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Sat, 10 Nov 2012 14:20:02 +0200Sat, 10 Nov 2012 14:20:02 +0200Multi-level Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3319
In this paper, we propose multi-level Monte Carlo(MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multi-phase flow and transport. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. We consider two types of ensemble level mixed multiscale finite element methods, (1) the no-local-solve-online ensemble level method (NLSO) and (2) the local-solve-online ensemble level method (LSO). Both mixed multiscale methods use a number of snapshots of the permeability media to generate a multiscale basis.
As a result, in the offline stage, we construct multiple basis functions for
each coarse region where basis functions correspond to different realizations.
In the no-local-solve-online ensemble level method one uses the whole set of pre-computed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method one uses the pre-computed functions to construct a multiscale basis for a particular realization. With this basis the solution corresponding to this
particular realization is approximated in LSO mixed MsFEM. In both approaches
the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to pre-compute a
multiscale basis. We note that LSO approaches share similarities with reduced basis methods [11, 21, 22].
In multi-level Monte Carlo methods ([14, 13]), more accurate (and expensive) forward simulations are run with fewer samples while less accurate(and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations carefully, one can show that MLMC methods can provide better accuracy
at the same cost as MC methods. In our simulations, our goal is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. In particular, we show that NLSO
mixed MsFEM is more accurate compared to LSO mixed MsFEM. Further, we use both approaches in the context of MLMC to speed-up MC
calculations. We present basic aspects of the algorithm and numerical
results for coupled flow and transport in heterogeneous porous media.Y. Efendiev; O. Iliev; C. Kronsbeinreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3319Thu, 11 Oct 2012 14:20:02 +0200A direction splitting approach for incompressible Brinkmann flow
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3318
The direction splitting approach proposed earlier in [6], aiming at the efficient solution of Navier-Stokes equations, is extended and adopted here to solve the Navier-Stokes-Brinkman equations describing incompressible flows in plain and in porous media. The resulting pressure equation is a perturbation of the
incompressibility constrained using a direction-wise factorized operator as proposed in [6]. We prove that this approach is unconditionally stable for the unsteady Navier-Stokes-Brinkman problem. We also provide numerical illustrations of the method's accuracy and efficiency.T. Gornak; J. L. Guermond; O. Iliev; P. D. Minevreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3318Thu, 11 Oct 2012 14:16:40 +0200An overview on the usage of some model reduction approaches for simulations of Li-ion transport in batteries
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2990
In this work, some model reduction approaches for performing simulations
with a pseudo-2D model of Li-ion battery are presented. A full pseudo-2D model of processes in Li-ion batteries is presented following
[3], and three methods to reduce the order of the full model are considered. These are: i) directly reduce the model order using proper
orthogonal decomposition, ii) using fractional time step discretization in order to solve the equations in decoupled way, and iii) reformulation
approaches for the diffusion in the solid phase. Combinations of above
methods are also considered. Results from numerical simulations are presented, and the efficiency and the accuracy of the model reduction approaches are discussed.O. Iliev; A. Latz; J. Zausch; S. Zhangreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2990Thu, 19 Apr 2012 07:47:12 +0200Multiscale Finite Element Coarse Spaces for the Analysis of Linear Elastic Composites
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2988
In this work we extend the multiscale finite element method (MsFEM)
as formulated by Hou and Wu in [14] to the PDE system of linear elasticity.
The application, motivated from the multiscale analysis of highly heterogeneous
composite materials, is twofold. Resolving the heterogeneities on
the finest scale, we utilize the linear MsFEM basis for the construction of
robust coarse spaces in the context of two-level overlapping Domain Decomposition
preconditioners. We motivate and explain the construction
and present numerical results validating the approach. Under the assumption
that the material jumps are isolated, that is they occur only in the
interior of the coarse grid elements, our experiments show uniform convergence
rates independent of the contrast in the Young's modulus within the
heterogeneous material. Elsewise, if no restrictions on the position of the
high coefficient inclusions are imposed, robustness can not be guaranteed
any more. These results justify expectations to obtain coefficient-explicit
condition number bounds for the PDE system of linear elasticity similar to
existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner
and Scheichl [12]. Furthermore, we numerically observe the properties
of the MsFEM coarse space for linear elasticity in an upscaling framework.
