Isogeometric Analysis Toward Integration of

"The authors are the originators of isogeometric analysis, are excellent scientists and good educators. It is very original. There is no other book on this topic." Rene de Borst, Eindhoven University of TechnologyWritten by leading experts in the field and featuring fully integrated colour throughout, Isogeometric Analysis provides a groundbreaking solution for the integration of CAD and FEA technologies. Tom Hughes and his researchers present their pioneering isogeometric approach, which aims to integrate the two techniques of CAD and FEA using precise NURBS geometry in the FEA application. This technology offers the potential to revolutionise automobile, ship and airplane design and analysis by allowing models to be designed, tested and adjusted in one integrative stage.Providing a systematic approach to the topic, the authors begin with a tutorial introducing the foundations of Isogeometric Analysis, before advancing to a comprehensive coverage of the most recent developments in the technique. The authors offer a clear explanation as to how to add isogeometric capabilities to existing finite element computer programs, demonstrating how to implement and use the technology. Detailed programming examples and datasets are included to impart a thorough knowledge and understanding of the material. Provides examples of different applications, showing the reader how to implement isogeometric models Addresses readers on both sides of the CAD/FEA divide Describes Non-Uniform Rational B-Splines (NURBS) basis functions

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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Isogeometric Analysis Toward Integration of CAD and FEA

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