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J. Austin Cottrell, Thomas J. R. Hughes & Yuri Basilievs, University of Texas at Austin, USA
J. Austin Cottrell is a postdoctoral scholar at the University of Texas at Austin, having received his PhD in Computational and Applied Mathematics in 2007. Isogeometric analysis is a topic pioneered by his graduate research under the supervision of Tom Hughes.
Tom Hughes was a leading professor of mechanical engineering at Stanford University before being lured to join the University of Texas at Austin in 2002 as Professor of Aerospace Engineering and Engineering Mechanics within the Institute for Computational Engineering and Sciences. He is co-editor of the International Journal of Computer Methods in Applied Mechanics and Engineering, a founder and past President of USACM and IACM, and past Chairman of the Applied Mechanics Division of ASME. A world leader in the development of the finite element method, he has received the Walter L. Huber Civil Engineering Research Prize from ASCE, the Melville Medal from ASME, the Computational Mechanics Award from the Japan Society of Mechanical Engineers, the von Neumann Medal from USACM, the Gauss-Newton Medal from IACM, and the Worcester Reed Warner Medal from ASME. Dr. Hughes is a member of the National Academy of Engineering.
Yuri Basilievs also obtained his PhD from the University of Texas at Austin in 2007 under the supervision of Tom Hughes.
Preface | p. xi |
From CAD and FEA to Isogeometric Analysis: An Historical Perspective | p. 1 |
Introduction | p. 1 |
The need for isogeometric analysis | p. 1 |
Computational geometry | p. 7 |
The evolution of FEA basis functions | p. 8 |
The evolution of CAD representations | p. 12 |
Things you need to get used to in order to understand NURBS-based isogeometric analysis | p. 16 |
Notes | p. 18 |
NURBS as a Pre-analysis Tool: Geometric Design and Mesh Generation | p. 19 |
B-splines | p. 19 |
Knot vectors | p. 19 |
Basis functions | p. 21 |
B-spline geometries | p. 28 |
Refinement | p. 36 |
Non-Uniform Rational B-Splines | p. 47 |
The geometric point of view | p. 47 |
The algebraic point of view | p. 50 |
Multiple patches | p. 52 |
Generating a NURBS mesh: a tutorial | p. 54 |
Preliminary considerations | p. 56 |
Selection of polynomial orders | p. 59 |
Selection of knot vectors | p. 60 |
Selection of control points | p. 61 |
Notation | p. 65 |
Data for the bent pipe | p. 66 |
Notes | p. 68 |
NURBS as a Basis for Analysis: Linear Problems | p. 69 |
The isoparametric concept | p. 69 |
Defining functions on the domain | p. 71 |
Boundary value problems (BVPs) | p. 72 |
Numerical methods | p. 72 |
Galerkin | p. 73 |
Collocation | p. 78 |
Least-squares | p. 81 |
Meshless methods | p. 83 |
Boundary conditions | p. 84 |
Dirichlet boundary conditions | p. 84 |
Neumann boundary conditions | p. 86 |
Robin boundary conditions | p. 86 |
Multiple patches revisited | p. 87 |
Local refinement | p. 87 |
Arbitrary topologies | p. 91 |
Comparing isogeometric analysis with classical finite element analysis | p. 92 |
Code architecture | p. 94 |
Similarities and differences | p. 97 |
Shape function routine | p. 97 |
Error estimates | p. 103 |
Notes | p. 106 |
Linear Elasticity | p. 109 |
Formulating the equations of elastostatics | p. 110 |
Strong form | p. 111 |
Weak form | p. 111 |
Galerkin's method | p. 112 |
Assembly | p. 113 |
Infinite plate with circular hole under constant in-plane tension | p. 116 |
Thin-walled structures modeled as solids | p. 120 |
Thin cylindrical shell with fixed ends subjected to constant internal pressure | p. 120 |
The shell obstacle course | p. 123 |
Hyperboloidal shell | p. 131 |
Hemispherical shell with a stiffener | p. 136 |
Geometrical data for the hemispherical shell | p. 142 |
Geometrical data for a cylindrical pipe | p. 142 |
Element assembly routine | p. 144 |
Notes | p. 147 |
Vibrations and Wave Propagation | p. 149 |
Longitudinal vibrations of an elastic rod | p. 149 |
Formulating the problem | p. 149 |
Results: NURBS vs. FEA | p. 151 |
Analytically computing the discrete spectrum | p. 155 |
Lumped mass approaches | p. 159 |
Rotation-free analysis of the transverse vibrations of a Bernoulli-Euler beam | p. 164 |