Therefore, we present experimental results showing the approximation errors
of the multiscale coarse space w.r.t. the fine-scale solution.M. Buck; O. Iliev; H. Andräreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2988Thu, 19 Apr 2012 07:42:13 +0200A Two-Dimensional Model of the Pressing Section of a Paper Machine Including Dynamic Capillary Effects
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2987
The paper production is a problem with significant importance for the society
and it is a challenging topic for scientific investigations. This study is concerned
with the simulations of the pressing section of a paper machine. A two-dimensional
model is developed to account for the water flow within the pressing zone. Richards’
type equation is used to describe the flow in the unsaturated zone. The dynamic capillary
pressure–saturation relation proposed by Hassanizadeh and co-workers (Hassanizadeh
et al., 2002; Hassanizadeh, Gray, 1990, 1993a) is adopted for the paper
production process.
The mathematical model accounts for the co-existence of saturated and unsaturated
zones in a multilayer computational domain. The discretization is performed
by the MPFA-O method. The numerical experiments are carried out for parameters
which are typical for the production process. The static and dynamic capillary
pressure–saturation relations are tested to evaluate the influence of the dynamic
capillary effect.O. Iliev; G. Printsypar; S. Riefreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2987Thu, 19 Apr 2012 07:39:47 +0200A one-dimensional model of the pressing section of a paper machine including dynamic capillary effects
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2322
This work presents the dynamic capillary pressure model (Hassanizadeh, Gray, 1990, 1993a) adapted for the needs of paper manufacturing process simulations. The dynamic capillary pressure-saturation relation is included in a one-dimensional simulation model for the pressing section of a paper machine. The one-dimensional model is derived from a two-dimensional model by averaging with respect to the vertical direction. Then, the model is discretized by the finite volume method and solved by Newton’s method. The numerical experiments are carried out for parameters typical for the paper layer. The dynamic capillary pressure-saturation relation shows significant influence on the distribution of water pressure. The behaviour of the solution agrees with laboratory experiments (Beck, 1983).O. Iliev; G. Printsypar; S. Riefreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2322Thu, 12 May 2011 14:09:24 +0200Finite volume discretization of equations describing nonlinear diffusion in Li-Ion batteries
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2225
Numerical modeling of electrochemical process in Li-Ion battery is an emerging topic of great practical interest. In this work we present a Finite Volume discretization of electrochemical diffusive processes occurring during the operation of Li-Ion batteries. The system of equations is a nonlinear, time-dependent diffusive system, coupling the Li concentration and the electric potential. The system is formulated at length-scale at which two different types of domains are distinguished, one for the electrolyte and one for the active solid particles in the electrode. The domains can be of highly irregular shape, with electrolyte occupying the pore space of a porous electrode. The material parameters in each domain differ by several orders of magnitude and can be non-linear functions of Li ions concentration and/or the electrical potential. Moreover, special interface conditions are imposed at the boundary separating the electrolyte from the active solid particles. The field variables are discontinuous across such an interface and the coupling is highly non- linear, rendering direct iteration methods ineffective for such problems. We formulate a Newton iteration for an purely implicit Finite Volume discretization of the coupled system. A series of numerical examples are presented for different type of electrolyte/electrode configurations and material parameters. The convergence of the Newton method is characterized both as function of nonlinear material parameters as well as the nonlinearity in the interface conditions.P. Popov; Y. Vutov; S. Margenov; O. Ilievreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2225Tue, 10 Aug 2010 09:23:20 +0200Modeling of species and charge transport in Li-Ion Batteries based on non-equilibrium thermodynamics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2219
In order to improve the design of Li ion batteries the complex interplay of various physical phenomena in the active particles of the electrodes and in the electrolyte has to be balanced. The separate transport phenomena in the electrolyte and in the active particle as well as their coupling due to the electrochemical reactions at the interfaces between the electrode particles and the electrolyte will inuence the performance and the lifetime of a battery. Any modeling of the complex phenomena during the usage of a battery has therefore to be based on sound physical and chemical principles in order to allow for reliable predictions for the response of the battery to changing load conditions. We will present a modeling approach for the transport processes in the electrolyte and the electrodesbased on non-equilibrium thermodynamics and transport theory. The assumption of local charge neutrality, which is known to be valid in concentrated electrolytes, is explicitly used to identify the independent thermodynamic variables and uxes. The theory guarantees strictly positive entropy production. Dierences to other theories will be discussed.A. Latz; J. Zausch; O. Ilievreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2219Wed, 21 Jul 2010 14:31:57 +0200On a numerical subgrid upscaling algorithm for Stokes-Brinkman equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2218
