Showing papers on "Finite element method published in 2002"

Mesh Free Methods: Moving Beyond the Finite Element Method

Abstract: Preliminaries Physical Problems in Engineering Solid Mechanics: A Fundamental Engineering Problem Numerical Techniques: Practical Solution Tools Defining Meshfree Methods Need for Meshfree Methods The Ideas of Meshfree Methods Basic Techniques for Meshfree Methods Outline of the Book Some Notations and Default Conventions Remarks Meshfree Shape Function Construction Basic Issues for Shape Function Construction Smoothed Particle Hydrodynamics Approach Reproducing Kernel Particle Method Moving Least Squares Approximation Point Interpolation Method Radial PIM Radial PIM with Polynomial Reproduction Weighted Least Square (WLS) Approximation Polynomial PIM with Rotational Coordinate Transformation Comparison Study via Examples Compatibility Issues: An Analysis Other Methods Function Spaces for Meshfree Methods Function Spaces Useful Spaces in Weak Formulation G Spaces: Definition G1h Spaces: Basic Properties Error Estimation Concluding Remarks Strain Field Construction Why Construct Strain Field? Historical Notes How to Construct? Admissible Conditions for Constructed Strain Fields Strain Construction Techniques Concluding Remarks Weak and Weakened Weak Formulations Introduction to Strong and Weak Forms Weighted Residual Method A Weak Formulation: Galerkin A Weakened Weak Formulation: GS-Galerkin The Hu-Washizu Principle The Hellinger-Reissner Principle The Modified Hellinger-Reissner Principle Single-Field Hellinger-Reissner Principle The Principle of Minimum Complementary Energy The Principle of Minimum Potential Energy Hamilton's Principle Hamilton's Principle with Constraints Galerkin Weak Form Galerkin Weak Form with Constraints A Weakened Weak Formulation: SC-Galerkin Parameterized Mixed Weak Form Concluding Remarks Element Free Galerkin Method EFG Formulation with Lagrange Multipliers EFG with Penalty Method Summary Meshless Local Petrov-Galerkin Method MLPG Formulation MLPG for Dynamic Problems Concluding Remarks Point Interpolation Methods Node-Based Smoothed Point Interpolation Method (NS-PIM) NS-PIM Using Radial Basis Functions (NS-RPIM) Upper Bound Properties of NS-PIM and NS-RPIM Edge-Based Smoothed Point Interpolation Methods (ES-PIMs) A Combined ES/NS Point Interpolation Methods (ES/NS-PIM) Strain-Constructed Point Interpolation Method (SC-PIM) A Comparison Study Summary Meshfree Methods for Fluid Dynamics Problem Introduction Navier-Stokes Equations Smoothed Particle Hydrodynamics Method Gradient Smoothing Method (GSM) Adaptive Gradient Smoothing Method (A-GSM) A Discussion on GSM for Incompressible Flows Other Improvements on GSM Meshfree Methods for Beams PIM Shape Function for Thin Beams Strong Form Equations Weak Formulation: Galerkin Formulation A Weakened Weak Formulation: GS-Galerkin Three Models Formulation for NS-PIM for Thin Beams Formulation for Dynamic Problems Numerical Examples for Static Analysis Numerical Examples: Upper Bound Solution Numerical Examples for Free Vibration Analysis Concluding Remarks Meshfree Methods for Plates Mechanics for Plates EFG Method for Thin Plates EFG Method for Thin Composite Laminates EFG Method for Thick Plates ES-PIM for Plates Meshfree Methods for Shells EFG Method for Spatial Thin Shells EFG Method for Thick Shells ES-PIM for Thick Shells Summary Boundary Meshfree Methods RPIM Using Polynomial Basis RPIM Using Radial Function Basis Remarks Meshfree Methods Coupled with Other Methods Coupled EFG/BEM Coupled EFG and Hybrid BEM Remarks Meshfree Methods for Adaptive Analysis Triangular Mesh and Integration Cells Node Numbering: A Simple Approach Bucket Algorithm for Node Searching Relay Model for Domains with Irregular Boundaries Techniques for Adaptive Analysis Concluding Remarks MFree2D(c) Overview Techniques Used in MFree2D Preprocessing in MFree2D Postprocessing in MFree2D Index References appear at the end of each chapter.

