Showing papers on "Finite element method published in 2014"

Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Abstract: The intended readership includes graduate students and researchers in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. The publisher describes the book as follows: * An excellent introduction to finite elements, iterative linear solvers and scientific computing * Contains theoretical problems and practical exercises * All methods and examples use freely available software * Focuses on theory and computation, not theory for computation * Describes approximation methods and numerical linear algebra

The Hitchhiker's Guide to the Virtual Element Method

Abstract: We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

Finite Element Solution of Boundary Value Problems: Theory and Computation

Abstract: Finite Element Solution of Boundary Value Problems: Theory and Computation provides a thorough, balanced introduction to both the theoretical and the computational aspects of the finite element method for solving boundary value problems for partial differential equations. Although significant advances have been made in the finite element method since this book first appeared in 1984, the basics have remained the same, and this classic, well-written text explains these basics and prepares the reader for more advanced study. Useful as both a reference and a textbook, complete with examples and exercies, it remains as relevant today as it was when originally published. This book is written for advanced undergraduate and graduate students in the areas of numerical analysis, mathematics, and computer science, as well as for theoretically inclined practitioners in engineering and the physical science.

A weak Galerkin mixed finite element method for second order elliptic problems

Abstract: . A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H and L norms are established for the corresponding weak Galerkin mixed finite element solutions.

Localization of elliptic multiscale problems

Abstract: This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size $ H$, patches of diameter $ H\log (1/H)$ are sufficient to preserve a linear rate of convergence in $ H$ without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. - See more at: http://www.ams.org/journals/mcom/2014-83-290/S0025-5718-2014-02868-8/#sthash.z2CCFXIg.dpuf

Finite Element Analysis of Structures through Unified Formulation

Abstract: This book deals with the Finite Element Method for the analysis of elastic structures such as beams, plates, shells and solids. The modern approach of Unified Formulation (UF), as proposed by the lead author, deals with the consideration of one-dimensional (beams), two-dimensional (plates and shells) and three-dimensional (solids) elements. Applications are given for structures which are typically employed in civil, mechanical, and aerospace engineering fields. Additional topics include mixed order elements, extension to layered composite structures, and the analysis of multifield problems involving mechanical, electrical and thermal loadings.

Basic principles of mixed Virtual Element Methods

Abstract: The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H (div)-conforming vector fields (or, more generally, of (n − 1) − Cochains ). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).

Introduction to finite and spectral element methods using MATLAB

Abstract: The Finite Element Method in One Dimension. Further Applications in One Dimension. High-Order and Spectral Elements in One Dimension. The Finite Element Method in Two Dimensions. Quadratic and Spectral Elements in Two Dimensions. Applications in Mechanics. Viscous Flow. Finite and Spectral Element Methods in Three Dimensions. Appendices. References. Index.

A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media

Abstract: SUMMARY A hierarchical multiscale framework is proposed to model the mechanical behaviour of granular media. The framework employs a rigorous hierarchical coupling between the FEM and the discrete element method (DEM). To solve a BVP, the FEM is used to discretise the macroscopic geometric domain into an FEM mesh. A DEM assembly with memory of its loading history is embedded at each Gauss integration point of the mesh to serve as the representative volume element (RVE). The DEM assembly receives the global deformation at its Gauss point from the FEM as input boundary conditions and is solved to derive the required constitutive relation at the specific material point to advance the FEM computation. The DEM computation employs simple physically based contact laws in conjunction with Coulomb's friction for interparticle contacts to capture the loading-history dependence and highly nonlinear dissipative response of a granular material. The hierarchical scheme helps to avoid the phenomenological assumptions on constitutive relation in conventional continuum modelling and retains the computational efficiency of FEM in solving large-scale BVPs. The hierarchical structure also makes it ideal for distributed parallel computing to fully unleash its predictive power. Importantly, the framework offers rich information on the particle level with direct link to the macroscopic material response, which helps to shed lights on cross-scale understanding of granular media. The developed framework is first benchmarked by a simulation of single-element drained test and is then applied to the predictions of strain localisation for sand subject to monotonic biaxial compression, as well as the liquefaction and cyclic mobility of sand in cyclic simple shear tests. It is demonstrated that the proposed method may reproduce interesting experimental observations that are otherwise difficult to be captured by conventional FEM or pure DEM simulations, such as the inception of shear band under smooth symmetric boundary conditions, non-coaxial granular response, large dilation and rotation at the edges of shear band and critical state reached within the shear band. Copyright © 2014 John Wiley & Sons, Ltd.