This paper discusses a numerical subgrid resolution approach for solving the Stokes-Brinkman system of equations, which is describing coupled ow in plain and in highly porous media. Various scientic and industrial problems are described by this system, and often the geometry and/or the permeability vary on several scales. A particular target is the process of oil ltration. In many complicated lters, the lter medium or the lter element geometry are too ne to be resolved by a feasible computational grid. The subgrid approach presented in the paper is aimed at describing how these ne details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid. This is done via a systematic and a careful procedure of modifying and updating the coecients of the Stokes-Brinkman system in chosen cells. This numerical subgrid approach is motivated from one side from homogenization theory, from which we borrow the formulations for the so called cell problem, and from the other side from the numerical upscaling approaches, such as Multiscale Finite Volume, Multiscale Finite Element, etc. Results on the algorithm's eciency, both in terms of computational time and memory usage, are presented. Comparison with solutions on full ne grid (when possible) are presented in order to evaluate the accuracy. Advantages and limitations of the considered subgrid approach are discussed.O. Iliev; Z. Lakdawala; V. Starikoviciusreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2218Wed, 21 Jul 2010 14:30:10 +0200Variational multiscale Finite Element Method for flows in highly porous media
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2210
We present a two-scale finite element method for solving Brinkman’s and Darcy’s equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes’ equations byWang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy’s equations. In order to reduce the “resonance error” and to ensure convergence to the global fine solution the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems.O. Iliev; R. Lazarov; J. Willemsreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2210Wed, 21 Jul 2010 14:08:14 +0200Design of pleated filters by computer simulations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2083
Four aspects are important in the design of hydraulic lters. We distinguish between two cost factors and two performance factors. Regarding performance, filter eciencynd lter capacity are of interest. Regarding cost, there are production considerations such as spatial restrictions, material cost and the cost of manufacturing the lter. The second type of cost is the operation cost, namely the pressure drop. Albeit simulations should and will ultimately deal with all 4 aspects, for the moment our work is focused on cost. The PleatGeo Module generates three-dimensional computer models of a single pleat of a hydraulic lter interactively. PleatDict computes the pressure drop that will result for the particular design by direct numerical simulation. The evaluation of a new pleat design takes only a few hours on a standard PC compared to days or weeks used for manufacturing and testing a new prototype of a hydraulic lter. The design parameters are the shape of the pleat, the permeabilities of one or several layers of lter media and the geometry of a supporting netting structure that is used to keep the out ow area open. Besides the underlying structure generation and CFD technology, we present some trends regarding the dependence of pressure drop on design parameters that can serve as guide lines for the design of hydraulic lters. Compared to earlier two-dimensional models, the three-dimensional models can include a support structure.A. Wiegmann; L. Cheng; E. Glatt; O. Iliev; S. Riefreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2083Wed, 29 Apr 2009 12:21:31 +0200A Graph-Laplacian approach for calculating the effective thermal conductivity of complicated fiber geometries
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2004
Abstract. An efficient approach to the numerical upscaling of thermal conductivities of fibrous media, e.g. insulation materials, is considered. First, standard cell problems for a second order elliptic equation are formulated for a proper piece of random fibrous structure, following homogenization theory. Next, a graph formed by the fibers is considered, and a second order elliptic equation with suitable boundary conditions is solved on this graph only. Replacing the boundary value problem for the full cell with an auxiliary problem with special boundary conditions on a connected subdomain of highly conductive material is justified in a previous work of the authors. A discretization on the graph is presented here, and error estimates are provided. The efficient implementation of the algorithm is discussed. A number of numerical experiments is presented in order to illustrate the performance of the proposed method.O. Iliev; R. Lazarov; J. Willemsreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2004Wed, 25 Jun 2008 13:01:53 +0200An efficient approach for upscaling properties of composite materials with high contrast of coefficients
