Download or read book An Adaptive Finite Element Method for Initial Boundary Value Problems for Partial Differential Equations written by Stephen F. Davis and published by . This book was released on 1981 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt: A finite element method is developed to solve initial-boundary value problems for vector systems of partial differential equations in one space dimension and time. The method automatically adapts the computational mesh as the solution progresses in time and is thus able to follow and resolve relatively sharp transitions such as mild boundary layers, shock layers, or wave fronts. This permits an accurate solution to be calculated with fewer mesh points than would be necessary with a uniform mesh. The overall method contains two parts, a solution algorithm and a mesh selection algorithm. The solution algorithm is a finite element-Galerkin method on trapezoidal space-time elements, using either piecewise linear or cubic polynomial approximations and the mesh selection algorithm builds upon similar work for variable knot spline interpolation. A computer code implementing these algorithms has been written and applied to a number of problems. These computations confirm that the theoretical error estimates are attained and demonstrate the utility of variable mesh methods for partial differential equations. (Author).
Download or read book Adaptive Finite Element Method I Solution Algorithm and Computational Examples written by and published by . This book was released on 1994 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt: An adaptive finite element method is developed to solve initial boundary value problems for vector systems of parabolic partial differential equations in one space dimension and time. The differential equations are discretized in space using piecewise linear finite element approximations. Superconvergence properties and quadratic polynomials are used to derive a computation ally inexpensive approximation to the spatial component of the error. This technique is coupled with time integration schemes of successively higher orders to obtain an approximation of the temporal and total discretization errors. These approximate errors are used to control an adaptive mesh refinement strategy. Refinement is performed in space, time, or both space and time depending on the dominant component of the error estimate. A computer code coupling this refinement strategy and stable mesh movement has been written and applied to a number of problems. These computations confirm that proper mesh movement can reduce the computational efforts associated with mesh refinement.
Download or read book Adaptive Finite Element Methods for Parabolic Partial Differential Equations written by J. E. Flaherty and published by . This book was released on 1983 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors discuss a finite element method for solving initial-boundary value problems for vector systems of partial differential equations in one space dimension and time. The method automatically adjusts the computational mesh as the solution evolves in time so as to approximately minimize the local discretization error. They are thus able to calculate accurate solutions with fewer elements than would be necessary with a uniform mesh. This overall method contains two distinct steps: a solution step and a mesh selection step. They solve the partial differential equations using a finite element-Galerkin method on trapezoidal space-time-elements with either piecewise linear or cubic Hermits polynomial approximations. A variety of mesh selection strategies are discussed and analyzed. Results are presented for several computational examples.
Download or read book Error controlled Adaptive Finite Elements in Solid Mechanics written by Ekkehard Ramm and published by John Wiley & Sons. This book was released on 2003-08-01 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book The Finite Element Method written by A. J. Davies and published by Oxford University Press. This book was released on 2011-09-08 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the application of the finite element method to the solution of boundary and initial-value problems posed in terms of partial differential equations. Contains worked examples throughout and each chapter has a set of exercises with detailed solutions.
Download or read book Finite Element Solution of Boundary Value Problems written by O. Axelsson and published by SIAM. This book was released on 2001-01-01 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: a thorough, balanced introduction to both the theoretical and the computational aspects of the topic.
Download or read book Space Time Methods written by Ulrich Langer and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-09-23 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.
Download or read book Finite Element Methods written by Jonathan Whiteley and published by Springer. This book was released on 2017-01-26 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents practical applications of the finite element method to general differential equations. The underlying strategy of deriving the finite element solution is introduced using linear ordinary differential equations, thus allowing the basic concepts of the finite element solution to be introduced without being obscured by the additional mathematical detail required when applying this technique to partial differential equations. The author generalizes the presented approach to partial differential equations which include nonlinearities. The book also includes variations of the finite element method such as different classes of meshes and basic functions. Practical application of the theory is emphasised, with development of all concepts leading ultimately to a description of their computational implementation illustrated using Matlab functions. The target audience primarily comprises applied researchers and practitioners in engineering, but the book may also be beneficial for graduate students.
Download or read book The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations written by A. K. Aziz and published by Academic Press. This book was released on 2014-05-10 with total page 814 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations is a collection of papers presented at the 1972 Symposium by the same title, held at the University of Maryland, Baltimore County Campus. This symposium relates considerable numerical analysis involved in research in both theoretical and practical aspects of the finite element method. This text is organized into three parts encompassing 34 chapters. Part I focuses on the mathematical foundations of the finite element method, including papers on theory of approximation, variational principles, the problems of perturbations, and the eigenvalue problem. Part II covers a large number of important results of both a theoretical and a practical nature. This part discusses the piecewise analytic interpolation and approximation of triangulated polygons; the Patch test for convergence of finite elements; solutions for Dirichlet problems; variational crimes in the field; and superconvergence result for the approximate solution of the heat equation by a collocation method. Part III explores the many practical aspects of finite element method. This book will be of great value to mathematicians, engineers, and physicists.
