Efficient numerical methods for the Biot poroelasticity system in multilayered domains

  • In this thesis, the quasi-static Biot poroelasticity system in bounded multilayered domains in one and three dimensions is studied. In more detail, in the one-dimensional case, a finite volume discretization for the Biot system with discontinuous coefficients is derived. The discretization results in a difference scheme with harmonic averaging of the coefficients. Detailed theoretical analysis of the obtained discrete model is performed. Error estimates, which establish convergence rates for both primary as well as flux unknowns are derived. Besides, modified and more accurate discretizations, which can be applied when the interface position coincides with a grid node, are obtained. These discretizations yield second order convergence of the fluxes of the problem. Finally, the solver for the solution of the produced system of linear equations is developed and extensively tested. A number of numerical experiments, which confirm the theoretical considerations are performed. In the three-dimensional case, the finite volume discretization of the system involves construction of special interpolating polynomials in the dual volumes. These polynomials are derived so that they satisfy the same continuity conditions across the interface, as the original system of PDEs. This technique allows to obtain such a difference scheme, which provides accurate computation of the primary as well as of the flux unknowns, including the points adjacent to the interface. Numerical experiments, based on the obtained discretization, show second order convergence for auxiliary problems with known analytical solutions. A multigrid solver, which incorporates the features of the discrete model, is developed in order to solve efficiently the linear system, produced by the finite volume discretization of the three-dimensional problem. The crucial point is to derive problem-dependent restriction and prolongation operators. Such operators are a well-known remedy for the scalar PDEs with discontinuous coefficients. Here, these operators are derived for the system of PDEs, taking into account interdependence of different unknowns within the system. In the derivation, the interpolating polynomials from the finite volume discretization are employed again, linking thus the discretization and the solution processes. The developed multigrid solver is tested on several model problems. Numerical experiments show that, due to the proper problem-dependent intergrid transfer, the multigrid solver is robust with respect to the discontinuities of the coefficients of the system. In the end, the poroelasticity system with discontinuous coefficients is used to model a real problem. The Biot model, describing this problem, is treated numerically, i.e., discretized by the developed finite volume techniques and then solved by the constructed multigrid solver. Physical characteristics of the process, such as displacement of the skeleton, pressure of the fluid, components of the stress tensor, are calculated and then presented at certain cross-sections.

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Author:Anna Naumovich
Advisor:Oleg Iliev
Document Type:Doctoral Thesis
Language of publication:English
Year of Completion:2007
Year of first Publication:2007
Publishing Institution:Technische Universität Kaiserslautern
Granting Institution:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2007/05/02
Date of the Publication (Server):2007/05/16
Tag:finite volume method; interface problem; multigrid method; poroelasticity
Faculties / Organisational entities:Kaiserslautern - Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M06 Finite difference methods
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M55 Multigrid methods; domain decomposition
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Lxx Special subfields of solid mechanics / 74L10 Soil and rock mechanics
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Sxx Numerical methods [See also 65-XX, 74G15, 74H15] / 74S10 Finite volume methods
76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX) / 76Sxx Flows in porous media; filtration; seepage / 76S05 Flows in porous media; filtration; seepage
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011