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Multi Scale Finite Element Approximation For Transport In Heterogeneous Porous Media
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Book Synopsis Multi-Scale Finite Element Approximation for Transport in Heterogeneous Porous Media by :
Download or read book Multi-Scale Finite Element Approximation for Transport in Heterogeneous Porous Media written by and published by . This book was released on 2002 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main objective of this study is to develop an efficient multiscale coarse grid method which can be used as a competitive algorithm in studying composite materials and flow transport in strongly heterogeneous porous media. On one hand, we have explored the possibility of using adaptive mesh to reduce the modeling error introduced by the traditional moment average technique. On the other hand, we found that in the case of high aspect ratio permeability tensor, the modeling error in ignoring high order moments (3rd order or higher) could be very large. To overcome this difficulty, we have investigated an alternative approach that uses two-scale homogenization analysis to derive a coarse grid model in a systematic way. Finally, we have made some progress in developing numerical methods to solve multiscale nonlinear stochastic partial differential equations by using Wiener-Chaos expansions. These methods will reduce the problem of solving stochastic PDEs to solving a set of deterministic PDEs. This numerical method can be combined with our multiscale computational method, and can be used to compute accurately high order statistical quantities more efficiently than the traditional Monte-Carlo method.
Book Synopsis Multiscale Finite Element Methods by : Yalchin Efendiev
Download or read book Multiscale Finite Element Methods written by Yalchin Efendiev and published by Springer Science & Business Media. This book was released on 2009-01-10 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this monograph is to describe the main concepts and recent - vances in multiscale ?nite element methods. This monograph is intended for thebroaderaudienceincludingengineers,appliedscientists,andforthosewho are interested in multiscale simulations. The book is intended for graduate students in applied mathematics and those interested in multiscale compu- tions. It combines a practical introduction, numerical results, and analysis of multiscale ?nite element methods. Due to the page limitation, the material has been condensed. Each chapter of the book starts with an introduction and description of the proposed methods and motivating examples. Some new techniques are introduced using formal arguments that are justi?ed later in the last chapter. Numerical examples demonstrating the signi?cance of the proposed methods are presented in each chapter following the description of the methods. In the last chapter, we analyze a few representative cases with the objective of demonstrating the main error sources and the convergence of the proposed methods. A brief outline of the book is as follows. The ?rst chapter gives a general introductiontomultiscalemethodsandanoutlineofeachchapter.Thesecond chapter discusses the main idea of the multiscale ?nite element method and its extensions. This chapter also gives an overview of multiscale ?nite element methods and other related methods. The third chapter discusses the ext- sion of multiscale ?nite element methods to nonlinear problems. The fourth chapter focuses on multiscale methods that use limited global information.
Book Synopsis Finite Element Modeling of Multiscale Transport Phenomena by : Vahid Nassehi
Download or read book Finite Element Modeling of Multiscale Transport Phenomena written by Vahid Nassehi and published by World Scientific. This book was released on 2011 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: Complex multiscale systems such as combined free or porous flow regimes and transport processes governed by combined diffusion, convection and reaction mechanisms, which cannot be readily modeled using traditional methods, can be solved by multiscale or stabilized finite element schemes. Due to the importance of the described multiscale processes in applications such as separation processes, reaction engineering and environmental systems analysis, a sound knowledge of such methods is essential for many researchers and design engineers who wish to develop reliable solutions for industrially relevant problems. The main scope of this book is to provide an authoritative description of recent developments in the field of finite element analysis, with a particular emphasis on the multiscale finite element modeling of transport phenomena and flow problem.
Book Synopsis Multiscale Modeling and Simulation in Science by : Björn Engquist
Download or read book Multiscale Modeling and Simulation in Science written by Björn Engquist and published by Springer Science & Business Media. This book was released on 2009-02-11 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most problems in science involve many scales in time and space. An example is turbulent ?ow where the important large scale quantities of lift and drag of a wing depend on the behavior of the small vortices in the boundarylayer. Another example is chemical reactions with concentrations of the species varying over seconds and hours while the time scale of the oscillations of the chemical bonds is of the order of femtoseconds. A third example from structural mechanics is the stress and strain in a solid beam which is well described by macroscopic equations but at the tip of a crack modeling details on a microscale are needed. A common dif?culty with the simulation of these problems and many others in physics, chemistry and biology is that an attempt to represent all scales will lead to an enormous computational problem with unacceptably long computation times and large memory requirements. On the other hand, if the discretization at a coarse level ignoresthe?nescale informationthenthesolutionwillnotbephysicallymeaningful. The in?uence of the ?ne scales must be incorporated into the model. This volume is the result of a Summer School on Multiscale Modeling and S- ulation in Science held at Boso ¤n, Lidingo ¤ outside Stockholm, Sweden, in June 2007. Sixty PhD students from applied mathematics, the sciences and engineering parti- pated in the summer school.
