Page 3 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONTENTS
viii
3.2.3
3.2.4
Area coordinates .. .. ... .. .. .. .. .. .. ... .. .. .. 52
Quadratic triangular elements . . . . . . . . . . . . . . . . . . . . . 54
3.2.5 Two-dimensional linear triangular elements . . . . . . . . . . . . . . 48
3.2.6 Two-dimensional quadrilateral elements . . . . . . . . . . . . . . . 57
3.2.7 Isoparametric elements . ... .. .. .. .. .. .. ... .. .. .. 62
3.2.8 Three-dimensional elements . .. .. .. .. .. .. ... .. .. .. 70
3.3 Formulation (Element Characteristics) .. .. .. .. .. .. ... .. .. .. 75
3.3.1 Ritz method (Heat balance integral method—Goodman’s method) . 76
3.3.2 Rayleigh–Ritz method (Variational method) . . . . . . . . . . . . . 78
3.3.3 The method of weighted residuals .. .. .. .. .. ... .. .. .. 80
3.3.4 Galerkin finite element method . . . . . . . . . . . . . . . . . . . . 85
3.4 Formulation for the Heat Conduction Equation . . . . . . . . . . . . . . . . 87
3.4.1 Variational approach .. ... .. .. .. .. .. .. ... .. .. .. 88
3.4.2 The Galerkin method .. ... .. .. .. .. .. .. ... .. .. .. 91
3.5 Requirements for Interpolation Functions . . . . . . . . . . . . . . . . . . . 92
3.6 Summary .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 98
3.7 Exercise. .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 98
Bibliography . .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 100

4 Steady State Heat Conduction in One Dimension 102
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Plane Walls . .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 102
4.2.1 Homogeneous wall . .. ... .. .. .. .. .. .. ... .. .. .. 102
4.2.2 Composite wall . .. .. ... .. .. .. .. .. .. ... .. .. .. 103
4.2.3 Finite element discretization . .. .. .. .. .. .. ... .. .. .. 105
4.2.4 Wall with varying cross-sectional area . .. .. .. ... .. .. .. 107
4.2.5 Plane wall with a heat source: solution by linear elements . . . . . 108
4.2.6 Plane wall with a heat source: solution by quadratic elements . . . . 112
4.2.7 Plane wall with a heat source: solution by modified quadratic
equations (static condensation) . . . . . . . . . . . . . . . . . . . . 114
4.3 Radial Heat Flow in a Cylinder ... .. .. .. .. .. .. ... .. .. .. 115
4.3.1 Cylinder with heat source .. .. .. .. .. .. .. ... .. .. .. 117
4.4 Conduction–Convection Systems . . . . . . . . . . . . . . . . . . . . . . . 120
4.5 Summary .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 123
4.6 Exercise. .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 123
Bibliography . .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 125

5 Steady State Heat Conduction in Multi-dimensions 126
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Two-dimensional Plane Problems . .. .. .. .. .. .. .. . .. .. .. . 127
5.2.1 Triangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3 Rectangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4 Plate with Variable Thickness . ... .. .. .. .. .. .. ... .. .. .. 139