Table Of ContentPetr Krysl
Thermal and Stress Analysis
with the
Finite Element Method
FAESOR
Accompanied by the MATLABr toolbox
December 13, 2010
Pressure Cooker Press
San Diego
c 2010 Petr Krysl
(cid:13)
Contents
1 Model of a Taut Wire ........................................................ 1
1.1 Deriving the PDE model .................................................... 1
1.2 Balance equation ........................................................... 1
1.3 Boundary conditions ....................................................... 2
1.4 Boundary conditions (in space)............................................... 2
Exercise 1 ................................................................. 3
Exercise 2 ................................................................. 4
1.5 Initial conditions (boundary conditions in time) ................................ 5
1.6 Initial Boundary Value Problem.............................................. 5
1.7 Examples.................................................................. 5
Exercise 3 ................................................................. 5
Exercise 4 ................................................................. 7
Exercise 5 ................................................................. 10
2 The Method of Galerkin ..................................................... 13
2.1 Residual of the balance equation ............................................. 13
2.2 Integral test of the residual .................................................. 14
2.3 Test function .............................................................. 14
2.4 Trial function .............................................................. 15
2.5 Shifting derivatives ......................................................... 16
2.6 Essential boundary condition ................................................ 16
2.7 Natural boundary condition ................................................. 16
2.8 Stiffness matrix and load vector .............................................. 18
Exercise 6 ................................................................. 19
Exercise 7 ................................................................. 20
Exercise 8 ................................................................. 21
Exercise 9 ................................................................. 23
Exercise 10 ................................................................ 25
2.9 Piecewise linear basis functions............................................... 26
Exercise 11 ................................................................ 28
Exercise 12 ................................................................ 30
Exercise 13 ................................................................ 31
Exercise 14 ................................................................ 32
2.10 Bookkeeping in the finite element method ..................................... 34
2.11 Finite element Galerkin method .............................................. 39
2.12 Element-by-element computations ............................................ 40
2.12.1 Elementwise quantities................................................ 42
2.13 Prescribed displacements .................................................... 45
Exercise 15 ................................................................ 47
4 Contents
2.14 Partitioned form ........................................................... 48
2.14.1 Derivation of the partitioned form ...................................... 50
2.15 Principle of superposition.................................................... 51
3 Taut wire dynamics with the Galerkin method ............................... 53
3.1 Residual of the balance equation ............................................. 53
3.2 Integral test of the residual .................................................. 53
3.3 Weighted residual manipulations ............................................. 54
3.4 Mass matrix and load vector................................................. 55
3.5 Elementwise mass matrix.................................................... 56
3.6 Initial conditions ........................................................... 57
Exercise 16 ................................................................ 57
3.7 Free vibration.............................................................. 60
Exercise 17 ................................................................ 61
4 Further refinements of the Galerkin finite element method ................... 63
4.1 Numerical quadrature....................................................... 63
Exercise 18 ................................................................ 64
Exercise 19 ................................................................ 66
Exercise 20 ................................................................ 67
4.2 Gauss quadrature .......................................................... 68
Exercise 21 ................................................................ 69
4.3 Derivatives of basis functions ................................................ 71
Exercise 22 ................................................................ 72
5 More about Boundary Conditions ............................................ 73
5.1 Mixed essential and natural boundary conditions ............................... 73
5.2 Essential boundary conditions only ........................................... 74
5.3 Natural boundary conditions only ............................................ 74
5.4 Concentrated forces in the interior............................................ 75
Exercise 23 ................................................................ 78
5.5 Elementwise stiffness matrix properties........................................ 79
5.6 Removing rigid body modes ................................................. 81
5.6.1 Adding pin support................................................... 82
5.6.2 Adding spring support ................................................ 83
5.7 Using springs to enforce essential boundary conditions .......................... 84
Exercise 24 ................................................................ 87
6 Statics and Dynamics of Taut Wire with the FEM toolbox ................... 91
6.1 Statics: uniform load........................................................ 91
6.2 Sparse matrices ............................................................ 94
6.3 Free vibration.............................................................. 97
Exercise 25 ................................................................ 97
Exercise 26 ................................................................ 99
6.4 Integration of transient motion............................................... 101
6.4.1 Using built-in Matlab solver ........................................... 102
6.4.2 Using the Trapezoidal integrator ....................................... 103
7 Model of Heat Conduction ................................................... 107
7.1 Balance equation ........................................................... 107
7.2 Constitutive equation ....................................................... 109
7.3 Boundary conditions........................................................ 110
7.3.1 On the sufficiency of boundary conditions................................ 111
7.4 Example of Boundary Condition formulation................................... 111
Contents 5
7.5 Initial condition ............................................................ 112
7.6 Summary of the PDE model of heat conduction ................................ 113
7.7 Parallels between the taut wire and the heat conduction model................... 113
8 Galerkin Method for the Model of Heat Conduction ......................... 117
8.1 Weighted residual formulation................................................ 117
8.2 One-dimensional heat conduction model....................................... 119
8.3 Comparison with the prestressed wire ......................................... 121
8.4 Heat conduction 1D FEM ................................................... 121
Exercise 27 ................................................................ 122
