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| Preface | |
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| From CAD and FEA to Isogeometric Analysis: An Historical Perspective | |
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| Introduction | |
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| The need for isogeometric analysis | |
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| Computational geometry | |
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| The evolution of FEA basis functions | |
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| The evolution of CAD representations | |
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| Things you need to get used to in order to understand NURBS-based isogeometric analysis | |
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| Notes | |
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| NURBS as a Pre-analysis Tool: Geometric Design and Mesh Generation | |
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| B-splines | |
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| Knot vectors | |
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| Basis functions | |
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| B-spline geometries | |
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| Refinement | |
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| Non-Uniform Rational B-Splines | |
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| The geometric point of view | |
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| The algebraic point of view | |
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| Multiple patches | |
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| Generating a NURBS mesh: a tutorial | |
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| Preliminary considerations | |
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| Selection of polynomial orders | |
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| Selection of knot vectors | |
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| Selection of control points | |
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| Notation | |
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| Data for the bent pipe | |
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| Notes | |
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| NURBS as a Basis for Analysis: Linear Problems | |
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| The isoparametric concept | |
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| Defining functions on the domain | |
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| Boundary value problems (BVPs) | |
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| Numerical methods | |
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| Galerkin | |
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| Collocation | |
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| Least-squares | |
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| Meshless methods | |
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| Boundary conditions | |
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| Dirichlet boundary conditions | |
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| Neumann boundary conditions | |
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| Robin boundary conditions | |
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| Multiple patches revisited | |
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| Local refinement | |
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| Arbitrary topologies | |
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| Comparing isogeometric analysis with classical finite element analysis | |
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| Code architecture | |
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| Similarities and differences | |
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| Shape function routine | |
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| Error estimates | |
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| Notes | |
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| Linear Elasticity | |
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| Formulating the equations of elastostatics | |
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| Strong form | |
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| Weak form | |
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| Galerkin's method | |
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| Assembly | |
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| Infinite plate with circular hole under constant in-plane tension | |
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| Thin-walled structures modeled as solids | |
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| Thin cylindrical shell with fixed ends subjected to constant internal pressure | |
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| The shell obstacle course | |
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| Hyperboloidal shell | |
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| Hemispherical shell with a stiffener | |
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| Geometrical data for the hemispherical shell | |
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| Geometrical data for a cylindrical pipe | |
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| Element assembly routine | |
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| Notes | |
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| Vibrations and Wave Propagation | |
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| Longitudinal vibrations of an elastic rod | |
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| Formulating the problem | |
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| Results: NURBS vs. FEA | |
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| Analytically computing the discrete spectrum | |
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| Lumped mass approaches | |
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| Rotation-free analysis of the transverse vibrations of a Bernoulli-Euler beam | |
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| Transverse vibrations of an elastic membrane | |
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| Linear and nonlinear parameterizations revisited | |
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| Formulation and results | |
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| Rotation-free analysis of the transverse vibrations of a Poisson-Kirchhoff plate | |
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| Vibrations of a clamped thin circular plate using three-dimensional solid elements | |
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| Formulating the problem | |
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| Results | |
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| The NASA aluminum testbed cylinder | |
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| Wave propagation | |
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| Dispersion analysis | |
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| Duality principle | |
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| Kolmogorov n-widths | |
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| Notes | |
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| Time-Dependent Problems | |
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| Elastodynamics | |
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| Semi-discrete methods | |
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| Matrix formulation | |
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| Viscous damping | |
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| Predictor/multicorrector Newmark algorithms | |
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| Space-time finite elements | |
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| Nonlinear Isogeometric Analysis | |
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| The Newton-Raphson method | |
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| Isogeometric analysis of nonlinear differential equations | |
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| Nonlinear heat conduction | |
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| Applying the Newton-Raphson method | |
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| Nonlinear finite element analysis | |
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| Nonlinear time integration: The generalized-a method | |
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| Note | |
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| Nearly Incompressible Solids | |
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| B formulation for linear elasticity using NURBS | |
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| An intuitive look at mesh locking | |
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| Strain projection and the B method | |
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| B, the projection operator, and NURBS | |
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| Infinite plate with circular hole under in-plane tension | |
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| F formulation for nonlinear elasticity | |
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| Constitutive equations | |
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| Pinched torus | |
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| Notes | |
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| Fluids | |
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| Dispersion analysis | |
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| Pure advection: the first-order wave equation | |
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| Pure diffusion: the heat equation | |
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| The variational multiscale (VMS) method | |
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| Numerical example: linear advection-diffusion | |
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| The Green's operator | |
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| A multiscale decomposition | |
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| The variational multiscale formulation | |
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| Reconciling Galerkin's method with VMS | |
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| Advection-diffusion equation | |
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| Formulating the problem | |
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| The streamline upwind/Petrov-Galerkin (SUPG) method | |
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| Numerical example: advection-diffusion in two dimensions, revisited | |
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| Turbulence | |
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| Incompressible Navier-Stokes equations | |
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| Multiscale residual-based formulation of the incompressible Navier-Stokes equations employing the advective form | |
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| Turbulent channel flow | |
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| Notes | |
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| Fluid-Structure Interaction and Fluids on Moving Domains | |
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| The arbitrary Lagrangian-Eulerian (ALE) formulation | |
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| Inflation of a balloon | |
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| Flow in a patient-specific abdominal aorta with aneurysm | |
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| Construction of the arterial cross-section | |
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| Numerical results | |
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| Rotating components | |
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| Coupling of the rotating and stationary domains | |
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| Numerical example: two propellers spinning in opposite directions | |
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| A geometrical template for arterial blood flow modeling | |
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| Higher-order Partial Differential Equations | |
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| The Cahn-Hilliard equation | |
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| The strong form | |
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| The dimensionless strong form | |
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| The weak form | |
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| Numerical results | |
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| A two-dimensional example | |
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| A three-dimensional example | |
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| The continuous/discontinuous Galerkin (CDG) method | |
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| Note | |
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| Some Additional Geometry | |
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| The polar form of polynomials | |
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| Bezier curves and the de Casteljau algorithm | |
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| Continuity of piecewise curves | |
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| The polar form of B-splines | |
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| Knot vectors and control points | |
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| Knot insertion and the de Boor algorithm | |
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| Bezier decomposition and function subdivision | |
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| Note | |
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| State-of-the-Art and Future Directions | |
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| State-of-the-art | |
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| Future directions | |
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| Connectivity Arrays | |
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| The INC Array | |
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| The IEN array | |
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| The ID array | |
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| The scalar case | |
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| The vector case | |
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| The LM array | |
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| Note | |
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| References | |
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| Index | |