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1987
An efficient approach for calculating the effective heat conductivity for a class of industrial composite materials, such as metal foams, fibrous glass materials, and the like, is discussed. These materials, used in insulation or in advanced heat exchangers, are characterized by a low volume fraction of the highly conductive material (glass or metal) having a complex, network-like structure and by a large volume fraction of the insulator (air). We assume that the composite materials have constant macroscopic thermal conductivity tensors, which in principle can be obtained by standard up-scaling techniques, that use the concept of representative elementary volumes (REV), i.e. the effective heat conductivities of composite media can be computed by post-processing the solutions of some special cell problems for REVs. We propose, theoretically justify, and numerically study an efficient approach for calculating the effective conductivity for media for which the ratio of low and high conductivities satisfies 1. In this case one essentially only needs to solve the heat equation in the region occupied by the highly conductive media. For a class of problems we show, that under certain conditions on the microscale geometry, the proposed approach produces an upscaled conductivity that is O() close to the exact upscaled permeability. A number of numerical experiments are presented in order to illustrate the accuracy and the limitations of the proposed method. Applicability of the presented approach to upscaling other similar problems, e.g. flow in fractured porous media, is also discussed.R. Ewing; O. Iliev; R. Lazarov; I. Rybak; J. Willemsreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1987Wed, 18 Jun 2008 15:28:21 +0200An analysis of one regularization approach for solution of pure Neumann problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1992
In this paper, the analysis of one approach for the regularization of pure Neumann problems for second order elliptical equations, e.g., Poisson’s equation and linear elasticity equations, is presented. The main topic under consideration is the behavior of the condition number of the regularized problem. A general framework for the analysis is presented. This allows to determine a form of regularization term which leads to the “natural” asymptotic of the condition number of the regularized problem with respect to mesh parameter. Some numerical results, which support theoretical analysis are presented as well. The main motivation for the presented research is to develop theoretical background for an efficient and robust implementation of the solver for pure Neumann problems for the linear elasticity equations. Such solvers usually are needed in a number of domain decomposition methods, e.g. FETI. Developed approaches are planed to be used in software, developing in ITWM, e.g. KneeMech simulation software.E. Savenkov; H. Andrä; O. Ilievreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1992Wed, 18 Jun 2008 15:27:25 +0200On two-level preconditioners for flow in porous media
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1975
Two-level domain decomposition preconditioner for 3D flows in anisotropic highly heterogeneous porous media is presented. Accurate finite volume discretization based on multipoint flux approximation (MPFA) for 3D pressure equation is employed to account for the jump discontinuities of full permeability tensors. DD/MG type preconditioner for above mentioned problem is developed. Coarse scale operator is obtained from a homogenization type procedure. The influence of the overlapping as well as the influence of the smoother and cell problem formulation is studied. Results from numerical experiments are presented and discussed.R. Ewing; O. Iliev; R. Lazarov; I. Rybakreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1975Wed, 28 May 2008 10:24:09 +0200On upscaling heat conductivity for a class of industrial problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1974
Calculating effective heat conductivity for a class of industrial problems is discussed. The considered composite materials are glass and metal foams, fibrous materials, and the like, used in isolation or in advanced heat exchangers. These materials are characterized by a very complex internal structure, by low volume fraction of the higher conductive material (glass or metal), and by a large volume fraction of the air. The homogenization theory (when applicable), allows to calculate the effective heat conductivity of composite media by postprocessing the solution of special cell problems for representative elementary volumes (REV). Different formulations of such cell problems are considered and compared here. Furthermore, the size of the REV is studied numerically for some typical materials. Fast algorithms for solving the cell problems for this class of problems, are presented and discussed.O. Iliev; I. Rybak; J. Willemsreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1974Wed, 28 May 2008 10:24:01 +0200On approximation property of multipoint flux approximation method
