Adaptive Error Control In The Finite Element Method For Time Dependent Problems
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Adaptive Error Control In The Finite Element Method For Time Dependent Problems
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Author : Peter K. Jimack
language : en
Publisher:
Release Date : 1992
Adaptive Error Control In The Finite Element Method For Time Dependent Problems written by Peter K. Jimack and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Finite element method categories.
The general approach is then applied to a class of convection- diffusion problems where a particular estimate of this new spatial error is discussed. This estimate is based upon obtaining an improved approximation to the time derivative, in this case by using a richer but less regular trial space, which is then compared with the approximation that is actually used. A number of numerical examples are included which demonstrate the success of the strategy in conjunction with an adaptive h- refinement and derefinement procedure. This adaptivity algorithm is described in some detail for the case of a mesh of triangles in 2-d."
Adaptive Finite Elements In The Discretization Of Parabolic Problems
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Author : Christian A. Möller
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2011
Adaptive Finite Elements In The Discretization Of Parabolic Problems written by Christian A. Möller and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
Adaptivity is a crucial tool in state-of-the-art scientific computing. However, its theoretical foundations are only understood partially and are subject of current research. This self-contained work provides theoretical basics on partial differential equations and finite element discretizations before focusing on adaptive finite element methods for time dependent problems. In this context, aspects of temporal adaptivity and error control are considered in particular. Based on the gained insights, a specific adaptive algorithm is designed and analyzed thoroughly. Most importantly, it is proven that the presented adaptive method terminates within any demanded error tolerance. Moreover, the developed algorithm is analyzed from a numerical point of view and its performance is compared to well-known standard methods. Finally, it is applied to the real-life problem of concrete carbonation, where two different discretizations are compared.
Error Controlled Adaptive Finite Elements In Solid Mechanics
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Author : Ekkehard Ramm
language : en
Publisher: John Wiley & Sons
Release Date : 2003-08-01
Error Controlled Adaptive Finite Elements In Solid Mechanics written by Ekkehard Ramm and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-08-01 with Technology & Engineering categories.
Finite Element Methods are used for numerous engineering applications where numerical solutions of partial differential equations are needed. As computers can now deal with the millions of parameters used in these methods, automatic error estimation and automatic adaptation of the utilised method (according to this error estimation), has become a hot research topic. This text offers comprehensive coverage of this new field of automatic adaptation and error estimation, bringing together the work of eight outstanding researchers in this field who have completed a six year national research project within the German Science Foundation. The result is a state-of-the-art work in true reference style. Each chapter is self-contained and covers theoretical, algorithmic and software presentations as well as solved problems. A main feature consists of several carefully elaborated benchmarks of 2D- and 3D- applications. * First book to go beyond the Finite Element Method in itself * Covers material from a new research area * Presents benchmarks of 2D- and 3D- applications * Fits with the new trend for genetic strategies in engineering
Adaptive Finite Elements In Linear And Nonlinear Solid And Structural Mechanics
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Author : Erwin Stein
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-04-02
Adaptive Finite Elements In Linear And Nonlinear Solid And Structural Mechanics written by Erwin Stein 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 2007-04-02 with Technology & Engineering categories.
This course with 6 lecturers intends to present a systematic survey of recent re search results of well-known scientists on error-controlled adaptive finite element methods in solid and structural mechanics with emphasis to problem-dependent concepts for adaptivity, error analysis as well as h- and p-adaptive refinement techniques including meshing and remeshing. Challenging applications are of equal importance, including elastic and elastoplastic deformations of solids, con tact problems and thin-walled structures. Some major topics should be pointed out, namely: (i) The growing importance of goal-oriented and local error estimates for quan tities of interest—in comparison with global error estimates—based on dual finite element solutions; (a) The importance of the p-version of the finite element method in conjunction with parameter-dependent hierarchical approximations of the mathematical model, for example in boundary layers of elastic plates; (Hi) The choice of problem-oriented error measures in suitable norms, consider ing residual, averaging and hierarchical error estimates in conjunction with the efficiency of the associated adaptive computations; (iv) The importance of implicit local postprocessing with enhanced test spaces in order to get constant-free, i. e. absolute-not only relative-discretizati- error estimates; (v) The coupling of error-controlled adaptive discretizations and the mathemat ical modeling in related subdomains, such as boundary layers. The main goals of adaptivity are reliability and efficiency, combined with in sight and access to controls which are independent of the applied discretization methods. By these efforts, new paradigms in Computational Mechanics should be realized, namely verifications and even validations of engineering models.
