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Lecture Introduction to the finite element method

Lecture Introduction to the finite element method has contents: Introduction, finite element equations for heat transfer, fem for solid mechanics problems, finite elements, discretization, assembly and solution. | I NTRODUCTION TO THE F INITE E LEMENT M ETHOD G. P. Nikishkov 2004 Lecture Notes. University of Aizu, Aizu-Wakamatsu 965-8580, Japan niki@u-aizu.ac.jp 2 Updated 2004-01-19 Contents 1 5 1.1 What is the finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 How the FEM works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Formulation of finite element equations . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 2 Introduction Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 11 2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Finite element discretization of heat transfer equations . . . . . . . . . . . . . . . . . 12 2.3 3 Finite Element Equations for Heat Transfer Different Type Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 15 3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Finite element equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 4 FEM for Solid Mechanics Problems Assembly of the global equation system . . . . . . . . . . . . . . . . . . . . . . . . . 18 Finite Elements 21 4.1 Two-dimensional triangular element . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Two-dimensional isoparametric elements . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.1 Shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.2 Strain-displacement matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.3 Element properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.4 Integration in quadrilateral elements . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.5 Calculation of