INTRODUCTION TO THE FINITE ELEMENT METHOD G.P.NIKISHKOV

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posté par nassima

INTRODUCTION TO THE FINITE ELEMENT METHOD G.P.NIKISHKOV

1Introduction

1.1 What is the finite element method

1.2 How the FEM works

1.3 Formulation of finite element equations

1.3.1 Galerkin method

1.3.2 Variational formulation

2 Finite Element Equations for Heat Transfer

2.1 Problem Statement

2.2 Finite element discretization of heat transfer equations

2.3 Different Type Problems

3 FEM for Solid Mechanics Problems

3.1 Problem statement

3.2 Finite element equations

3.3 Assembly of the global equation system

4 Finite Elements

4.1 Two-dimensional triangular element

4.2 Two-dimensional isoparametric elements

4.2.1 Shape functions

4.2.2 Strain-displacement matrix

4.2.3 Element properties

4.2.4 Integration in quadrilateral elements

4.2.5 Calculation of strains and stresses

4.3 Three-dimensional isoparametric elements

4.3.1 Shape functions

4.3.2 Strain-displacement matrix

4.3.3 Element properties

4.3.4 Efficient computation of the stiffness matrix

4.3.5 Integration of the stiffness matrix

4.3.6 Calculation of strains and stresses

4.3.7 Extrapolation of strains and stresses

5 Discretization

5.1 Discrete model of the problem

5.2 Mesh generation

5.2.1 Mesh generators

5.2.2 Mapping technique

5.2.3 Delaunay triangulation

6 Assembly and Solution

6.1 Disassembly and assembly

6.2 Disassembly algorithm

6.3 Assembly

6.3.1 Assembly algorithm for vectors

6.3.2 Assembly algorithm for matrices

6.4 Displacement boundary conditions

6.4.1 Explicit specification of displacement BC

6.4.2 Method of large number

6.5 Solution of Finite Element Equations

6.5.1 Solution methods

6.5.2 Direct LDU method with profile matrix

6.5.3 Preconditioned conjugate gradient method

7 Nonlinear Problems

7.1 Physical and geometrical nonlinearity

7.2 Constitutive relations for elastic-plastic material

7.3 Computing finite stress increments

7.3.1 Determining elastic fraction of stress increment

7.3.2 Subincrementation for computing stress increment

7.3.3 Midpoint integration of constitutive relations

7.4 Nonlinear solution procedure

7.4.1 Newton-Raphson method

7.4.2 Initial stress method

7.4.3 Convergence criteria

8 Visualization

8.1 Introduction

8.2 Overview of algorithm

8.3 Geometry and results data

8.3.1 Geometry relations for 20-node element

8.3.2 Surface of the finite element model

8.3.3 Continuous field of secondary results

8.4 Subdivision of quadratic surfaces

8.4.1 Curvature-based subdivision

8.4.2 Results-based subdivision

8.4.3 Delaunay triangulation

8.5 Using Java3D API for visualization

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