Download or read book Multi scale Problems an Improved Non intrusive Algorithm that Enhances FEA Platforms with the Generalized Finite Element Method An Improved Preconditioned Conjugate Gradient Solver for Hierarchical Generalized Finite Element Systems of Equations written by Travis Fillmore and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Numerical Methods for Mixed Finite Element Problems written by Jean Deteix and published by Springer Nature. This book was released on 2022-09-24 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on iterative solvers and preconditioners for mixed finite element methods. It provides an overview of some of the state-of-the-art solvers for discrete systems with constraints such as those which arise from mixed formulations. Starting by recalling the basic theory of mixed finite element methods, the book goes on to discuss the augmented Lagrangian method and gives a summary of the standard iterative methods, describing their usage for mixed methods. Here, preconditioners are built from an approximate factorisation of the mixed system. A first set of applications is considered for incompressible elasticity problems and flow problems, including non-linear models. An account of the mixed formulation for Dirichlet’s boundary conditions is then given before turning to contact problems, where contact between incompressible bodies leads to problems with two constraints. This book is aimed at graduate students and researchers in the field of numerical methods and scientific computing.

Download or read book Solution of Finite Element Problems by Preconditioned Conjugate Gradient and Lanczos Methods written by Robert Leroy Taylor and published by . This book was released on 1984 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt: The solution of nonlinear, transient finite element problems may be achieved by using step-by-step integration of the equations of motion combined with a Newton solution of the resulting nonlinear algebraic equations. The use of Newton type methods leads to a set of linear simultaneous algebraic equations whose solution gives the iterate. For very large problems the solution of the large set of linearized equations may be a formidable task - often consuming more than half of the computing effort when performed by a direct method based upon Gauss elimination. Accordingly, it is of considerable importance to investigate alternative methods to solve the problem. The present study presents results obtained by using a Preconditioned Conjugate Gradient Method (PCG) described in (7) and a Preconditioned Lanczos Method (PLM) described in (6) to solve a variety of numerical examples. Based upon results obtained it is evident that a significant reduction in overall effort, compared to direct solutions, may be achieved using the preconditioned methods.

Download or read book Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems written by Clemens Pechstein and published by Springer Science & Business Media. This book was released on 2012-12-14 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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.

Download or read book Non Intrusive Extension of a Generalized Finite Element Method for Multiscale Problems to the Abaqus Analysis Platform written by Julia A. Plews and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Several classes of important engineering problems 0́3 in this case, problems exhibiting sharp thermal gradients 0́3 have solution features spanning multiple spatial scales of interest and, therefore, necessitate advanced hp finite element discretizations. Although hp-FEM is unavailable off-the-shelf in many predominant commercial analysis software packages, a novel method is proposed herein which is used to introduce these capabilities via the generalized FEM with global-local enrichments (GFEMgl) non-intrusively in Abaqus, a popular, general-purpose FEA platform. Numerical results show that the techniques utilized allow for accurate resolution of localized thermal features on structural-scale meshes without hp-adaptivity or the ability to account for very localized loads in the FEM framework itself. This methodology enables the user to take advantage of all the benefits of both hp-FEM discretizations and the appealing features of many available CAE/FEA software packages in order to obtain optimal convergence for challenging multiscale problems.

Download or read book Finite Element Solution of Boundary Value Problems written by O. Axelsson and published by . This book was released on 1984 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite Element Solution of Boundary Value Problems.

Download or read book A Fast Solver Free of Fill In for Finite Element Problems written by M. R. Li and published by . This book was released on 1981 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new algorithm for solving FEM problems is presented. It blends a preconditioned conjugate gradient iteration into a direct factorization method. The goal was to reduce fill to a negligible level and thus reduce storage requirements but it turned out to be faster than its rivals for an important class of problems. (Author).

Download or read book Finite Element Methods written by Michel Krizek and published by Routledge. This book was released on 2017-11-22 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: ""Based on the proceedings of the first conference on superconvergence held recently at the University of Jyvaskyla, Finland. Presents reviewed papers focusing on superconvergence phenomena in the finite element method. Surveys for the first time all known superconvergence techniques, including their proofs.

Download or read book A Multi scale Generalized Finite Element Method for Sharp Transient Thermal Gradients written by Patrick J. O'Hara and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this research, heat transfer problems exhibiting sharp thermal gradients are analyzed using the generalized finite element method. Convergence studies show that low order (linear and quadratic) elements require strongly refined meshes for acceptable accuracy. The high mesh density leads to small allowable time-step sizes, and significant increase in the computational cost. When mesh refinement and unrefinement is required between time-steps the mapping of solution vectors and state-dependent variables becomes difficult. A generalized FEM with global-local enrichments is proposed for the class of problems investigated in this research. In this procedure, a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized FEM with solutions of local boundary value problems. The local problems are defined using the same procedure as in the global-local FEM, where boundary conditions are provided by a coarse-scale global solution. Coarse, uniform, global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size. Convergence on these discretizations is achieved even when no or limited convergence is observed in the local problems. The two-way information transfer provided by the proposed generalized FEM is appealing to several classes of problems, especially those involving multiple spatial scales. The proposed methodology brings the benefits of generalized FEM to problems where limited or no information about the solution is known a-priori. The proposed methodology is formulated for, and applied to transient problems, where local domains at time t^{n+1} obtain their boundary conditions from the global domain at t^{n}. No transient effects need to be considered in the local domain. The method has shown the ability to produce accurate and efficient transient simulations in situations where traditional FEM analyses would lead to difficult re-meshing, and solution mapping issues. With the proposed methodology, the enrichment functions are added hierarchically to the stiffness matrix. As such, large portions of the coarse, global meshes may be assembled and factorized only once. The factorizations can then be re-used for multiple loading scenarios, or multiple time-steps so as to significantly improve the computational efficiency of the simulations. The issue of prohibitively small time-step sizes dictated by high mesh density in traditional FEM analyses is also addressed. With the use of appropriate shape functions, sufficient accuracy is obtained without the requirement of highly refined meshes. The resulting critical time-steps are less restrictive, making transient analyses more computationally feasible.

