Isogeometric Analysis

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-? 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