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1973
Approximation property of multipoint flux approximation (MPFA) approach for elliptic equations with discontinuous full tensor coefficients is discussed here. Finite volume discretization of the above problem is presented in the case of jump discontinuities for the permeability tensor. First order approximation for the fluxes is proved. Results from numerical experiments are presented and discussed.O. Iliev; I. Rybakreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1973Wed, 28 May 2008 10:23:55 +0200On numerical upscaling for flows in heterogeneous porous media
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1972
A numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of auxil- iary problems in grid blocks and formal upscaling of the obtained re sults to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous multiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage as MPFA as a discretiza- tion method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. The main aim of this paper is to compare NU with the MsFEM. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied.O. Iliev; I. Rybakreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1972Wed, 28 May 2008 10:23:47 +0200On parallel numerical algorithms for simulating industrial filtration problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1968
The performance of oil filters used in the automotive industry can be significantly improved, especially when computer simulation is an essential component of the design process. In this paper, we consider parallel numerical algorithms for solving mathematical models describing the process of filtration, filtering out solid particles from liquid oil. The Navier-Stokes-Brinkmann system of equations is used to describe the laminar flow of incompressible isothermal oil. The space discretization in the complicated filter geometry is based on the finite-volume method. Special care is taken for an accurate approximation of velocity and pressure on the interface between the fluid and the porous media. The time discretization used here is a proper modification of the fractional time step discretization (cf. Chorin scheme) of the Navier-Stokes equations, where the Brinkmann term is considered at both, prediction and correction substeps. A data decomposition method is used to develop a parallel algorithm, where the domain is distributed among processors by using a structured reference grid. The MPI library is used to implement the data communication part of the algorithm. A theoretical model is proposed for the estimation of the complexity of the given parallel algorithm and a scalability analysis is done on the basis of this model. Results of computational experiments are presented, and the accuracy and efficiency of the parallel algorithm is tested on real industrial geometries.R. Ciegis; O. Iliev; Z. Lakdawalareporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1968Wed, 28 May 2008 10:23:18 +0200Numerical study of two-grid preconditioners for 1d elliptic problems with highly oscillating discontinuous coefficients
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1964
Abstract — Various advanced two-level iterative methods are studied numerically and compared with each other in conjunction with finite volume discretizations of symmetric 1-D elliptic problems with highly oscillatory discontinuous coefficients. Some of the methods considered rely on the homogenization approach for deriving the coarse grid operator. This approach is considered here as an alternative to the well-known Galerkin approach for deriving coarse grid operators. Different intergrid transfer operators are studied, primary consideration being given to the use of the so-called problemdependent prolongation. The two-grid methods considered are used as both solvers and preconditioners for the Conjugate Gradient method. The recent approaches, such as the hybrid domain decomposition method introduced by Vassilevski and the globallocal iterative procedure proposed by Durlofsky et al. are also discussed. A two-level method converging in one iteration in the case where the right-hand side is only a function of the coarse variable is introduced and discussed. Such a fast convergence for problems with discontinuous coefficients arbitrarily varying on the fine scale is achieved by a problem-dependent selection of the coarse grid combined with problem-dependent prolongation on a dual grid. The results of the numerical experiments are presented to illustrate the performance of the studied approaches.O. Iliev; R. Lazarov; J. Willemsreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1964Wed, 28 May 2008 10:22:58 +0200On 3D Numerical Simulations of Viscoelastic Fluids
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1743
D. Niedziela; O. Iliev; A. Latzreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1743Wed, 31 May 2006 14:03:04 +0200On Efficent Simulation of Non-Newtonian Flow in Saturated Porous Media with a Multigrid Adaptive Refinement Solver
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1683
Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Efficient solution of the resulting second order nonlinear elliptic equation is discussed here. The equation is discretized by a finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. A special implementation approach is used, which allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here, as an essential part of the accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid - adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.W. Dörfler; O. Iliev; D. Stoyanov; D. Vassilevareporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1683Wed, 23 Nov 2005 14:29:15 +0100On convergence of certain finite difference discretizations for 1D poroelasticity interface problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1599
Finite difference discretizations of 1D poroelasticity equations with discontinuous coefficients are analyzed. A recently suggested FD discretization of poroelasticity equations with constant coefficients on staggered grid, [5], is used as a basis. A careful treatment of the interfaces leads to harmonic averaging of the discontinuous coefficients. Here, convergence for the pressure and for the displacement is proven in certain norms for the scheme with harmonic averaging (HA). Order of convergence 1.5 is proven for arbitrary located interface, and second order convergence is proven for the case when the interface coincides with a grid node. Furthermore, following the ideas from [3], modified HA discretization are suggested for particular cases. The velocity and the stress are approximated with second order on the interface in this case. It is shown that for wide class of problems, the modified discretization provides better accuracy. Second order convergence for modified scheme is proven for the case when the interface coincides with a displacement grid node. Numerical experiments are presented in order to illustrate our considerations.R. Ewing; O. Iliev; R. Lazarov; A. Naumovichreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1599Mon, 31 Jan 2005 16:32:30 +0100On numerical solution of 1-D poroelasticity equations in a multilayered domain
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1579
In soil mechanics assumption of only vertical subsidence is often invoked and this leads to the one-dimensional model of poroelasticity. The classical model of linear poroelasticity is obtained by Biot [1], detailed derivation can be found e.g., in [2]. This model is applicable also to modelling certain processes in geomechanics, hydrogeology, petroleum engineering (see, e.g., [3, 8], in biomechanics (e.g., [9, 10]), in filtration (e.g., filter cake formation, see [15, 16, 17]), in paper manufacturing (e.g., [11, 12]), in printing (e.g., [13]), etc. Finite element and finite difference methods were applied by many authors for numerical solution of the Biot system of PDEs, see e.g. [3, 4, 5] and references therein. However, as it is wellknown, the standard FEM and FDM methods are subject to numerical instabilities at the first time steps. To avoid this, discretization on staggered grid was suggested in [4, 5]. A single layer deformable porous medium was considered there. This paper can be viewed as extension of [4, 5] to the case of multilayered deformable porous media. A finite volume discretization to the interface problem for the classical one-dimensional Biot model of consolidation process is applied here. Following assumptions are supposed to be valid: each of the porous layers is composed of incompressible solid matrix, it is homogeneous and isotropic. Furthermore, one of two following assumptions is valid: porous medium is not completely saturated and ﬂuid is incompressible or porous medium is completely saturated and fluid is slightly compressible. The reminder of the paper is organised as follows. Next section presents the mathematical model. Third section is devoted to the dicsretization of the continuous problem. Fourth section contains the results from the numerical experiments.F. Gaspar; O. Iliev; F. Lisbona; A. Naumovich; P. Vabishchevichreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1579Wed, 27 Oct 2004 17:17:21 +0200Fluid structure interaction problems in deformable porous media: Toward permeability of deformable porous media
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1578
In this work the problem of fluid flow in deformable porous media is studied. First, the stationary fluid-structure interaction (FSI) problem is formulated in terms of incompressible Newtonian fluid and a linearized elastic solid. The flow is assumed to be characterized by very low Reynolds number and is described by the Stokes equations. The strains in the solid are small allowing for the solid to be described by the Lame equations, but no restrictions are applied on the magnitude of the displacements leading to strongly coupled, nonlinear fluid-structure problem. The FSI problem is then solved numerically by an iterative procedure which solves sequentially fluid and solid subproblems. Each of the two subproblems is discretized by finite elements and the fluid-structure coupling is reduced to an interface boundary condition. Several numerical examples are presented and the results from the numerical computations are used to perform permeability computations for different geometries.O. Iliev; A. Mikelic; P. Popovreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1578Wed, 27 Oct 2004 17:16:34 +0200