Multiscale Model Reduction For High Contrast Flow Problems
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Multiscale Model Reduction For High Contrast Flow Problems
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Author : Guanglian Li
language : en
Publisher:
Release Date : 2015
Multiscale Model Reduction For High Contrast Flow Problems written by Guanglian Li and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.
Many applications involve media that contain multiple scales and physical properties that vary in orders of magnitude. One example is a rock sample, which has many micro-scale features. Most multiscale problems are often parameter-dependent, where the parameters represent variations in medium properties, randomness, or spatial heterogeneities. Because of disparity of scales in multiscale problems, solving such problems is prohibitively expensive. Among the most popular and developed techniques for efficiently solving the global system arising from a finite element approximation of the underlying problem on a very fine mesh are multigrid methods, multilevel methods, and domain decomposition techniques. More recently, a new large class of accurate reduced-order methods has been introduced and used in various applications. These include Galerkin multiscale finite element methods, mixed multiscale finite element methods, multiscale finite volume methods, and mortar multiscale methods, and so on. In this dissertation, a multiscale finite element method is studied for the computation of heterogeneous problems involving high-contrast, no-scale separation, parameter dependency and nonlinearities. A general formulation of the elliptic heterogeneous problems is discussed, including an oversampling strategy and randomized snapshots generation for a more efficient and accurate computation. Furthermore, a multiscale adaptive algorithm is proposed and analyzed to reduce the computational cost. Then, this multiscale finite element method is extended to the nonlinear high-contrast elliptic problems. Specifically, both continuous and discontinuous Galerkin formulations are considered. In the end, an application to high-contrast heterogeneous Brinkman flow is analyzed. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/155073
Multiscale Model Reduction
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Author : Eric Chung
language : en
Publisher: Springer Nature
Release Date : 2023-06-07
Multiscale Model Reduction written by Eric Chung and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-06-07 with Mathematics categories.
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.
Computational Mathematics Algorithms And Data Processing
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Author : Daniele Mortari
language : en
Publisher: MDPI
Release Date : 2020-12-07
Computational Mathematics Algorithms And Data Processing written by Daniele Mortari and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-07 with Technology & Engineering categories.
“Computational Mathematics, Algorithms, and Data Processing” of MDPI consists of articles on new mathematical tools and numerical methods for computational problems. Topics covered include: numerical stability, interpolation, approximation, complexity, numerical linear algebra, differential equations (ordinary, partial), optimization, integral equations, systems of nonlinear equations, compression or distillation, and active learning.
Numerical Analysis Of Multiscale Problems
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Author : Ivan G. Graham
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-01-05
Numerical Analysis Of Multiscale Problems written by Ivan G. Graham and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-05 with Mathematics categories.
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.
Snapshot Based Methods And Algorithms
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Author : Peter Benner
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2020-12-16
Snapshot Based Methods And Algorithms written by Peter Benner and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-16 with Mathematics categories.
An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science.
A New Adaptive Multiscale Finite Element Method With Applications To High Contrast Interface Problems
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Author : Raymond Millward
language : en
Publisher:
Release Date : 2011
A New Adaptive Multiscale Finite Element Method With Applications To High Contrast Interface Problems written by Raymond Millward and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
In this thesis we show that the finite element error for the high contrast elliptic interface problem is independent of the contrast in the material coefficient under certain assumptions. The error estimate is proved using a particularly technical proof with construction of a specific function from the finite dimensional space of piecewise linear functions. We review the multiscale finite element method of Chu, Graham and Hou to give clearer insight. We present some generalisations to extend their work on a priori contrast independent local boundary conditions, which are then used to find multiscale basis functions by solving a set of local problems. We make use of their regularity result to prove a new relative error estimate for both the standard finte element method and the multiscale finite element method that is completely coefficient independent The analytical results we explore in this thesis require a complicated construction. To avoid this we present an adaptive multiscale finite element method as an enhancement to the adaptive local-global method of Durlofsky, Efendiev and Ginting. We show numerically that this adaptive method converges optimally as if the coefficient were smooth even in the presence of singularities as well as in the case of a realisation of a random field. The novel application of this thesis is where the adaptive multiscale finite element method has been applied to the linear elasticity problem arising from the structural optimisation process in mechanical engineering. We show that a much smoother sensitivity profile is achieved along the edges of a structure with the adaptive method and no additional heuristic smoothing techniques are needed. We finally show that the new adaptive method can be efficiently implemented in parallel and the processing time scales well as the number of processors increases. The biggest advantage of the multiscale method is that the basis functions can be repeatedly used for additional problems with the same high contrast material coefficient.
Homogenization Of Differential Operators And Integral Functionals
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Author : V.V. Jikov
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Homogenization Of Differential Operators And Integral Functionals written by V.V. Jikov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.
Generalized Discontinuous Multiscale Methods For Flows In Highly Heterogeneous Porous Media
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Author : Minam Moon
language : en
Publisher:
Release Date : 2015
Generalized Discontinuous Multiscale Methods For Flows In Highly Heterogeneous Porous Media written by Minam Moon and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.
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
Generalization Of Mixed Multiscale Finite Element Methods With Applications
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Author :
language : en
Publisher:
Release Date : 2016
Generalization Of Mixed Multiscale Finite Element Methods With Applications written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with categories.
Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms. For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces. Two enrichment approaches are proposed, one is based on generalized multiscale finte element metthods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (rAMGe). The former one, which is called mixed GMsFEM, is developed for both Darcy's flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor. The latter enrichment approach is based on rAMGe. The algorithm differs from GMsFEM in that both of the velocity and pressure spaces are coarsened. Due the multigrid nature of the algorithm, recursive application is available, which results in an efficient multilevel construction of the coarse spaces. Stability, convergence analysis, and exhaustive numerical experiments are carried out to validate the proposed enrichment approaches. iii.
Tchebycheff Systems
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Author : Samuel Karlin
language : en
Publisher:
Release Date : 1966
Tchebycheff Systems written by Samuel Karlin and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1966 with Approximation theory categories.