Computational Contact Mechanics

Abstract: Preface. Introduction. Introduction to Contact Mechanics. Continuum Solid Mechanics and Weak Forms. Contact Kinematics. Constitutive Equations for Contact Interfaces. Contact Boundary Value Problem and Weak Form. Discretization of the Continuum. Discretization, Small Deformation Contact. Discretization, Large Deformation Contact. Solution Algorithms. Thermo--mechanical Contact. Beam Contact. Adaptive Finite Element Methods for Contact Problems. Computation of Critical Points with Contact Constraints. Appendix A: Gauss Integration Rules. Appendix B: Convective Coordinates. Appendix C: Parameter Identification for Friction Materials. References. Index.

Energy principles and variational methods in applied mechanics

Abstract: Preface xv 1 Introduction 1 2 Mathematical Preliminaries 8 3 Review Of Equations Of Solid Mechanics 48 4 Work, Energy, And Variational Calculus 79 5 Energy Principles Of Structural 133 6 Dynamical Systems: Hamilton's Principle 177 7 Direct Variational Methods 204 8 Theory And Analysis Of Plates 299 9 The Finite Element Method 433 10 Mixed Variational Formulations 502 Answers / Solutions to Selected Problems 544 Index 583 About the Author 591

Finite elements in computational electromagnetism

Abstract: This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shells

Abstract: This work is an overview of available theories and finite elements that have been developed for multilayered, anisotropic, composite plate and shell structures. Although a comprehensive description of several techniques and approaches is given, most of this paper has been devoted to the so called axiomatic theories and related finite element implementations. Most of the theories and finite elements that have been proposed over the last thirty years are in fact based on these types of approaches. The paper has been divided into three parts. Part I, has been devoted to the description of possible approaches to plate and shell structures: 3D approaches, continuum based methods, axiomatic and asymptotic two-dimensional theories, classical and mixed formulations, equivalent single layer and layer wise variable descriptions are considered (the number of the unknown variables is considered to be independent of the number of the constitutive layers in the equivalent single layer case). Complicating effects that have been introduced by anisotropic behavior and layered constructions, such as high transverse deformability, zig-zag effects and interlaminar continuity, have been discussed and summarized by the acronimC -Requirements. Two-dimensional theories have been dealt with in Part II. Contributions based on axiomatic, asymtotic and continuum based approaches have been overviewed. Classical theories and their refinements are first considered. Both case of equivalent single-layer and layer-wise variables descriptions are discussed. The so-called zig-zag theories are then discussed. A complete and detailed overview has been conducted for this type of theory which relies on an approach that is entirely originated and devoted to layered constructions. Formulas and contributions related to the three possible zig-zag approaches, i.e. Lekhnitskii-Ren, Ambartsumian-Whitney-Rath-Das, Reissner-Murakami-Carrera ones have been presented and overviewed, taking into account the findings of a recent historical note provided by the author. Finite Element FE implementations are examined in Part III. The possible developments of finite elements for layered plates and shells are first outlined. FEs based on the theories considered in Part II are discussed along with those approaches which consist of a specific application of finite element techniques, such as hybrid methods and so-called global/local techniques. The extension of finite elements that were originally developed for isotropic one layered structures to multilayerd plates and shells are first discussed. Works based on classical and refined theories as well as on equivalent single layer and layer-wise descriptions have been overviewed. Development of available zig-zag finite elements has been considered for the three cases of zig-zag theories. Finite elements based on other approches are also discussed. Among these, FEs based on asymtotic theories, degenerate continuum approaches, stress resultant methods, asymtotic methods, hierarchy-p,_-s global/local techniques as well as mixed and hybrid formulations have been overviewed.

Finite Element Method Magnetics

Abstract: Acknowledgements Thanks to the following people for their valuable contributions to FEMM: • Si Hang, for writing fast point location routines that greatly increase the speed at which FEMM evaluates line integrals; • Robin Cornelius, for adding Lua extensions for FEMM for scripting / batch run capabilities ; • Keith Gregory, for his detailed help and comments.

Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics

Abstract: Error estimates for Galerkin proper orthogonal decomposition (POD) methods for nonlinear parabolic systems arising in fluid dynamics are proved For the time integration the backward Euler scheme is considered The asymptotic estimates involve the singular values of the POD snapshot set and the grid-structure of the time discretization as well as the snapshot locations

Non‐planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model

Abstract: A methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near-tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems. Copyright © 2002 John Wiley & Sons, Ltd.

Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers

Abstract: A full-vectorial imaginary-distance beam propagation method based on a finite element scheme is newly formulated and is effectively applied to investigating the problem of leakage due to a finite number of arrays of air holes in photonic-crystal holey fibers (HFs). In order to treat arbitrarily shaped air holes and to avoid spurious solutions, a curvilinear edge/nodal hybrid element is introduced. Furthermore, in order to evaluate propagation characteristics of not only bound modes but leaky modes in HFs, an anisotropic perfectly matched layer is also employed as a boundary condition at computational window edges. It is confirmed from numerical results that the propagation loss increases rapidly with increasing wavelength, especially for HFs with one ring of smaller air holes, and that the propagation loss is drastically reduced by adding one more ring of air holes to the cladding region.

Non-planar 3d crack growth by the extended finite element and level sets- part ii: level set update

Abstract: We present a level set method for treating the growth of non-planar three-dimensional cracks.The crack is defined by two almost-orthogonal level sets (signed distance functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space, and the second is used to describe the one-dimensional crack front, which is the intersection of the two level sets. A Hamilton–Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity. This displacement field is constructed directly in terms of the level sets, so the discretization by finite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with significant changes in topology. Copyright © 2002 John Wiley & Sons, Ltd.

Coupling Fluid Flow with Porous Media Flow

Abstract: The transport of substances back and forth between surface water and groundwater is a very serious problem. We study herein the mathematical model of this setting consisting of the Stokes equations in the fluid region coupled with the Darcy equations in the porous medium, coupled across the interface by the Beavers--Joseph--Saffman conditions. We prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems. This is important because there are many "legacy" codes available which have been optimized for uncoupled porous media and fluid flow.

Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality

Abstract: We give an overview of recent developments concerning the use of adjoint methods in two areas: the a posteriori error analysis of finite element methods for the numerical solution of partial differential equations where the quantity of interest is a functional of the solution, and superconvergent extraction of integral functionals by postprocessing.

The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods

Abstract: A comparison study of the efficiency and ac- curacy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are pre- sented in this paper. In all these six MLPG methods, absolutely no meshes are needed either for the interpo- lation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells. Because complicated shape functions for the trial function are inevitable at the present stage, in order to develop a fast and robust meshless method, we explore ways to avoid the use of a domain integral in the weak-form, by choosing an appropriate test function. The MLPG5 method (wherein the local, nodal-based test function, over a local sub-domain Ω s (or Ω te) centered at a node, is the Heaviside step function) avoids the need for both a domain integral in the attendant symmetric weak-form as well as a singular integral. Convergence studies in the numerical examples show that all of the MLPG methods possess excellent rates of convergence, for both the unknown variables and their derivatives. An analysis of computational costs shows that the MLPG5 method is less expensive, both in computational costs as well as definitely in human-labor costs, than the FEM, or BEM. Thus, due to its speed, accuracy and robustness, the MLPG5 method may be expected to replace the FEM, in the near future.

Stable real-time deformations

Abstract: The linear strain measures that are commonly used in real-time animations of deformable objects yield fast and stable simulations However, they are not suitable for large deformations Recently, more realistic results have been achieved in computer graphics by using Green's non-linear strain tensor, but the non-linearity makes the simulation more costly and introduces numerical problemsIn this paper, we present a new simulation technique that is stable and fast like linear models, but without the disturbing artifacts that occur with large deformations As a precomputation step, a linear stiffness matrix is computed for the system At every time step of the simulation, we compute a tensor field that describes the local rotations of all the vertices in the mesh This field allows us to compute the elastic forces in a non-rotated reference frame while using the precomputed stiffness matrix The method can be applied to both finite element models and mass-spring systems Our approach provides robustness, speed, and a realistic appearance in the simulation of large deformations