Book Synopsis Multiscale Mortar Mixed Finite Element Methods for Flow Problems in Highly Heterogeneous Porous Media by : Hailong Xiao
Download or read book Multiscale Mortar Mixed Finite Element Methods for Flow Problems in Highly Heterogeneous Porous Media written by Hailong Xiao and published by . This book was released on 2013 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: We use Darcy's law and conservation of mass to model the flow of a fluid through a porous medium. It is a second order elliptic system with a heterogeneous coefficient. We consider the equations written in mixed form. In the heterogeneous case, we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom. In the case of a locally periodic heterogeneous coefficient of period epsilon, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when epsilon is small. Moreover, we present numerical examples to assess its performance when the coefficient is not obviously locally periodic. We show that the new mortar method works well, and better than polynomial mortar spaces. On the other hand, we also propose to use multiscale mortars as a coarse component to construct a two-level preconditioner for the saddle point linear system arising from the fine scale discretization of the mixed finite element system. The two-level preconditioners are constructed based on the interfaces. We propose a framework to define the interpolation operators for the face based two-level preconditioners for different combination of coarse and fine scale mortar spaces for matching and nonmatching grids. In this dissertation, we show that for quasi-homogeneous problems and matching grids, the condition number of the preconditioned interface operator is bounded by (log(H/h))2, which is the same as the traditional two-level preconditioners, for quasi-homogeneous problems. We show several numerical examples to demonstrate that for the strongly heterogeneous porous media, it is often desirable and even necessary to use a higher dimensional coarse mortar space to construct the coarse preconditioner to achieve convergence. We apply our ideas to study slightly compressible single phase and two-phase flow in a porous medium. We find that for the nonlinear single phase problem, the two-level preconditioners could be successfully applied to the symmetrized linear system. For the two-phase problem, using the fine scale, instead of multiscale, velocity solutions from the flow problem can greatly benefit the transport problem.
Book Synopsis Generalized Discontinuous Multiscale Methods for Flows in Highly Heterogeneous Porous Media by : Minam Moon
Download or read book Generalized Discontinuous Multiscale Methods for Flows in Highly Heterogeneous Porous Media written by Minam Moon and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation is devoted to the development, study and testing of numerical methods for elliptic and parabolic equations with heterogeneous coefficients. The motivation for this study is to meet the need for fast and robust methods for numerical upscaling and simulation of single and multi-phase fluid flow in highly heterogeneous porous media. We consider the multiscale model reduction technique in the framework of the discontinuous Galerkin (DG) and the hybridizable discontinuous Galerkin (HDG) finite element methods. First, we design multiscale finite element methods for second order elliptic equations by applying the symmetric interior penalty discontinuous Galekin finite element method. We propose two different types of finite element spaces on the coarse mesh within DG framework. The first type of spaces is based on a local spectral problem that uses an interior weighted L2-norm and a boundary weighted L2-norm for computing the mass matrix. The second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension. Second, we develop multiscale model reduction methods within the HDG framework. We provide construction of several multiscale finite element spaces (related to the coarse-mesh edges) that guarantee a reasonable approximation on a reduced dimensional space of the numerical traces. In these approaches, we use local snapshot spaces and local spectral decomposition following the concept of Generalized Multiscale Finite Element Methods. We also provide a general framework for systematic construction of multiscale spaces. By using local snapshots we were able to add local features to the solution space and to avoid high dimensional representation of trace spaces. Further, we extend multiscale finite element methods within HDG method to nonlinear and/or time-dependent problems. These extensions demonstrate the potential of the proposed constructions for some advanced and more practical applications. For most of the proposed methods, we investigate their stability and derive error estimates for the approximate solutions. Furthermore we study the performance of all proposed methods on a representative number of numerical examples. In the numerical tests, we use various permeability data of highly heterogeneous porous media and contrasts ranging from 103 to 106. Since the exact solution is in general unknown, we first generate solutions on a very fine mesh and use them as reference solutions in our tests. The numerical results confirm the theoretical study of the accuracy of the proposed methods and their robustness with respect to the media contrast. Our numerical experiments also show that the proposed methods could be implemented in a practical and efficient way. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/155430
Book Synopsis Numerical Analysis of Multiscale Problems by : Ivan G. Graham
Download or read book Numerical Analysis of Multiscale Problems written by Ivan G. Graham and published by Springer Science & Business Media. This book was released on 2012-01-05 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 91st London Mathematical Society Durham Symposium took place from July 5th to 15th 2010, with more than 100 international participants attending. The Symposium focused on Numerical Analysis of Multiscale Problems and this book contains 10 invited articles from some of the meeting's key speakers, covering a range of topics of contemporary interest in this area. Articles cover the analysis of forward and inverse PDE problems in heterogeneous media, high-frequency wave propagation, atomistic-continuum modeling and high-dimensional problems arising in modeling uncertainty. Novel upscaling and preconditioning techniques, as well as applications to turbulent multi-phase flow, and to problems of current interest in materials science are all addressed. As such this book presents the current state-of-the-art in the numerical analysis of multiscale problems and will be of interest to both practitioners and mathematicians working in those fields.