Phase field approximation of dynamic brittle fracture

Abstract: Numerical methods that are able to predict the failure of technical structures due to fracture are important in many engineering applications. One of these approaches, the so-called phase field method, represents cracks by means of an additional continuous field variable. This strategy avoids some of the main drawbacks of a sharp interface description of cracks. For example, it is not necessary to track or model crack faces explicitly, which allows a simple algorithmic treatment. The phase field model for brittle fracture presented in Kuhn and Muller (Eng Fract Mech 77(18):3625---3634, 2010) assumes quasi-static loading conditions. However dynamic effects have a great impact on the crack growth in many practical applications. Therefore this investigation presents an extension of the quasi-static phase field model for fracture from Kuhn and Muller (Eng Fract Mech 77(18):3625---3634, 2010) to the dynamic case. First of all Hamilton's principle is applied to derive a coupled set of Euler-Lagrange equations that govern the mechanical behaviour of the body as well as the crack growth. Subsequently the model is implemented in a finite element scheme which allows to solve several test problems numerically. The numerical examples illustrate the capabilities of the developed approach to dynamic fracture in brittle materials.

The finite element method in solid mechanics

Abstract: The book focuses on topics that are at the core of the Finite Element Method (FEM) for the mechanics of deformable solids and structures.Its main objective is to provide the reader, who is assumed to be familiar with standard continuum solid mechanics, with a clear grasp of the essentials, sufficient background for reading and exploiting the research literature on computational solid mechanics, and a working knowledge of the main implementational issues of the FEM.This book arises from a course taught since 2004 to last-year students of Ecole Polytechnique (France). It is intended for Master and PhD students, as well as scientists and engineers looking for a rigorous introduction to FEM theory and programming for linear and non-linear analyses in solid mechanics.As a distinguishing feature, in addition to sections devoted to theory and concepts presented in general terms, each chapter also features other sections (interspersed with the former) devoted to detailed description of specific features (e.g. the construction of a specific finite element), annotated Matlab code and/or numerical examples produced with it, or worked-out analytical examples.

Nitsche's method for two and three dimensional NURBS patch coupling

Abstract: We present a Nitche's method to couple non-conforming two and three-dimensional non uniform rational b-splines (NURBS) patches in the context of isogeometric analysis. We present results for linear elastostatics in two and and three-dimensions. The method can deal with surface-surface or volume-volume coupling, and we show how it can be used to handle heterogeneities such as inclusions. We also present preliminary results on modal analysis. This simple coupling method has the potential to increase the applicability of NURBS-based isogeometric analysis for practical applications.

Geometric imperfections and lower-bound methods used to calculate knock-down factors for axially compressed composite cylindrical shells

Abstract: The important role of geometric imperfections on the decrease of the buckling load for thin-walled cylinders had been recognized already by the first authors investigating the theoretical approaches on this topic. However, there are currently no closed-form solutions to take imperfections into account already during the early design phases, forcing the analysts to use lower-bound methods to calculate the required knock-down factors (KDF). Lower-bound methods such as the empirical NASA SP-8007 guideline are commonly used in the aerospace and space industries, while the approaches based on the Reduced Stiffness Method (RSM) have been used mostly in the civil engineering field. Since 1970s a considerable number of experimental and numerical investigations have been conducted to develop new stochastic and deterministic methods for calculating less conservative KDFs. Among the deterministic approaches, the single perturbation load approach (SPLA), proposed by Huhne, will be further investigated for axially compressed fiber composite cylindrical shells and compared with four other methods commonly used to create geometric imperfections: linear buckling mode-shaped, geometric dimples, axisymmetric imperfections and measured geometric imperfections from test articles. The finite element method using static analysis with artificial damping is used to simulate the displacement controlled compression tests up to the post-buckled range of loading. The implementation of each method is explained in details and the different KDFs obtained are compared. The study is part of the European Union (EU) project DESICOS, whose aim is to combine stochastic and deterministic approaches to develop less conservative guidelines for the design of imperfection sensitive structures.

A Simple Introduction to the Mixed Finite Element Method

Abstract: Introduction.- The Babuska-Brezzi Theory.- Raviart-Thomas Spaces.- Mixed Finite Element Methods.

XLME interpolants, a seamless bridge between XFEM and enriched meshless methods

Abstract: In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition, we keep the advantages of the LME shape functions, such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes near the crack such as blending or shifting.