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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 | |