Isogeometric Analysis – Toward Integration of CAD and FEA

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 1 From CAD and FEA to Isogeometric Analysis: An Historical Perspective 1.1 Introduction 1.2 The evolution of FEA basis functions 1.3 The evolution of CAD representations 1.4 Things you need to get used to in order to understand NURBS-based isogeometric analysis Notes 2 NURBS as a Pre-analysis Tool: Geometric Design and Mesh Generation 2.1 B-splines 2.2 Non-Uniform Rational B-Splines 2.3 Multiple patches 2.4 Generating a NURBS mesh: a tutorial 2.5 Notation Appendix 2.A: Data for the bent pipe Notes 3 NURBS as a Basis for Analysis: Linear Problems 3.1 The isoparametric concept 3.2 Boundary value problems 3.3 Numerical methods 3.4 Boundary conditions 3.5 Multiple patches revisited 3.6 Comparing isogeometric analysis with classical finite element analysis Appendix 3.A: Shape function routine Appendix 3.B: Error estimates Notes 4 Linear Elasticity 4.1 Formulating the equations of elastostatics 4.2 Infinite plate with circular hole under constant in-plane tension 4.3 Thin-walled structures modeled as solids Appendix 4.A: Geometrical data for the hemispherical shell Appendix 4.B: Geometrical data for a cylindrical pipe Appendix 4.C: Element assembly routine Notes 5 Vibrations and Wave Propagation 5.1 Longitudinal vibrations of an elastic rod 5.2 Rotation-free analysis of the transverse vibrations of a Bernoulli-Euler beam 5.3 Transverse vibrations of an elastic membrane 5.4 Rotation-free analysis of the transverse vibrations of a Poisson-Kirchhoff plate 5.5 Vibrations of a clamped thin circular plate using three-dimensional solid elements 5.6 The NASA aluminum testbed cylinder 5.7 Wave propagation Appendix 5.A: Kolmogorov n -widths Notes 6 Time-Dependent Problems 6.1 Elastodynamics 6.2 Semi-discrete methods 6.3 Space-time finite elements 7 Nonlinear Isogeometric Analysis 7.1 The Newton-Raphson method 7.2 Isogeometric analysis of nonlinear differential equations 7.3 Nonlinear time integration: The generalized- alpha method Note 8 Nearly Incompressible Solids 8.1 B formulation for linear elasticity using NURBS 8.2 F formulation for nonlinear elasticity Notes 9 Fluids 9.1 Dispersion analysis 9.2 The variational multiscale (VMS) method 9.3 Advection-diffusion equation 9.4 Turbulence Notes 10 Fluid-Structure Interaction and Fluids on Moving Domains 10.1 The arbitrary Lagrangian-Eulerian (ALE) formulation 10.2 Inflation of a balloon 10.3 Flow in a patient-specific abdominal aorta with aneurysm 10.4 Rotating components Appendix 10.A: A geometrical template for arterial blood flow modeling 11 Higher-order Partial Differential Equations 11.1 The Cahn-Hilliard equation 11.2 Numerical results 11.3 The continuous/discontinuous Galerkin (CDG) method Note 12 Some Additional Geometry 12.1 The polar form of polynomials 12.2 The polar form of B-splines Note 13 State-of-the-Art and Future Directions 13.1 State-of-the-art 13.2 Future directions Appendix A: Connectivity Arrays A.1 The INC Array A.2 The IEN array A.3 The ID array A.3.1 The scalar case A.3.2 The vector case A.4 The LM array Note References Index