Isogeometric Analysis and Applications 2018

Foreword
Preface
Contents
Contributors
Generating Star-Shaped Blocks for Scaled Boundary Multipatch IGA
1 Introduction
2 Scaled Boundary Isogeometric Analysis
2.1 Tensor Product B-Splines
2.2 Scaled Boundary (SB) Parametrization
2.3 SB-Parametrizations in Isogeometric Analysis
2.4 Multipatch Geometries
3 Quadtree Decomposition
3.1 Star-Shapedness
3.2 Recursion Scheme
3.3 Intersection of Rays and Splines
3.4 Quadtree-Based Refinement Algorithm
3.5 Setup of Interfaces for Numerical Treatment
3.6 Benefits, Termination and Difficulties
4 Art Gallery Decomposition
4.1 Fisk's Algorithm
4.2 Voronoi Diagrams for Exterior Visibility
4.3 Robustness and Higher Dimensions
5 Numerical Simulations
5.1 The Rotor with Five Wings
5.2 The Yeti Footprint
5.3 Convergence on an Annulus
6 Conclusions
References
Approximation Power of C1-Smooth Isogeometric Splines on Volumetric Two-Patch Domains
1 Introduction
2 Preliminaries
3 Basis Construction
4 Approximation Properties
5 Conclusion
References
A Novel Approach to Fluid-Structure Interaction Simulations Involving Large Translation and Contact
1 Introduction
2 Governing Equations of Fluid Dynamics
2.1 Governing Equations of Fluid Dynamics
2.2 Deforming-Spatial-Domain/Stabilized Space-Time Method
2.3 Elastic Mesh Update Method
3 The Phantom Domain Mesh Deformation Method
4 Computational Results
4.1 2D Poiseuille Flow on Moving Background Mesh
4.2 Falling Ring in a Fluid-Filled Container
5 Discussion
References
An IGA Framework for PDE-Based Planar Parameterization on Convex Multipatch Domains
1 Introduction
2 Problem Formulation
3 Solution Strategy
3.1 Single Patch Parameterizations
3.2 Multipatch
4 Numerical Experiments
4.1 L-Bend
4.2 Tube-Like Shaped Geometry
4.3 Multipatch Problem: The Bat Geometry
5 Conclusion
References
Preconditioning for Linear Systems Arising from IgA Discretized Incompressible Navier–Stokes Equations
1 Introduction
2 Problem Formulation
2.1 Discretization and Linearization
2.2 Motivational Problem
3 Solution Methods
3.1 Block Triangular Preconditioners
3.1.1 Least-Squares Commutator Preconditioner
3.1.2 Augmented Lagrangian Preconditioner
3.2 SIMPLE-Type Preconditioners
4 Numerical Experiments
4.1 Backward Step 2D
4.2 Periodic Domain 2D
5 Conclusions
References
Solving 2D Heat Transfer Problems with the Aidof a BEM-Isogeometric Solver
1 Introduction
1.1 Using the Design Optimization Framework
1.2 Chapter Outline
2 Heat Transfer Problem Formulation
2.1 Continuous Formulation
2.2 IGABEM Formulation
2.3 IGABEM Solver Efficiency
3 Parametric Model
3.1 Uniformly Distributed Control Points
3.2 Protrusions and Interfaces with Fixed Parts
4 Optimization Environment and Examples
4.1 Forward Design
4.1.1 Heat Transfer Maximization Subject to Area Constraints
4.1.2 Heat Transfer Maximization Subject to Perimeter Constraints
4.1.3 Heat Transfer Maximization Subject to Area and Perimeter Constraints
4.2 Inverse Design
4.2.1 Interface Shape for a Given Heat Transfer Value
4.2.2 Interface Shape for a Given Temperature Distribution
5 Conclusions and Future Work
References
Isogeometric Methods for Free Boundary Problems
1 Introduction
2 Free Boundary Problem
2.1 Weak Formulation
2.2 Very-Weak Formulation
3 Linearising the FBP
3.1 Shape Derivatives
3.2 Linearisation of the Weak Formulation
3.3 Linearisation of the Very-Weak Formulation
4 Numerical Schemes
4.1 B-Splines Based Isogeometric Analysis
4.2 Isogeometric Galerkin Methods
4.3 Isogeometric Collocation Method
5 Numerical Results
5.1 Test 1: Parabolic Boundary, Dirichlet b.c.
5.2 Test 2: Sinusoidal Boundary, Dirichlet b.c.
5.3 Test 3: Sinusoidal Boundary, Periodic b.c.
6 Conclusions
References
Approximately C1-Smooth Isogeometric Functions on Two-Patch Domains
1 Introduction
2 Preliminaries
3 Results for B1
3.1 Construction of Approximately Smooth Functions
3.2 Properties of the Function Space
4 Results for B2
4.1 Construction of Approximately Smooth Functions
4.2 Properties of the Function Space
5 Numerical Examples
5.1 Approximation Power
5.1.1 Least Squares Approximation
5.1.2 Poisson Problem
5.1.3 Biharmonic Equation
5.2 Dimension of the Space
6 Conclusion
References
Properties of Spline Spaces Over Structured Hierarchical BoxPartitions
1 Introduction
2 Condition Numbers and Knotline Multiplicity at Domain Boundary
2.1 Boundary Knotline Multiplicities
2.1.1 Observation for the Mass Matrix
3 Box Partitions, Meshes, and Spline Spaces
3.1 Tensor Product Splines
3.2 Locally Refined Spline Spaces
3.2.1 LR-Splines
3.2.2 Truncated Hierarchical B-Splines
3.2.3 T-Splines
4 Local Modification of Meshes and the Reduction of Overloading
5 Numerical Experiments
5.1 L2-Projection
5.2 The Poisson Equation
5.3 Condition Numbers
5.4 Numerical Results
5.4.1 Central Refinement
5.4.2 Diagonal Refinements
6 Conclusion
References
Efficient p-Multigrid Based Solvers for Isogeometric Analysis on Multipatch Geometries
1 Introduction
2 Model Problem
3 p-Multigrid Method
3.1 Prolongation and Restriction
3.2 Smoother
3.3 Coarse Grid Operator
4 Spectral Analysis
5 Numerical Results
5.1 ILUT as a Solver
5.2 Comparison h- and hp-Multigrid
6 Conclusion
References
The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms
1 Introduction
2 B-Spline Spaces
2.1 Primal B-Spline Spaces
2.2 Dual Representations
3 Numerical Test
3.1 Poisson Problem
3.2 Manufactured Solution Test Case
4 Conclusions
References
Manifold-Based B-Splines on Unstructured Meshes
1 Introduction
2 Review of Manifold-Based Basis Functions
2.1 Basic Approach
2.2 Mesh-Based Approach
3 Reproduction of B-Splines
3.1 Blending Function Choices
3.1.1 Piecewise Linear C0 Continuous Blending Functions
3.1.2 Higher Order Cp-1 Continuous B-Spline Blending Functions
3.1.3 Rational B-Spline Blending Functions
3.1.4 Linear Combinations of B-Splines as Blending Functions
3.1.5 Comparison of Different Blending Function Choices
3.2 Local Approximants
4 Conclusions
References