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A pragmatic introduction to the finite element method for thermal and stress analysis : with the matlab toolkit SOFEA PDF

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Preview A pragmatic introduction to the finite element method for thermal and stress analysis : with the matlab toolkit SOFEA

Petr 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.)

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