1 Introduction

1

Page References

2

How to use this Book

3

Preliminaries

2 Graphical Facilities

5

Downloading the Free FlexPDE program

5

The FlexPDE Editor

6

Plotting Functions of One Variable

7

Running the Problem Descriptor File

8

Plotting Functions of Two Variables

9

Help and Manual

15

3 "Curly" Velocity Fields

17

Liquid, Rotating as a Disk

17

Non-Constant w

19

More General Velocity Fields

24

4 Fields of Gravitation

26

Earth and Moon as Point Masses

26

Planets of Finite Size

28

Divergence of g

33

5 Laplace and Poisson Equations

35

Known Values on Boundaries

35

Values and Derivatives on Boundaries

37

Multiplying through the PDE

38

Derivative Boundary Conditions Only

39

Solution over a Quarter-Circle

40

The Poisson Equation

41

Elastic Deformation

6 Elementary Elasticity in (x,y) Space

43

Plane Stress

44

Boundary Conditions

46

Steel Plate Under Compressive Strain

48

Strained Plate, No Sliding

52

Plate Subjected to Pressure at One End

54

Plate with Tension Applied at both Ends

55

Plate Subjected to Shear Forces

56

7 Straight Beams

59

Beam with Uniform Load

59

Comparison with an Analytic Solution

61

Further Analysis of Stress

62

Beam Loaded on a Line

67

Gravity Bending

68

Forces and Moments

70

Beam with Bending Moments at Ends

71

Elastic Energy

75

Plane Strain

76

8 Hooks, Rings and Tubes

80

Semi-Circular Beam

80

Compound Semi-Circular Beam

82

Tube with Internal and External Pressure

84

Comparison with Exact Solution

87

Tube with Tangential Load

88

Tube Strained between Two Edges

90

9 Elasticity in (r ,z) Space

96

Tube with Internal Pressure

98

Tube Pressurized over Part of its Length

101

Stress Concentration in a Shaft

103

10 Torsion

108

Twisted Circular Steel Shaft

109

Twisted Square Bar

111

General PDE for Torsion

113

Compound Bar

114

Hollow Bar

115

PDE for Warping

116

Warping of an Elliptical Shaft

119

11 Thermo-Elasticity

122

Uniformly Heated Semi-Circular Rod

123

Thermo-Elasticity in (x,y) Space

125

Steel Block, Strained by a Temperature Gradient

126

Bi-Metallic Block

128

Simultaneous PDEs for Stress and Temperature

131

PDEs for Axially Symmetric Problems

133

Axially Symmetric heating of a Cylinder

135

12 Deformation in Three Dimensions

139

Two Point Charges of Different Signs

139

Non-Linear Set of Point Charges

142

Laplace Equation in 3D

143

Elasticity in Three Dimensions

146

Include-File for 3D Elasticity

147

Capturing Descriptor Errors

148

Plate under Pressure from Top

148

Bending of a Beam under Gravity

150

Beam under Asymmetric Load

152

Tube under Asymmetric Load

154

Torsion and Warping of a Hollow Bar

156

Elastic Vibration

13 Waves Along a Beam

160

Complex Solution

161

Forced Shear Vibration

163

Animated Shear Vibration

165

Driving Force

167

Wavelength

167

Longitudinal Vibration

168

Bending Vibration

170

14 Eigenstates in Two Dimensions

173

Frequency Scan

173

Finding Eigenstates

176

Plate with Free Boundaries

178

Animation of a Vibrating Plate

180

15 Eigenstates of a Membrane

183

Circular Membrane

184

Triangular Drumhead

186

Fancy Drumhead

188

16 Eigenstates in Three Dimensions

190

Vibration of a Rectangular Block

191

Vibration of a Tube

193

Vibrating Circular Plate

195

Vibrating Plate with Free Boundaries

197

Vibrating Sledgehammer

199

17 Waves in Fluids

203

Complex Solution

204

Plane Acoustic Wave

206

Resonance

208

Constricted Cavity

209

Cylindrical Resonator

211

Appendix

Principles of Finite Element Analysis

215

Interpolation and Differentiation in 2D

215

Interpolating to Obtain a Solution in 2D

217

Solving for Node Values in 2D

219

Natural Boundary Conditions

221

Conclusion

223

References

224

Vocabulary of FlexPDE

225