Exercise 28 ................................................................ 123
Exercise 29 ................................................................ 125
8.5 Reducing the model dimension to two......................................... 126
8.6 Test and trial functions: basis functions on triangulations........................ 128
8.7 Basis functions on the standard triangle ....................................... 129
Exercise 30 ................................................................ 130
8.8 Direct construction of the T3 basis functions................................... 132
Exercise 31 ................................................................ 134
Exercise 32 ................................................................ 134
8.9 Discretizing the weighted residual equation .................................... 135
8.10 Derivatives of the basis functions; Jacobian .................................... 138
8.11 Numerical integration....................................................... 141
Exercise 33 ................................................................ 142
Exercise 34 ................................................................ 143
Exercise 35 ................................................................ 145
Exercise 36 ................................................................ 146
8.12 Conductivity matrix and heat loads........................................... 147
Exercise 37 ................................................................ 151
Exercise 38 ................................................................ 152
Exercise 39 ................................................................ 154
Exercise 40 ................................................................ 155
8.13 Surface heat transfer matrix and load ......................................... 157
Exercise 41 ................................................................ 158
Exercise 42 ................................................................ 159
Exercise 43 ................................................................ 160
Exercise 44 ................................................................ 162
Exercise 45 ................................................................ 163
Exercise 46 ................................................................ 164
9 Steady-state Heat Conduction Solutions ..................................... 167
9.1 Steady-state heat conduction equation ........................................ 167
9.2 Thick-walled tube .......................................................... 167
9.3 Orthotropic insert .......................................................... 169
9.4 The T4 NAFEMS Benchmark................................................ 172
Exercise 47 ................................................................ 174
Exercise 48 ................................................................ 178
10 Transient Heat Conduction Solutions ........................................ 181
10.1 Discretization in time for transient heat conduction............................. 181
10.2 The T3 NAFEMS Benchmark................................................ 183
10.3 Transient cooling in a shrink-fitting application................................. 185
Exercise 49 ................................................................ 188
6 Contents
11 Expanding the Library of Element Types .................................... 191
11.1 Quadratic triangle T6....................................................... 191
Exercise 50 ................................................................ 193
11.2 Quadratic 1-D element L3 ................................................... 197
Exercise 51 ................................................................ 198
Exercise 52 ................................................................ 200
Exercise 53 ................................................................ 203
Exercise 54 ................................................................ 204
11.3 Point element P1........................................................... 205
11.4 Integrating over m-dimensional domains....................................... 206
11.5 Tetrahedron T4 ............................................................ 209
11.6 Simplex elements........................................................... 210
Exercise 55 ................................................................ 211
Exercise 56 ................................................................ 212
11.7 QuadrilateralQ4 ........................................................... 214
11.8 Hexahedron H8 ............................................................ 215
Exercise 57 ................................................................ 215
Exercise 58 ................................................................ 216
Exercise 59 ................................................................ 218
Exercise 60 ................................................................ 220
11.9 Extracting the mesh boundary ............................................... 223
12 Discretization Error, Error Control, and Convergence ........................ 225
12.1 Motivating example......................................................... 225
Exercise 61 ................................................................ 225
Exercise 62 ................................................................ 227
12.2 Interpolation errors......................................................... 231
12.2.1 Interpolation error for temperature ..................................... 232
12.2.2 Interpolation error for temperature gradient ............................. 234
12.2.3 Controlling the error; Convergence rate ................................. 235
12.3 Richardson extrapolation.................................................... 237
Exercise 63 ................................................................ 238
Exercise 64 ................................................................ 241
12.4 The T4 NAFEMS Benchmark revisited........................................ 242
12.5 Graded meshes............................................................. 243
12.6 Shrink fitting revisited ...................................................... 243
12.7 Representing functions by interpolation ....................................... 245
Exercise 65 ................................................................ 246
Exercise 66 ................................................................ 247
Exercises ...................................................................... 249
13 Model of Elastodynamics .................................................... 251
13.1 Balance of linear momentum................................................. 251
13.2 Stress..................................................................... 253
Exercise 67 ................................................................ 256
Exercise 68 ................................................................ 258
13.2.1 Balance of angular momentum and stress symmetry. ...................... 259
Exercise 69 ................................................................ 259
Exercise 70 ................................................................ 260
Exercise 71 ................................................................ 261
13.3 Local equilibrium........................................................... 261
13.3.1 Change of linear momentum ........................................... 262
13.3.2 Stress divergence ..................................................... 262
13.3.3 All together now ..................................................... 264
Contents 7
13.4 Strains and displacements ................................................... 265
13.5 Constitutive equation ....................................................... 267
13.6 Boundary conditions........................................................ 268
13.6.1 Example: concrete dam ............................................... 268
13.6.2 Example: rigid punch ................................................. 269
13.6.3 Formal definition of the boundary conditions............................. 270