Book Synopsis Multiscale Model Reduction by : Eric Chung
Download or read book Multiscale Model Reduction written by Eric Chung and published by Springer Nature. This book was released on 2023-06-07 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.
Book Synopsis Multi-scale Phenomena in Complex Fluids by : Thomas Y. Hou
Download or read book Multi-scale Phenomena in Complex Fluids written by Thomas Y. Hou and published by World Scientific. This book was released on 2009 with total page 379 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multi-Scale Phenomena in Complex Fluids is a collection of lecture notes delivered during the ªrst two series of mini-courses from "Shanghai Summer School on Analysis and Numerics in Modern Sciences," which was held in 2004 and 2006 at Fudan University, Shanghai, China. This review volume of 5 chapters, covering various fields in complex fluids, places emphasis on multi-scale modeling, analyses and simulations. It will be of special interest to researchers and graduate students who want to work in the field of complex fluids.
Book Synopsis Multiscale Numerical Methods for Partial Differential Equations Using Limited Global Information and Their Applications by : Lijian Jiang
Download or read book Multiscale Numerical Methods for Partial Differential Equations Using Limited Global Information and Their Applications written by Lijian Jiang and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation we develop, analyze and implement effective numerical methods for multiscale phenomena arising from flows in heterogeneous porous media. The main purpose is to develop innovative numerical and analytical methods that can capture the effect of small scales on the large scales without resolving the small scale details on a coarse computational grid. This research activity is strongly motivated by many important practical applications arising in contaminant transport in heterogeneous porous media, oil reservoir simulations and subsurface characterization. In the work, we investigate three main multiscale numerical methods, i.e., multiscale finite element method, partition of unity method and mixed multiscale finite element method. These methods employ limited single or multiple global information. We apply these numerical methods to partial differential equations (elliptic, parabolic and wave equations) with continuum scales. To compute the solution of partial differential equations on a coarse grid, we define global fields such that the solution smoothly depends on these fields. The global fields typically contain non-local information required for achieving a convergence independent of small scales. We present a rigorous analysis and show that the proposed global multiscale numerical methods converge independent of small scales. In particular, a global mixed multiscale finite element method is extensively studied and applied to two-phase flows. We present some numerical results for two-phase simulations on coarse grids. The numerical results demonstrate that the global multiscale numerical methods achieve high accuracy.
Book Synopsis Mathematical Modeling for Flow and Transport Through Porous Media by : Gedeon Dagan
Download or read book Mathematical Modeling for Flow and Transport Through Porous Media written by Gedeon Dagan and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main aim of this paper is to present some new and general results, ap plicable to the the equations of two phase flow, as formulated in geothermal reservoir engineering. Two phase regions are important in many geothermal reservoirs, especially at depths of order several hundred metres, where ris ing, essentially isothermal single phase liquid first begins to boil. The fluid then continues to rise, with its temperature and pressure closely following the saturation (boiling) curve appropriate to the fluid composition. Perhaps the two most interesting theoretical aspects of the (idealised) two phase flow equations in geothermal reservoir engineering are that firstly, only one component (water) is involved; and secondly, that the densities of the two phases are so different. This has led to the approximation of ignoring capillary pressure. The main aim of this paper is to analyse some of the consequences of this assumption, especially in relation to saturation changes within a uniform porous medium. A general analytic treatment of three dimensional flow is considered. Pre viously, three dimensional modelling in geothermal reservoirs have relied on numerical simulators. In contrast, most of the past analytic work has been restricted to one dimensional examples.
Book Synopsis Multiscale Problems: Theory, Numerical Approximation And Applications by : Alain Damlamian
Download or read book Multiscale Problems: Theory, Numerical Approximation And Applications written by Alain Damlamian and published by World Scientific. This book was released on 2011-10-13 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this is on the latest developments related to the analysis of problems in which several scales are presented. After a theoretical presentation of the theory of homogenization in the periodic case, the other contributions address a wide range of applications in the fields of elasticity (asymptotic behavior of nonlinear elastic thin structures, modeling of junction of a periodic family of rods with a plate) and fluid mechanics (stationary Navier-Stokes equations in porous media). Other applications concern the modeling of new composites (electromagnetic and piezoelectric materials) and imperfect transmission problems. A detailed approach of numerical finite element methods is also investigated.