Adaptive Finite Element Methods For Differential Equations
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Author : Wolfgang Bangerth
language : en
Publisher: Birkhäuser
Release Date : 2013-11-11
Adaptive Finite Element Methods For Differential Equations written by Wolfgang Bangerth and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.
These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.
Finite Element Analysis With Error Estimators
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Author : J. E. Akin
language : en
Publisher: Elsevier
Release Date : 2005-06-22
Finite Element Analysis With Error Estimators written by J. E. Akin and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-06-22 with Technology & Engineering categories.
This key text is written for senior undergraduate and graduate engineering students. It delivers a complete introduction to finite element methods and to automatic adaptation (error estimation) that will enable students to understand and use FEA as a true engineering tool. It has been specifically developed to be accessible to non-mathematics students and provides the only complete text for FEA with error estimators for non-mathematicians. Error estimation is taught on nearly half of all FEM courses for engineers at senior undergraduate and postgraduate level; no other existing textbook for this market covers this topic. The only introductory FEA text with error estimation for students of engineering, scientific computing and applied mathematics Includes source code for creating and proving FEA error estimators
A Moving Finite Element Method For Time Dependent Partial Differential Equations With Error Estimation And Refinement
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Author : S. Adjerid
language : en
Publisher:
Release Date : 1984
A Moving Finite Element Method For Time Dependent Partial Differential Equations With Error Estimation And Refinement written by S. Adjerid and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with categories.
The authors discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension. The mesh is moved so as to equidistribute the spatial component of the discretization error in H1. They present a method of estimating this error by using p-hierarchic finite elements. The error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement. The authors discretize the partial differential equations in space using a Galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error. A system of ordinary differential equations for mesh velocities are used to control element motions. The authors use existing software for stiff ordinary differential equations for the temporal integration of the solution, the error estimate, and the mesh motion. Computational results using a code based on this method are presented for several examples.
A Local Refinement Finite Element Method For Time Dependent Partial Differential Equations
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Author : J. E. Flaherty
language : en
Publisher:
Release Date : 1984
A Local Refinement Finite Element Method For Time Dependent Partial Differential Equations written by J. E. Flaherty and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with categories.
The authors discuss an adaptive local refinement finite element method for solving initial-boundary value problems for vector systems of partial differential equations in one space dimension and time. The method ues piecewise bilinear rectangular space-time finite elements. For each time step, grids are automatically added to regions where the local discretization error is estimated as being larger than a prescribed tolerance. The authors discuss several aspects oof their algorithm, including the tree structure that is used to represent the finite element solution and grids, an error estimation technique, and initial boundary conditions at coarse-fine mesh interfaces. The authors also present computational results for a simple linear hyperbolic problem, a problem involving Burger's equation, and a model combustion problem. Originator-supplied keywords include: Adaptive methods, Finite element methods, Local refinement, and Time dependent problems.
An Adaptive Finite Element Method For Pointwise Error Control
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Author : Renu Dhadwal
language : en
Publisher:
Release Date : 2000
An Adaptive Finite Element Method For Pointwise Error Control written by Renu Dhadwal and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with categories.
Adaptive Finite Element Methods
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Author : Wenbin Liu
language : en
Publisher: Alpha Science International Limited
Release Date : 2012
Adaptive Finite Element Methods written by Wenbin Liu and has been published by Alpha Science International Limited this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
Summary: "This book emphasizes the discussions of some unique issues from the adaptive finite element approximation of optimal control. The main idea used in the approximation error analysis (both a priori and a posteriori) is to first combine convex analysis and interpolation error estimations of suitable interpolators, which much depend on the structure of the control constraints, to derive the error estimates for the control via the variational inequalities in the optimality conditions, and then to apply the standard techniques to derive the error estimates for the state equations. The need, the framework and the techniques of using multi adaptive meshes in developing efficient numerical algorithms for optimal control have been emphasized throughout the book. The book starts from several typical examples of optimal control problems and then discusses existence and optimality conditions for some optimal control problems. It is believed that these discussions are especially useful for the researchers and students who first entered this area. Then the finite element approximation schemes for several typical optimal control problems are set up, their a priori and a posteriori error estimates are derived following the main idea mentioned, and their computational methods are studied."-- Publisher website, viewed 13th July, 2012.