A New Adaptive Multiscale Finite Element Method With Applications To High Contrast Interface Problems
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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.
Domain Decomposition Methods In Science And Engineering Xxi
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Author : Jocelyne Erhel
language : en
Publisher: Springer
Release Date : 2014-10-10
Domain Decomposition Methods In Science And Engineering Xxi written by Jocelyne Erhel and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-10 with Mathematics categories.
This volume contains a selection of papers presented at the 21st international conference on domain decomposition methods in science and engineering held in Rennes, France, June 25-29, 2012. Domain decomposition is an active and interdisciplinary research discipline, focusing on the development, analysis and implementation of numerical methods for massively parallel computers. Domain decomposition methods are among the most efficient solvers for large scale applications in science and engineering. They are based on a solid theoretical foundation and shown to be scalable for many important applications. Domain decomposition techniques can also naturally take into account multiscale phenomena. This book contains the most recent results in this important field of research, both mathematically and algorithmically and allows the reader to get an overview of this exciting branch of numerical analysis and scientific computing.
Finite And Boundary Element Tearing And Interconnecting Solvers For Multiscale Problems
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Author : Clemens Pechstein
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-14
Finite And Boundary Element Tearing And Interconnecting Solvers For Multiscale Problems written by Clemens Pechstein 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 2012-12-14 with Mathematics categories.
Tearing and interconnecting methods, such as FETI, FETI-DP, BETI, etc., are among the most successful domain decomposition solvers for partial differential equations. The purpose of this book is to give a detailed and self-contained presentation of these methods, including the corresponding algorithms as well as a rigorous convergence theory. In particular, two issues are addressed that have not been covered in any monograph yet: the coupling of finite and boundary elements within the tearing and interconnecting framework including exterior problems, and the case of highly varying (multiscale) coefficients not resolved by the subdomain partitioning. In this context, the book offers a detailed view to an active and up-to-date area of research.
Multiscale Finite Element Methods
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Author : Yalchin Efendiev
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-01-10
Multiscale Finite Element Methods written by Yalchin Efendiev 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 2009-01-10 with Technology & Engineering categories.
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.
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.
Mechatronics And Intelligent Materials Iii
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Author : Ran Chen
language : en
Publisher: Trans Tech Publications Ltd
Release Date : 2013-06-13
Mechatronics And Intelligent Materials Iii written by Ran Chen and has been published by Trans Tech Publications Ltd this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-13 with Technology & Engineering categories.
Selected, peer reviewed papers from the 2013 International Conference on Mechatronics and Intelligent Materials (MIM 2013), May 18-19, 2013, XiShuangBanNa, China
Operator Adapted Wavelets Fast Solvers And Numerical Homogenization
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Author : Houman Owhadi
language : en
Publisher: Cambridge University Press
Release Date : 2019-10-24
Operator Adapted Wavelets Fast Solvers And Numerical Homogenization written by Houman Owhadi and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-24 with Mathematics categories.
Presents interplays between numerical approximation and statistical inference as a pathway to simple solutions to fundamental problems.
Domain Decomposition Methods In Science And Engineering Xxv
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Author : Ronald Haynes
language : en
Publisher: Springer Nature
Release Date : 2020-10-24
Domain Decomposition Methods In Science And Engineering Xxv written by Ronald Haynes and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-24 with Mathematics categories.
These are the proceedings of the 25th International Conference on Domain Decomposition Methods in Science and Engineering, which was held in St. John's, Newfoundland, Canada in July 2018. Domain decomposition methods are iterative methods for solving the often very large systems of equations that arise when engineering problems are discretized, frequently using finite elements or other modern techniques. These methods are specifically designed to make effective use of massively parallel, high-performance computing systems. The book presents both theoretical and computational advances in this domain, reflecting the state of art in 2018.
The Immersed Interface Method
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Author : Zhilin Li
language : en
Publisher: SIAM
Release Date : 2006-01-01
The Immersed Interface Method written by Zhilin Li and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-01 with Mathematics categories.
This book provides an introduction to the immersed interface method (IIM), a powerful numerical method for solving interface problems and problems defined on irregular domains for which analytic solutions are rarely available. This book gives a complete description of the IIM, discusses recent progress in the area, and describes numerical methods for a number of classic interface problems. It also contains many numerical examples that can be used as benchmark problems for numerical methods designed for interface problems on irregular domains.