B and Wall WA (2014) A new face-oriented stabilized xfem approach for 2d and 3d incompressible Navier-Stokes equations. Comput Methods Appl M 276: 233–265. doi: 10.1016/j.cma.2014.02.014
[30]	Settari A and Mourits FM (1998) A coupled reservoir and geomechanical simulation system. SPE J 3: 219–226. doi: 10.2118/50939-PA
[31]	Storvik E, Both JW, Kumar K, et al. (2019) On the optimization of the fixed-stress splitting for Biot's equations. Int J Numer Meth Eng 120: 179–194.. doi: 10.1002/nme.6130
[32]	Tuncay K, Park A and Ortoleva P (2000) Sedimentary basin deformation: an incremental stress approach. Tectonophysics 323: 77–104. doi: 10.1016/S0040-1951(00)00095-0
[33]	White JA, Castelletto N and Tchelepi HA (2016) Block-partitioned solvers for coupled poromechanics: A unified framework. Comput Method Appl M 303: 55–74. doi: 10.1016/j.cma.2016.01.008
[34]	Zunino P, Cattaneo L and Colciago CM (2011) An unfitted interface penalty method for the numerical approximation of contrast problems. Appl Numer Math 61: 1059–1076. doi: 10.1016/j.apnum.2011.06.005

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