Transverse vibrations of an elastic membrane | p. 165 |
Linear and nonlinear parameterizations revisited | p. 166 |
Formulation and results | p. 166 |
Rotation-free analysis of the transverse vibrations of a Poisson-Kirchhoff plate | p. 168 |
Vibrations of a clamped thin circular plate using three-dimensional solid elements | p. 169 |
Formulating the problem | p. 170 |
Results | p. 172 |
The NASA aluminum testbed cylinder | p. 172 |
Wave propagation | p. 173 |
Dispersion analysis | p. 178 |
Duality principle | p. 179 |
Kolmogorov n-widths | p. 180 |
Notes | p. 184 |
Time-Dependent Problems | p. 185 |
Elastodynamics | p. 185 |
Semi-discrete methods | p. 186 |
Matrix formulation | p. 186 |
Viscous damping | p. 187 |
Predictor/multicorrector Newmark algorithms | p. 188 |
Space-time finite elements | p. 191 |
Nonlinear Isogeometric Analysis | p. 197 |
The Newton-Raphson method | p. 197 |
Isogeometric analysis of nonlinear differential equations | p. 198 |
Nonlinear heat conduction | p. 198 |
Applying the Newton-Raphson method | p. 199 |
Nonlinear finite element analysis | p. 200 |
Nonlinear time integration: The generalized-a method | p. 202 |
Note | p. 209 |
Nearly Incompressible Solids | p. 211 |
B formulation for linear elasticity using NURBS | p. 212 |
An intuitive look at mesh locking | p. 213 |
Strain projection and the B method | p. 215 |
B, the projection operator, and NURBS | p. 216 |
Infinite plate with circular hole under in-plane tension | p. 220 |
F formulation for nonlinear elasticity | p. 221 |
Constitutive equations | p. 221 |
Pinched torus | p. 222 |
Notes | p. 225 |
Fluids | p. 227 |
Dispersion analysis | p. 227 |
Pure advection: the first-order wave equation | p. 227 |
Pure diffusion: the heat equation | p. 230 |
The variational multiscale (VMS) method | p. 231 |
Numerical example: linear advection-diffusion | p. 232 |
The Green's operator | p. 233 |
A multiscale decomposition | p. 235 |
The variational multiscale formulation | p. 237 |
Reconciling Galerkin's method with VMS | p. 238 |
Advection-diffusion equation | p. 239 |
Formulating the problem | p. 240 |
The streamline upwind/Petrov-Galerkin (SUPG) method | p. 240 |
Numerical example: advection-diffusion in two dimensions, revisited | p. 241 |
Turbulence | p. 243 |
Incompressible Navier-Stokes equations | p. 245 |
Multiscale residual-based formulation of the incompressible Navier-Stokes equations employing the advective form | p. 246 |
Turbulent channel flow | p. 248 |
Notes | p. 251 |
Fluid-Structure Interaction and Fluids on Moving Domains | p. 253 |
The arbitrary Lagrangian-Eulerian (ALE) formulation | p. 253 |
Inflation of a balloon | p. 254 |
Flow in a patient-specific abdominal aorta with aneurysm | p. 256 |
Construction of the arterial cross-section | p. 256 |
Numerical results | p. 261 |
Rotating components | p. 264 |
Coupling of the rotating and stationary domains | p. 266 |
Numerical example: two propellers spinning in opposite directions | p. 272 |
A geometrical template for arterial blood flow modeling | p. 275 |
Higher-order Partial Differential Equations | p. 279 |
The Cahn-Hilliard equation | p. 279 |
The strong form | p. 280 |
The dimensionless strong form | p. 281 |
The weak form | p. 281 |
Numerical results | p. 282 |
A two-dimensional example | p. 282 |
A three-dimensional example | p. 282 |
The continuous/discontinuous Galerkin (CDG) method | p. 283 |
Note | p. 285 |
Some Additional Geometry | p. 287 |
The polar form of polynomials | p. 287 |
Bezier curves and the de Casteljau algorithm | p. 288 |
Continuity of piecewise curves | p. 291 |
The polar form of B-splines | p. 293 |
Knot vectors and control points | p. 293 |
Knot insertion and the de Boor algorithm | p. 295 |
Bezier decomposition and function subdivision | p. 297 |
Note | p. 301 |
State-of-the-Art and Future Directions | p. 303 |
State-of-the-art | p. 303 |
Future directions | p. 305 |
Connectivity Arrays | p. 313 |
The INC Array | p. 313 |
The IEN array | p. 315 |
The ID array | p. 318 |
The scalar case | p. 318 |
The vector case | p. 318 |
The LM array | p. 319 |
Note | p. 321 |
References | p. 323 |
Index | p. 333 |
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