Download or read book Parallel Domain Decomposition Preconditioning for the Adaptive Finite Element Solution of Elliptic Problems in Three Dimensions written by Sarfraz Ahmad Nadeem and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Fixed Mesh ALE Method Applied to Multiphysics Problems Using Stabilized Formulations written by Joan Baiges Aznar and published by . This book was released on 2013 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: The finite element method is a tool very often employed to deal with the numerical simulation of multiphysics problems.Many times each of these problems can be attached to a subdomain in space which evolves in time. Fixed grid methods appear in order to avoid the drawbacks of remeshing in ALE (Arbitrary Lagrangian-Eulerian) methods when the domain undergoes very large deformations. Instead of having one mesh attached to each of the subdomains, one has a single mesh which covers the whole computational domain. Equations arising from the finite element analysis are solved in an Eulerian manner in this background mesh. In this work we present our particular approach to fixed mesh methods, which we call FM-ALE (Fixed-Mesh ALE). Our main concern is to properly account for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian-Eulerian framework, the distinctive feature being that at each time step results are projected onto a fixed, background mesh, that is where the problem is actually solved.We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. When dealing with certain physical problems, and depending on the finite element space used, the standard Galerkin finite element method fails and leads to unstable solutions. The variational multiscale method is often used to deal with this instability. We introduce a way to approximate the subgrid scales on the boundaries of the elements in a variational twoscale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. We then use the subscales on the element boundaries to improve transmition conditions between subdomains by introducing the subgrid scales between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid-structure interaction problems. The benefits in each case are respectively a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behaviour of the iterative algorithm to couple the fluid and the solid in fluid-structure interaction problems. We develop FELAP, a linear systems of equations solver package for problems arising from finite element analysis. The main features of the package are its capability to work with symmetric and unsymmetric systems of equations, direct and iterative solvers and various renumbering techniques. Performance is enhanced by considering the finite element mesh graph instead of the matrix graph, which allows to perform highly efficient block computations.

Download or read book A preconditioned conjugate gradient method for finite element equations is stable for rounding erroprs written by Owe Axelsson and published by . This book was released on 1979 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Parallel Version of the Conjugate Gradient Algorithm for Finite Element Problems written by Hatfield Polytechnic. Numerical Optimisation Centre and published by . This book was released on 1982 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Preconditioned Conjugate Gradient and Finite Element Methods for Massively Data parallel Architectures written by Nahil Atef Sobh and published by . This book was released on 1990 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Multilevel Approaches to Nonconforming Finite Element Discretizations of Linear Second Order Elliptic Boundary Value Problems written by Barbara Wohlmuth and published by . This book was released on 1993 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We consider adaptive multilevel techniques for nonconforming finite element discretizations of second order elliptic boundary value problems. In particular, we will focus on two basic ingredients of an efficient adaptive algorithm. The first one is the iterative solution of the arising linear system by preconditioned conjugate gradient methods and the second one is an a posteriori error estimator for the global discretization error. Both element-oriented and edge-oriented estimators will be investigated. Their local contributions will serve as an indicator within the refinement process. Finally, some numerical results will be presented. They illustrate the performance of the preconditioners as well as the refinement process."

Download or read book Canonical Dual Finite Element Method for Solving Nonconvex Mechanics and Topology Optimization written by Elaf Jaafar Ali and published by . This book was released on 2018 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m ≤ n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems. This thesis, first, presents a detailed study of large deformation problems in 2-D structural system. Based on the canonical duality theory, a canonical dual finite element method is applied to find a global minimization to the general nonconvex optimization problem using a new primal-dual semi-definite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam is investigated. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using the canonical dual finite element method, a new primal-dual semi-definite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolation are applied to obtain the closed dimensions between discretized displacement and discretized stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to the external load, thickness of the beam, numerical precision, and the size of finite elements. Finally, a mathematically rigorous and computationally powerful method for solving 3-D topology optimization problems is demonstrated. This method is based on CDT developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard Knapsack problem in topology optimization can be solved deterministically in polynomial-time via a canonical penalty-duality (CPD) method to obtain precise global optimal 0-1 density distribution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Finally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly addressed." -- Abstract.