A Finite Element Model of the Human Knee Joint for the Study of Tibio-Femoral Contact

Abstract: As a step towards developing a finite element model of the knee that can be used to study how the variables associated with a meniscal replacement affect tibio-femoral contact, the goals of this study were 1) to develop a geometrically accurate three-dimensional solid model of the knee joint with special attention given to the menisci and articular cartilage, 2) to determine to what extent bony deformations affect contact behavior, and 3) to determine whether constraining rotations other than flexion/extension affects the contact behavior of the joint during compressive loading. The model included both the cortical and trabecular bone of the femur and tibia, articular cartilage of the femoral condyles and tibial plateau, both the medial and lateral menisci with their horn attachments, the transverse ligament, the anterior cruciate ligament, and the medial collateral ligament. The solid models for the menisci and articular cartilage were created from surface scans provided by a noncontacting, laser-based, three-dimensional coordinate digitizing system with an root mean squared error (RMSE) of less than 8 microns. Solid models of both the tibia and femur were created from CT images, except for the most proximal surface of the tibia and most distal surface of the femur which were created with the three-dimensional coordinate digitizing system. The constitutive relation of the menisci treated the tissue as transversely isotropic and linearly elastic. Under the application of an 800 N compressive load at 0 degrees of flexion, six contact variables in each compartment (ie., medial and lateral) were computed including maximum pressure, mean pressure, contact area, total contact force, and coordinates of the center of pressure. Convergence of the finite element solution was studied using three mesh sizes ranging from an average element size of 5 mm by 5 mm to 1 mm by 1 mm. The solution was considered converged for an average element size of 2 mm by 2 mm. Using this mesh size, finite element solutions for rigid versus deformable bones indicated that none of the contact variables changed by more than 2% when the femur and tibia were treated as rigid. However, differences in contact variables as large as 19% occurred when rotations other than flexion/extension were constrained. The largest difference was in the maximum pressure. Among the principal conclusions of the study are that accurate finite element solutions of tibio-femoral contact behavior can be obtained by treating the bones as rigid. However, unrealistic constraints on rotations other than flexion/extension can result in relatively large errors in contact variables.

A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis

Abstract: In this paper we present a two-layer structural model suitable for predicting reliably the passive (unstimulated) time-dependent three-dimensional stress and deformation states of healthy young arterial walls under various loading conditions. It extends to the viscoelastic regime a recently developed constitutive framework for the elastic strain response of arterial walls (see Holzapfel et al. (2001)). The structural model is formulated within the framework of nonlinear continuum mechanics and is well-suited for a finite element implementation. It has the special merit that it is based partly on histological information, thus allowing the material parameters to be associated with the constituents of each mechanically-relevant arterial layer. As one essential ingredient from the histological information the constitutive model requires details of the directional organization of collagen fibers as commonly observed under a microscope. We postulate a fully automatic technique for identifying the orientations of cellular nuclei, these coinciding with the preferred orientations in the tissue. The biological material is assumed to behave incompressibly so that the constitutive function is decomposed locally into volumetric and isochoric parts. This separation turns out to be advantageous in avoiding numerical complications within the finite element analysis of incompressible materials. For the description of the viscoelastic behavior of arterial walls we employ the concept of internal variables. The proposed viscoelastic model admits hysteresis loops that are known to be relatively insensitive to strain rate, an essential mechanical feature of arteries of the muscular type. To enforce incompressibility without numerical difficulties, the finite element treatment adopted is based on a three-field Hu-Washizu variational approach in conjunction with an augmented Lagrangian optimization technique. Two numerical examples are used to demonstrate the reliability and efficiency of the proposed structural model for arterial wall mechanics as a basis for large scale numerical simulations.

Lower bound limit analysis using non-linear programming

Abstract: This paper describes a new formulation, based on linear finite elements and non-linear programming, for computing rigorous lower bounds in 1, 2 and 3 dimensions. The resulting optimization problem is typically very large and highly sparse and is solved using a fast quasi-Newton method whose iteration count is largely independent of the mesh refinement. For two-dimensional applications, the new formulation is shown to be vastly superior to an equivalent formulation that is based on a linearized yield surface and linear programming. Although it has been developed primarily for geotechnical applications, the method can be used for a wide range of plasticity problems including those with inhomogeneous materials, complex loading, and complicated geometry. Copyright © 2002 John Wiley & Sons, Ltd.