Download or read book Adaptive Finite Element Method II Error Estimation written by and published by . This book was released on 1994 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt: An adaptive finite element method is developed to solve initial boundary value problems for vector systems of parabolic partial differential equations in one space dimension and time. The differential equations are discretized in space using piecewise linear finite element approximations. Superconvergence properties and quadratic polynomials are used to derive a computation ally inexpensive approximation to the spatial component of the error. This technique is coupled with time integration schemes of successively higher orders to obtain an approximation of the temporal and total discretization errors. Computational results indicate that these approximations converge to the exact discretization errors as the mesh is refined.
Download or read book A Local Refinement Finite Element Method for Time Dependent Partial Differential Equations written by J. E. Flaherty and published by . This book was released on 1984 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book The Scaled Boundary Finite Element Method written by John P. Wolf and published by John Wiley & Sons. This book was released on 2003-03-14 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: A novel computational procedure called the scaled boundary finite-element method is described which combines the advantages of the finite-element and boundary-element methods : Of the finite-element method that no fundamental solution is required and thus expanding the scope of application, for instance to anisotropic material without an increase in complexity and that singular integrals are avoided and that symmetry of the results is automatically satisfied. Of the boundary-element method that the spatial dimension is reduced by one as only the boundary is discretized with surface finite elements, reducing the data preparation and computational efforts, that the boundary conditions at infinity are satisfied exactly and that no approximation other than that of the surface finite elements on the boundary is introduced. In addition, the scaled boundary finite-element method presents appealing features of its own : an analytical solution inside the domain is achieved, permitting for instance accurate stress intensity factors to be determined directly and no spatial discretization of certain free and fixed boundaries and interfaces between different materials is required. In addition, the scaled boundary finite-element method combines the advantages of the analytical and numerical approaches. In the directions parallel to the boundary, where the behaviour is, in general, smooth, the weighted-residual approximation of finite elements applies, leading to convergence in the finite-element sense. In the third (radial) direction, the procedure is analytical, permitting e.g. stress-intensity factors to be determined directly based on their definition or the boundary conditions at infinity to be satisfied exactly. In a nutshell, the scaled boundary finite-element method is a semi-analytical fundamental-solution-less boundary-element method based on finite elements. The best of both worlds is achieved in two ways: with respect to the analytical and numerical methods and with respect to the finite-element and boundary-element methods within the numerical procedures. The book serves two goals: Part I is an elementary text, without any prerequisites, a primer, but which using a simple model problem still covers all aspects of the method and Part II presents a detailed derivation of the general case of statics, elastodynamics and diffusion.
Download or read book Adaptive Finite Element Method IV Mesh Movement written by and published by . This book was released on 1995 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: An adaptive finite element method is developed to solve initial boundary value problems for vector systems of parabolic partial differential equations in one space dimension and time. The differential equations are discretized in space using piecewise linear finite element approximations. Superconvergence properties and quadratic polynomials are used to derive a computationally inexpensive approximation to the spatial component of the error. This technique is coupled with time integration schemes of successively higher orders to obtain an approximation of the temporal and total discretization error. The stability of several mesh equidistribution schemes for time dependent partial differential equations is studied. The schemes move a finite difference or finite element mesh so that a given quantity is uniform over the domain. Mesh moving methods that are based on solving a system of ordinary differential equations for the mesh velocities are considered and some of these methods are shown to be unstable with respect to an equidistributing mesh when the partial differential system is dissiptive. Simple criteria for determining the stability of a particular method are developed and the construction of stable differential systems for the mesh velocities is demonstrated. Several examples illustrating stable and unstable mesh motions are present.
Download or read book Scrapbook of Photographs Newspaper Clippings and Programs Concerning U S S Mason and Her Crew 1943 45 written by and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Adaptive Methods for Partial Differential Equations written by Ivo Babushka and published by SIAM. This book was released on 1989-01-01 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Proceedings of the Workshop on Adaptive Computational Methods for Partial Differential Equations, Rensselaer Polytechnic Institute, October 13-15, 1988"--T.p. verso.
Download or read book Numerical Solution of Partial Differential Equations by the Finite Element Method written by Claes Johnson and published by Courier Corporation. This book was released on 2012-05-23 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.
Download or read book Adaptive Finite Element Methods for Parabolic Systems in One and Two Space Dimensions written by Slimane Adjerid and published by . This book was released on 1987 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: Adaptive finite element methods are given for solving initial boundary value problems for vector systems of parabolic partial differential equations in one- and two-space dimensions. One-dimension systems are discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial error estimate is calculated using piecewise quadratic approximations that employ nodal superconvergence to increase computational efficiency. This error estimate is used to move and refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the finite element solution. Two-dimension systems are discretized using piecewise bilinear finite element approximations in space and backward difference software in time. A spatial error estimate is calculated using piecewise cubic approximations that take advantage of nodal superconvergence. This error estimate is used to locally refine a stationary finite element mesh in order to satisfy a prescribed spatial error tolerance.