13.6.4 Inadmissible “concentrated” boundary conditions......................... 270
13.6.5 Symmetry and anti-symmetry.......................................... 271
13.6.6 Example: a pure-traction problem ...................................... 273
13.6.7 Example: shaft under torsion .......................................... 275
13.6.8 Example: overspecified boundary conditions ............................. 275
13.7 Initial conditions ........................................................... 276
14 Galerkin Formulation for Elastodynamics .................................... 277
14.1 Manipulation of the residuals ................................................ 277
14.1.1 The first two steps.................................................... 277
14.1.2 Step 3: Preliminaries.................................................. 278
14.1.3 Step 3: Conclusion.................................................... 279
14.2 Method of weighted residuals as the principle of virtual work..................... 279
14.3 Discretizing................................................................ 280
14.3.1 The trial function .................................................... 280
14.3.2 The test function..................................................... 281
14.3.3 Producing the requisite equations ...................................... 282
14.4 The discrete equations: system of ODE’s ...................................... 283
14.4.1 Inertial term: Mass matrix............................................. 284
Exercise 72 ................................................................ 284
14.4.2 Body loads and traction loads.......................................... 287
Exercise 73 ................................................................ 287
Exercise 74 ................................................................ 287
14.4.3 Resisting forces: Stiffness matrix ....................................... 288
14.4.4 Summary of the elastodynamics ODE’s ................................. 289
14.5 Constitutive equations of linearly elastic materials .............................. 289
14.5.1 General anisotropic material. .......................................... 290
14.5.2 Orthotropic material. ................................................. 290
14.5.3 Transversely isotropic material. ........................................ 290
14.5.4 Isotropic material..................................................... 291
14.6 Imposed (thermal) strains ................................................... 292
14.7 Strain-displacement matrix .................................................. 293
Exercise 75 ................................................................ 294
Exercise 76 ................................................................ 296
Exercise 77 ................................................................ 298
14.8 Material directions and basis transformation ................................... 300
14.9 Stiffness matrix ............................................................ 301
14.10Pure-tractionproblems and singular stiffness................................... 303
Exercises ...................................................................... 304
15 Finite Elements for true 3-D Problems ....................................... 305
15.1 Modal analysis with the tetrahedron T4: the drum.............................. 305
15.2 Modal analysis with the tetrahedron T4: the composite rod...................... 307
15.3 Tetrahedron T10 ........................................................... 309
15.3.1 Example: the drum revisited........................................... 310
15.4 The composite rod with the tetrahedron T10................................... 311
15.5 Static analysis with hexahedra H8 and H20 .................................... 312
15.5.1 Hexahedron H8 ...................................................... 312
8 Contents
15.5.2 Dilatational locking................................................... 312
15.5.3 Shear locking ........................................................ 315
15.5.4 Thin clamped square plate with concentrated load........................ 315
15.5.5 Quadratic element H20................................................ 316
15.5.6 Quadratic element Q8................................................. 319
15.5.7 Pinched cylinder ..................................................... 320
15.5.8 Pinched sphere....................................................... 321
15.5.9 Beam deflection revisited.............................................. 321
15.6 Errors,validation, and verification............................................ 322
15.6.1 Verification and Prediction ............................................ 324
15.6.2 Validation........................................................... 324
15.6.3 Errors .............................................................. 325
15.6.4 Using modeling to make predictions .................................... 325
15.6.5 Using benchmarks.................................................... 326
Exercises ...................................................................... 326
16 Analyzing the Stresses ....................................................... 329
16.1 Singularities ............................................................... 329
16.2 Interpretation of stresses .................................................... 331
16.3 Stress concentrations........................................................ 334
16.4 Adaptive refinement ........................................................ 336
17 Plane Strain, Plane Stress, and Axisymmetric Models ....................... 341
17.1 Plane strain model reduction................................................. 341
17.2 Plane stress model reduction................................................. 343
17.3 Model reduction for axial symmetry .......................................... 344
17.4 Material stiffness for two-dimensional models .................................. 346
17.5 Strain-displacement matrices for two-dimensional models ........................ 347
17.6 Integration for two-dimensional models........................................ 348
17.7 Thermal strains in two-dimensional models .................................... 349
17.8 Examples.................................................................. 350
17.8.1 Thermal strains in a bimetallic assembly ................................ 350
17.8.2 Orthotropic balloon .................................................. 353
17.9 Transient dynamic analysis .................................................. 355
17.9.1 Centered difference time stepping....................................... 355
17.9.2 Example: stress wave propagation ...................................... 357
17.10Solved exercises ............................................................ 358
Exercise 78 ................................................................ 358
Exercise 79 ................................................................ 361
Exercises ...................................................................... 362
18 Consistency + Stability = Convergence ...................................... 363
18.1 Consistency................................................................ 363
18.1.1 Completeness ........................................................ 363
18.1.2 Compatibility........................................................ 364
18.2 Stability................................................................... 364
18.2.1 Conclusion .......................................................... 366
Exercises ...................................................................... 366
References........................................................................ 369
Index............................................................................. 371
1
Model of a Taut Wire
This chapter will formulate a relatively simple model for the so-called initial boundary value prob-
lem that describes the deflection or vibrations of a taut string. In the next chapter, we will seek
approximate solutions to this model with the Galerkin method.
