Isogeometric Analysis

J. Austin Cottrell, Thomas J. R. Hughes & Yuri Basilievs, University of Texas at Austin, USAJ. 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.

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• Preface 1 From CAD and FEA to Isogeometric Analysis: An Historical Perspective1.1 Introduction1.2 The evolution of FEA basis functions1.3 The evolution of CAD representations1.4 Things you need to get used to in order to understand NURBS-based isogeometric analysisNotes2 NURBS as a Pre-analysis Tool: Geometric Design and Mesh Generation2.1 B-splines2.2 Non-Uniform Rational B-Splines2.3 Multiple patches2.4 Generating a NURBS mesh: a tutorial2.5 NotationAppendix 2.A: Data for the bent pipeNotes3 NURBS as a Basis for Analysis: Linear Problems3.1 The isoparametric concept3.2 Boundary value problems3.3 Numerical methods3.4 Boundary conditions3.5 Multiple patches revisited3.6 Comparing isogeometric analysis with classical finite element analysisAppendix 3.A: Shape function routineAppendix 3.B: Error estimatesNotes4 Linear Elasticity4.1 Formulating the equations of elastostatics4.2 Infinite plate with circular hole under constant in-plane tension4.3 Thin-walled structures modeled as solidsAppendix 4.A: Geometrical data for the hemispherical shellAppendix 4.B: Geometrical data for a cylindrical pipeAppendix 4.C: Element assembly routineNotes5 Vibrations and Wave Propagation5.1 Longitudinal vibrations of an elastic rod5.2 Rotation-free analysis of the transverse vibrations of a Bernoulli–Euler beam5.3 Transverse vibrations of an elastic membrane5.4 Rotation-free analysis of the transverse vibrations of a Poisson–Kirchhoff plate5.5 Vibrations of a clamped thin circular plate using three-dimensional solid elements5.6 The NASA aluminum testbed cylinder5.7 Wave propagationAppendix 5.A: Kolmogorov n-widthsNotes6 Time-Dependent Problems6.1 Elastodynamics6.2 Semi-discrete methods6.3 Space–time finite elements7 Nonlinear Isogeometric Analysis7.1 The Newton–Raphson method7.2 Isogeometric analysis of nonlinear differential equations7.3 Nonlinear time integration: The generalized-α methodNote8 Nearly Incompressible Solids8.1 B formulation for linear elasticity using NURBS8.2 F formulation for nonlinear elasticityNotes9 Fluids9.1 Dispersion analysis9.2 The variational multiscale (VMS) method9.3 Advection–diffusion equation9.4 TurbulenceNotes10 Fluid–Structure Interaction and Fluids on Moving Domains10.1 The arbitrary Lagrangian–Eulerian (ALE) formulation10.2 Inflation of a balloon10.3 Flow in a patient-specific abdominal aorta with aneurysm10.4 Rotating componentsAppendix 10.A: A geometrical template for arterial blood flow modeling11 Higher-order Partial Differential Equations11.1 The Cahn–Hilliard equation11.2 Numerical results11.3 The continuous/discontinuous Galerkin (CDG) methodNote12 Some Additional Geometry12.1 The polar form of polynomials12.2 The polar form of B-splinesNote13 State-of-the-Art and Future Directions13.1 State-of-the-art13.2 Future directionsAppendix A: Connectivity ArraysA.1 The INC ArrayA.2 The IEN arrayA.3 The ID arrayA.3.1 The scalar caseA.3.2 The vector caseA.4 The LM arrayNoteReferencesIndex