A Matlab toolkit for three-dimensional electrical impedance tomography: a contribution to the Electrical Impedance and Diffuse Optical Reconstruction Software project

Abstract: The objective of the Electrical Impedance and Diffuse Optical Reconstruction Software project is to develop freely available software that can be used to reconstruct electrical or optical material properties from boundary measurements. Nonlinear and ill posed problems such as electrical impedance and optical tomography are typically approached using a finite element model for the forward calculations and a regularized nonlinear solver for obtaining a unique and stable inverse solution. Most of the commercially available finite element programs are unsuitable for solving these problems because of their conventional inefficient way of calculating the Jacobian, and their lack of accurate electrode modelling. A complete package for the two-dimensional EIT problem was officially released by Vauhkonen et al at the second half of 2000. However most industrial and medical electrical imaging problems are fundamentally three-dimensional. To assist the development we have developed and released a free toolkit of Matlab routines which can be employed to solve the forward and inverse EIT problems in three dimensions based on the complete electrode model along with some basic visualization utilities, in the hope that it will stimulate further development. We also include a derivation of the formula for the Jacobian (or sensitivity) matrix based on the complete electrode model.

Three-dimensional modeling of the friction stir-welding process

Abstract: This paper presents an attempt to model the stir-welding process using three-dimensional visco-plastic modeling. The scope of the project is focused on butt joints for aluminum thick plates. Parametric studies have been conducted to determine the effect of tool speeds on plate temperatures and to validate the model predictions with available measurements. In addition, forces acting on the tool have been computed for various welding and rotational speeds. It is found that pin forces increase with increasing welding speeds, but the opposite effect is observed for increasing rotational speeds. Numerical models such as the one presented here will be useful in designing welding tools which will yield desired thermal gradients and avoid tool breakage.

Computation of linear elastic properties from microtomographic images: Methodology and agreement between theory and experiment

Abstract: Elastic property‐porosity relationships are derived directly from microtomographic images. This is illustrated for a suite of four samples of Fontainebleau sandstone with porosities ranging from 7.5% to 22%. A finite‐element method is used to derive the elastic properties of digitized images. By estimating and minimizing several sources of numerical error, very accurate predictions of properties are derived in excellent agreement with experimental measurements over a wide range of the porosity. We consider the elastic properties of the digitized images under dry, water‐saturated, and oil‐saturated conditions. The observed change in the elastic properties due to fluid substitution is in excellent agreement with the exact Gassmann's equations. This shows both the accuracy and the feasibility of combining microtomographic images with elastic calculations to accurately predict petrophysical properties of individual rock morphologies. We compare the numerical predictions to various empirical, effective medium ...

Mixed finite elements for elasticity

Abstract: There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.

Upper bound limit analysis using linear finite elements and non-linear programming

Abstract: A new method for computing rigorous upper bounds on the limit loads for one-, two- and three-dimensional continua is described. The formulation is based on linear finite elements, permits kinematically admissible velocity discontinuities at all interelement boundaries, and furnishes a kinematically admissible velocity field by solving a non-linear programming problem. In the latter, the objective function corresponds to the dissipated power (which is minimized) and the unknowns are subject to linear equality constraints as well as linear and non-linear inequality constraints. Provided the yield surface is convex, the optimization problem generated by the upper bound method is also convex and can be solved efficiently by applying a two-stage, quasi-Newton scheme to the corresponding Kuhn–Tucker optimality conditions. A key advantage of this strategy is that its iteration count is largely independent of the mesh size. Since the formulation permits non-linear constraints on the unknowns, no linearization of the yield surface is necessary and the modelling of three-dimensional geometries presents no special difficulties. The utility of the proposed upper bound method is illustrated by applying it to a number of two- and three-dimensional boundary value problems. For a variety of two-dimensional cases, the new scheme is up to two orders of magnitude faster than an equivalent linear programming scheme which uses yield surface linearization. Copyright © 2001 John Wiley & Sons, Ltd.