1.1 Deriving the PDE model
Figure 1.1 illustrates an idealization of a taut wire. The wire is under prestress by the force P,
assumedtobe uniformalongthelengthofthewire.Theleft-handendisimmovablyfixed,while the
right-handendisheldinafixturewhichcanslideperpendicularlytotheaxisofthewire(occasionally
referred to as a “roller”). A transverse force F is applied at the movable end. In addition, there
L
may be some distributed force q (in physical units of force per unit length) acting along the length
(for instance gravity). The transverse displacement is a function of both the axial coordinate x and
the time t, w =w(x,t) . The transverse displacement is assumed to be very small compared to the
length of the wire. The deformation in Figure 1.1 is highly magnified in order to be apparent.
Fig. 1.1. Schematic of taut wire
1.2 Balance equation
We take a segmentof length ∆x of the wire (see Fig.1.2). The forcesacting onthe segmentare the
prestressing forces in either cross-section and the resultant of the distributed load. By assumption
the deflection is very small comparedto the span, w L, and we also assume that the slope of the
≪
wireisverysmall,w =∂w/∂x 1.Thesegeometricalfeaturesareintroducedintothebalanceofall
′
≪
the forces.In the horizontaldirectionwe havejust the two prestressingforces in opposite directions
andhence they cancel.Inthe verticaldirectionwe addupthe componentsofthe prestressingforces
in the verticaldirectionwith the transverseload,where we takethe Taylor-seriesapproximationfor
the slope at x+∆x
w(x+∆x) w (x)+w (x)∆x,
′ ′ ′′
≈
and
2 Thermal and Stress Analysis with the FEM
∂2w(x)
w (x)= ,
′′
∂x2
and we equate the resultant of the vertical forces to the inertial force (Newton’s law). This leads to
a balance equation for the taut wire
Pw +q =µw¨ , (1.1)
′′
∂2w
where w¨ = is the acceleration.
∂t2
Fig. 1.2. The forces acting on a segment of thetaut wire
1.3 Boundary conditions
The function w that describes the transverse deflection takes two arguments, x, and t. It is defined
on a rectangular domain shown in Fig. 1.3: 0 x L, and 0 t t¯. The deflection function needs
≤ ≤ ≤ ≤
to be determined to satisfy the balance equation (1.1). However,derivatives with the respect to the
variables x and t needs to be “integrated” in order to arrive at a solution, and that implies the
presence of integration “constants”. In order to determine the solution uniquely we need to resolve
the integrations,andforthatweneedadditionalequations.Indeed,there are otherthingswe would
require a solution to satisfy, namely the conditions at the boundaries of the domain rectangle.
How many pieces of information do we need to know? A reasonable answer is, ‘Enough to make
the solution unique.’ Using the definitions
∂w ∂w
v = , θ = ,
∂t ∂x
we may rewrite the balance equation that involves the second derivatives of the function w as a
system of first order partial differential equations
∂θ ∂v
=
∂t ∂x
∂θ ∂v
P +q µ = 0
∂x − ∂t
∂v ∂θ
Foreachderivative , ,oneboundarycondition(integrationconstant)willbeneeded.Similarly,
∂x ∂x
∂v ∂θ
for each of the time derivatives , and one boundary condition along the time axis will be
∂t ∂t
required.
1.4 Boundary conditions (in space)
The conditions on w along the edges of the domain rectangle parallel to the time axis are known
(for historical reasons)as the boundary conditions. (Perhaps also because they are applied along
the physical boundaries of the structure.)
