Finite Element Modeling for Materials Engineers Using MATLAB®

The finite element method is often used for numerical computation in the applied sciences. It makes a major contribution to the range of numerical methods used in the simulation of systems and irregular domains, and its importance today has made it an important subject of study for all engineering students. While treatments of the method itself can be found in many traditional finite element books, __Finite Element Modeling for Materials Engineers Using MATLAB®__ combines the finite element method with MATLAB to offer materials engineers a fast and code-free way of modeling for many materials processes. __Finite Element Modeling for Materials Engineers Using MATLAB®__ covers such topics as: * developing a weak formulation as a prelude to obtaining the finite element equation, * interpolation functions, * derivation of elemental equations, and * use of the Partial Differential Equation ToolboxTM. Exercises are given based on each example and m-files based on the examples are freely available to readers online. Researchers, advanced undergraduate and postgraduate students, and practitioners in the fields of materials and metallurgy will find __Finite Element Modeling for Materials Engineers Using MATLAB®__ a useful guide to using MATLAB for engineering analysis and decision-making. Cover 1 Finite Element Modeling for Materials Engineers Using MATLAB 4 ISBN 9780857296603 5 Preface 6 Contents 8 For the Reader 12 1 Introduction 14 2 The Weak Formulation 16 2.1...Nodal Finite Elements 16 2.2...Mesh Elements 16 2.3...The Finite Element Method Procedure 16 2.4...Weak Formulation of Governing Equations 17 2.5...Gradient and Divergence Theorems 17 2.5.1 The Gradient Theorem 18 2.5.2 Divergence Theorem 18 2.6...Integration by Parts 19 2.7...Weak Formulations 19 2.8...Exercises 24 References 25 3 Linear Interpolation Functions 26 3.1...Parameter Functions and Interpolating Functions 26 3.2...Interpolation, Weighting and Approximation Functions 26 3.3...Linear Interpolation Function for One-Dimensional Analysis 27 3.4...Linear Interpolation Functions for Two-Dimensional Analysis 29 3.4.1 The Linear Triangular Element 29 3.4.2 The Bilinear Element 32 3.5...Linear Interpolation Functions for Three-Dimensional Problems 34 3.5.1 Four-Node Tetrahedral Elements 34 3.5.2 Eight-Node Brick Elements 36 3.6...Other Coordinate Systems Used in Derivation of Shape Functions 36 3.6.1 Serendipity Coordinates 36 3.6.1.1 Serendipity Coordinates Applied to One-Dimensional Problems 37 3.6.1.2 Serendipity Coordinates Applied to Two-Dimensional Problems 37 3.6.1.3 Serendipity Coordinates Applied to Three-Dimensional Problems 38 3.6.2 Length Coordinates 39 3.6.3 Area Coordinates 39 3.6.4 Volume Coordinates 40 3.7...Isoparametric Elements 40 3.7.1 Linear Isoparametric Element 41 3.7.2 Triangular Isoparametric Element 42 3.7.3 Quadrilateral Isoparametric Element 44 3.8...Exercises 45 References 46 4 Derivation of Element Matrices, Assembly and Solution of the Finite Element Equation 48 4.1...Derivation of Element Matrix for One-Dimensional Problems Using the Galerkin Method, Assembly and Solution 48 4.1.1 Weak Formulation 48 4.1.2 Assembly of Element Equations 49 4.1.3 Imposition of Boundary Conditions 51 4.1.4 Obtaining Neumann Boundary Conditions at X = 0 and X = l 52 4.2...Derivation of Element Matrix for Two-Dimensional Problems Using the Galerkin Method 52 4.2.1 Using Triangular Discretization 52 4.2.2 Using Bilinear Elements 54 4.3...Derivation of Element Matrix for Three-Dimensional Problems Using the Galerkin Method 55 4.4...Transient Problems 56 4.4.1 Time Integration Method for Transient Problems 57 4.4.1.1 Backward Difference Method 58 4.5...Derivation of Matrix Equations for Axisymmetric Problems 59 4.6...Sample Solutions on Elements Matrix Computation, Assembly and Solution 62 4.6.1 Calculating the Column Vector 67 4.7...One-Dimensional Fourth Order Differential Equation (Beam Bending Problem) 68 4.8...The Use of Other Coordinate Systems in Derivation of Finite Element Equation 69 4.8.1 Length Coordinates 69 4.8.2 Area Coordinates 70 4.8.3 Volume Coordinates 70 4.9...Exercises 56 References 71 5 Steps to Modeling Using PDEtoolboxtrade Graphics Interface 72 5.1...Engineering and Modeling 72 5.2...Steps for Modeling with the PDEtoolbox 72 5.2.1 Starting the MATLAB PDEtool GUI 73 5.2.2 Specifying the Application Type 73 5.2.3 Drawing the Problem Geometry 74 5.2.4 Specifying the PDE 76 5.2.5 Specifying Boundary Conditions 76 5.2.6 Meshing the Domain and Mesh Refinement 77 5.2.7 Specifying Initial Conditions for Transient Problems 79 5.2.8 Solving the PDE 79 5.2.9 Extracting Values from Plots 80 5.3...Exercises 79 References 85 6 Application of PDEtoolboxtrade to Heat Transfer Problems 86 6.1...Setting-Up the GUI for Heat Transfer Problems 86 6.2...Example Problems on Heat Transfer in Materials Engineering 86 6.2.1 Steady-State Heat Transfer 87 6.2.2 Transient Problems (Heating and Cooling Problems) 88 6.2.2.1 Heating Problems 88 6.2.2.2 Cooling Problems 91 6.2.3 Transient Problem (Heat Generation in a Tubular Furnace) 93 6.3...Exercises 95 References 96 7 Application of PDEtoolbox”TM’ to Elasticity Problems 98 7.1...Basics of Elasticity in Finite Element Application 98 7.2...Using the PDEtoolbox in Modeling Elasticity Problems in Materials Engineering 101 7.3...Applications of PDEtoolbox in Modeling Elasticity Problems 101 7.4...Exercises 116 References 117 Appendix: Sample M-codes for Exercisesin the Text 118 0857296604,9780857296603 Springer 2011 Front Matter....Pages i-xi Introduction....Pages 1-1 The Weak Formulation....Pages 3-12 Linear Interpolation Functions....Pages 13-33 Derivation of Element Matrices, Assembly and Solution of the Finite Element Equation....Pages 35-58 Steps to Modeling Using PDEtoolboxTM Graphics Interface....Pages 59-72 Application of PDEtoolboxTM to Heat Transfer Problems....Pages 73-83 Application of PDEtoolboxTM to Elasticity Problems....Pages 85-104 Back Matter....Pages 105-120