Isogeometric analysis: toward integration of CAD and FEA

ISOGEOMETRIC\nANALYSIS......Page 7
Contents......Page 9
Preface......Page 15
1.1.1 The need for isogeometric analysis......Page 21
1.1.2 Computational geometry......Page 27
1.2 The evolution of FEA basis functions......Page 28
1.3 The evolution of CAD representations......Page 32
1.4 Things you need to get used to in order to understand NURBS-based isogeometric analysis......Page 36
Notes......Page 38
2.1.1 Knot vectors......Page 39
2.1.2 Basis functions......Page 41
2.1.3 B-spline geometries......Page 48
2.1.4 Refinement......Page 56
2.2.1 The geometric point of view......Page 67
2.2.2 The algebraic point of view......Page 70
2.3 Multiple patches......Page 72
2.4 Generating a NURBS mesh: a tutorial......Page 74
2.4.1 Preliminary considerations......Page 76
2.4.2 Selection of polynomial orders......Page 79
2.4.3 Selection of knot vectors......Page 80
2.4.4 Selection of control points......Page 81
2.5 Notation......Page 85
Appendix 2.A: Data for the bent pipe......Page 86
Notes......Page 88
3.1 The isoparametric concept......Page 89
3.1.1 Defining functions on the domain......Page 91
3.3 Numerical methods......Page 92
3.3.1 Galerkin......Page 93
3.3.2 Collocation......Page 98
3.3.3 Least-squares......Page 101
3.3.4 Meshless methods......Page 103
3.4.1 Dirichlet boundary conditions......Page 104
3.4.3 Robin boundary conditions......Page 106
3.5.1 Local refinement......Page 107
3.5.2 Arbitrary topologies......Page 111
3.6 Comparing isogeometric analysis with classical finite element analysis......Page 112
3.6.1 Code architecture......Page 114
Appendix 3.A: Shape function routine......Page 117
Appendix 3.B: Error estimates......Page 123
Notes......Page 126
4 Linear Elasticity......Page 129
4.1 Formulating the equations of elastostatics......Page 130
4.1.2 Weak form......Page 131
4.1.3 Galerkin’s method......Page 132
4.1.4 Assembly......Page 133
4.2 Infinite plate with circular hole under constant in-plane tension......Page 136
4.3.1 Thin cylindrical shell with fixed ends subjected to constant internal pressure......Page 140
4.3.2 The shell obstacle course......Page 143
4.3.3 Hyperboloidal shell......Page 151
4.3.4 Hemispherical shell with a stiffener......Page 156
Appendix 4.B: Geometrical data for a cylindrical pipe......Page 162
Appendix 4.C: Element assembly routine......Page 164
Notes......Page 167
5.1.1 Formulating the problem......Page 169
5.1.2 Results: NURBS vs. FEA......Page 171
5.1.3 Analytically computing the discrete spectrum......Page 175
5.1.4 Lumped mass approaches......Page 179
5.2 Rotation-free analysis of the transverse vibrations of a Bernoulli–Euler beam......Page 184
5.3 Transverse vibrations of an elastic membrane......Page 185
5.3.2 Formulation and results......Page 186
5.4 Rotation-free analysis of the transverse vibrations of a Poisson–Kirchhoff plate......Page 188
5.5 Vibrations of a clamped thin circular plate using three-dimensional solid elements......Page 189
5.5.1 Formulating the problem......Page 190
5.6 The NASA aluminum testbed cylinder......Page 192
5.7 Wave propagation......Page 193
5.7.1 Dispersion analysis......Page 198
5.7.2 Duality principle......Page 199
Appendix 5.A: Kolmogorov n-widths......Page 200
Notes......Page 204
6.1 Elastodynamics......Page 205
6.2.1 Matrix formulation......Page 206
6.2.2 Viscous damping......Page 207
6.2.3 Predictor/multicorrector Newmark algorithms......Page 208
6.3 Space–time finite elements......Page 211
7.1 The Newton–Raphson method......Page 217
7.2.1 Nonlinear heat conduction......Page 218
7.2.2 Applying the Newton–Raphson method......Page 219
7.2.3 Nonlinear finite element analysis......Page 220
7.3 Nonlinear time integration: The generalized-α method......Page 222
Note......Page 229
8 Nearly Incompressible Solids......Page 231
8.1 formulation for linear elasticity using NURBS......Page 232
8.1.1 An intuitive look at mesh locking......Page 233
8.1.2 Strain projection and the method......Page 235
8.1.3 B, the projection operator, and NURBS......Page 236
8.1.4 Infinite plate with circular hole under in-plane tension......Page 240
8.2.1 Constitutive equations......Page 241
8.2.2 Pinched torus......Page 242
Notes......Page 245
9.1.1 Pure advection: the first-order wave equation......Page 247
9.1.2 Pure diffusion: the heat equation......Page 250
9.2 The variational multiscale (VMS) method......Page 251
9.2.1 Numerical example: linear advection–diffusion......Page 252
9.2.2 The Green’s operator......Page 253
9.2.3 A multiscale decomposition......Page 255
9.2.4 The variational multiscale formulation......Page 257
9.2.5 Reconciling Galerkin’s method with VMS......Page 258
9.3 Advection–diffusion equation......Page 259
9.3.2 The streamline upwind/Petrov–Galerkin (SUPG) method......Page 260
9.3.3 Numerical example: advection–diffusion in two dimensions, revisited......Page 261
9.4 Turbulence......Page 263
9.4.1 Incompressible Navier–Stokes equations......Page 265
9.4.2 Multiscale residual-based formulation of the incompressible Navier–Stokes equations employing the advective form......Page 266
9.4.3 Turbulent channel flow......Page 268
Notes......Page 271
10.1 The arbitrary Lagrangian–Eulerian (ALE) formulation......Page 273
10.2 Inflation of a balloon......Page 274
10.3.1 Construction of the arterial cross-section......Page 276
10.3.2 Numerical results......Page 281
10.4 Rotating components......Page 284
10.4.1 Coupling of the rotating and stationary domains......Page 286
10.4.2 Numerical example: two propellers spinning in opposite directions......Page 292
Appendix 10.A: A geometrical template for arterial blood flow modeling......Page 295
11.1 The Cahn–Hilliard equation......Page 299
11.1.1 The strong form......Page 300
11.1.3 The weak form......Page 301
11.2.2 A three-dimensional example......Page 302
11.3 The continuous/discontinuous Galerkin (CDG) method......Page 303
Note......Page 305
12.1 The polar form of polynomials......Page 307
12.1.1 Bézier curves and the de Casteljau algorithm......Page 308
12.1.2 Continuity of piecewise curves......Page 311
12.2.1 Knot vectors and control points......Page 313
12.2.2 Knot insertion and the de Boor algorithm......Page 315
12.2.3 Bézier decomposition and function subdivision......Page 317
Note......Page 321
13.1 State-of-the-art......Page 323
13.2 Future directions......Page 325
A.1 The INC Array......Page 333
A.2 The IEN array......Page 335
A.3.2 The vector case......Page 338
A.4 The LM array......Page 339
Note......Page 341
References......Page 343
Index......Page 353