Book Synopsis Handbook of Geomathematics by : Willi Freeden
Download or read book Handbook of Geomathematics written by Willi Freeden and published by Springer Science & Business Media. This book was released on 2010-08-13 with total page 1371 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the last three decades geosciences and geo-engineering were influenced by two essential scenarios: First, the technological progress has changed completely the observational and measurement techniques. Modern high speed computers and satellite based techniques are entering more and more all geodisciplines. Second, there is a growing public concern about the future of our planet, its climate, its environment, and about an expected shortage of natural resources. Obviously, both aspects, viz. efficient strategies of protection against threats of a changing Earth and the exceptional situation of getting terrestrial, airborne as well as spaceborne data of better and better quality explain the strong need of new mathematical structures, tools, and methods. Mathematics concerned with geoscientific problems, i.e., Geomathematics, is becoming increasingly important. The ‘Handbook Geomathematics’ as a central reference work in this area comprises the following scientific fields: (I) observational and measurement key technologies (II) modelling of the system Earth (geosphere, cryosphere, hydrosphere, atmosphere, biosphere) (III) analytic, algebraic, and operator-theoretic methods (IV) statistical and stochastic methods (V) computational and numerical analysis methods (VI) historical background and future perspectives.
Download or read book Multiscale Methods written by Jacob Fish and published by Oxford University Press. This book was released on 2010 with total page 631 pages. Available in PDF, EPUB and Kindle. Book excerpt: Small scale features and processes occurring at nanometer and femtosecond scales have a profound impact on what happens at a larger scale and over an extensive period of time. The primary objective of this volume is to reflect the state-of-the-art in multiscale mathematics, modeling, and simulations and to address the following barriers: What is the information that needs to be transferred from one model or scale to another and what physical principles must be satisfied during thetransfer of information? What are the optimal ways to achieve such transfer of information? How can variability of physical parameters at multiple scales be quantified and how can it be accounted for to ensure design robustness?The multiscale approaches in space and time presented in this volume are grouped into two main categories: information-passing and concurrent. In the concurrent approaches various scales are simultaneously resolved, whereas in the information-passing methods the fine scale is modeled and its gross response is infused into the continuum scale. The issue of reliability of multiscale modeling and simulation tools which focus on a hierarchy of multiscale models and an a posteriori model of errorestimation including uncertainty quantification, is discussed in several chapters. Component software that can be effectively combined to address a wide range of multiscale simulations is also described. Applications range from advanced materials to nanoelectromechanical systems (NEMS), biologicalsystems, and nanoporous catalysts where physical phenomena operates across 12 orders of magnitude in time scales and 10 orders of magnitude in spatial scales.This volume is a valuable reference book for scientists, engineers and graduate students practicing in traditional engineering and science disciplines as well as in emerging fields of nanotechnology, biotechnology, microelectronics and energy.
Book Synopsis Flow and Transport in Porous Media and Fractured Rock by : Muhammad Sahimi
Download or read book Flow and Transport in Porous Media and Fractured Rock written by Muhammad Sahimi and published by John Wiley & Sons. This book was released on 2011-05-09 with total page 635 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this standard reference of the field, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible displacements in porous media and fractured rock are considered. Two different approaches are discussed and contrasted with each other. The first approach is based on the classical equations of flow and transport, called 'continuum models'. The second approach is based on modern methods of statistical physics of disordered media; that is, on 'discrete models', which have become increasingly popular over the past 15 years. The book is unique in its scope, since (1) there is currently no book that compares the two approaches, and covers all important aspects of porous media problems; and (2) includes discussion of fractured rocks, which so far has been treated as a separate subject. Portions of the book would be suitable for an advanced undergraduate course. The book will be ideal for graduate courses on the subject, and can be used by chemical, petroleum, civil, environmental engineers, and geologists, as well as physicists, applied physicist and allied scientists that deal with various porous media problems.
Book Synopsis Higher-Order Finite Element Methods by : Pavel Solin
Download or read book Higher-Order Finite Element Methods written by Pavel Solin and published by CRC Press. This book was released on 2003-07-28 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: The finite element method has always been a mainstay for solving engineering problems numerically. The most recent developments in the field clearly indicate that its future lies in higher-order methods, particularly in higher-order hp-adaptive schemes. These techniques respond well to the increasing complexity of engineering simulations and
Book Synopsis Extended Finite Element Method by : Amir R. Khoei
Download or read book Extended Finite Element Method written by Amir R. Khoei and published by John Wiley & Sons. This book was released on 2015-02-23 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics Explores the concept of partition of unity, various enrichment functions, and fundamentals of XFEM formulation. Covers numerous applications of XFEM including fracture mechanics, large deformation, plasticity, multiphase flow, hydraulic fracturing and contact problems Accompanied by a website hosting source code and examples
