This book collects selected contributions presented at the INdAM Workshop "Geometric Challenges in Isogeometric Analysis", held in Rome, Italy on January 27-31, 2020. It gives an overview of the forefront research on splines and their efficient use in isogeometric methods for the discretization of differential problems over complex and trimmed geometries. A variety of research topics in this context are covered, including (i) high-quality spline surfaces on complex and trimmed geometries, (ii) construction and analysis of smooth spline spaces on unstructured meshes, (iii) numerical aspects and benchmarking of isogeometric discretizations on unstructured meshes, meshing strategies and software. Given its scope, the book will be of interest to both researchers and graduate students working in the areas of approximation theory, geometric design and numerical simulation.
Chapter 10 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
Author(s): Carla Manni, Hendrik Speleers
Series: Springer INdAM Series, 49
Publisher: Springer
Year: 2022
Language: English
Pages: 384
City: Cham
521119_1_En_OFC
521119_1_En_BookFrontmatter_OnlinePDF
Preface
Contents
About the Editors
521119_1_En_1_Chapter_OnlinePDF
Quadrilateral Orbifold Splines
1 Introduction
2 The Hyperbolic Plane
3 Rational Patches and Their Bézier Forms
4 Smooth Joints
5 Orbifold Splines
6 Projective Structures
7 Quadrilateral Orbifold Splines with Bi-Projective Structures
8 Quadrilateral Orbifold Splines with Projective Structures
9 Conclusions
References
521119_1_En_2_Chapter_OnlinePDF
B-Symmetric Univariate Splines and Euler Numbers
1 Introduction
2 Preliminary Results
3 Euler Numbers
4 B-Symmetric Splines of Minimal Defect
5 Approximation Order
6 Conclusions
References
521119_1_En_4_Chapter_OnlinePDF
A Collocation IGA-BEM for 3D Potential Problems on Unbounded Domains
1 Introduction
2 Isogeometric Indirect BEMs on 3D Unbounded Domains
3 Cubature Rules
4 Numerical Simulations
4.1 Problem on an Unbounded Domain Exterior to a Torus
4.2 A Screen Problem for a Curved Obstacle
4.3 A Nearly Singular Potential Problem Exterior to a Torus
5 Conclusion
References
521119_1_En_3_Chapter_OnlinePDF
DPB-Splines: The Decoupled Basis of Patchwork Splines
1 Introduction
2 The Patchwork Hierarchy
2.1 Patches
2.2 Spline Spaces
2.3 The Patchwork Spline Space
3 Decoupled Patchwork B-Splines
3.1 Preliminaries
3.2 Recursive Selection
4 Properties of DPB-Splines
4.1 Assumptions and Main Result
4.2 Proof of Theorem 1
5 Comparison with THB- and TPB-Splines
6 Macro Element Refinement for Adaptive Surface Reconstruction
6.1 Sufficient Condition
6.2 Macro Elements
6.3 Iterative Refinement and its Application to Surface Approximation
7 Conclusion
References
521119_1_En_5_Chapter_OnlinePDF
Simplex-Splines on the Clough-Tocher Split with Arbitrary Smoothness
1 Introduction
2 Preliminaries
2.1 Bernstein Polynomials
2.2 Bivariate Simplex Splines
3 The Clough-Tocher Split and a Basis for mathbbS3rr
4 Marsden-Like Identity, Domain Points, Stability and mathcalCr-Connection
4.1 The mathcalC2 Elements, barΣ62
4.2 The mathcalC3 Elements, barΣ93= Σ93
5 Conclusion
References
521119_1_En_6_Chapter_OnlinePDF
Trimmed Spline Surfaces with Accurate Boundary Control
1 Introduction
2 Basic Principle
3 ABC-Spline Surfaces
3.1 Reparametrization
3.2 Weights
4 File Export
5 Examples
6 Conclusion
References
521119_1_En_7_Chapter_OnlinePDF
Fast Formation and Assembly of Isogeometric Galerkin Matrices for Trimmed Patches
1 Introduction
2 The Discrepancy Between Weighted Quadrature and Trimming
2.1 Fast Formation by Weighted Quadrature and Sum Factorization
2.2 Trimmed Patches
3 Integration of Cut Basis Functions
3.1 Treatment of Cut Elements
3.2 Hybrid Gauss Approach
3.3 Discontinuous Weighted Quadrature
4 Numerical Results
4.1 Mass Matrix Formation
4.2 L2-Projection
5 Conclusions
References
521119_1_En_8_Chapter_OnlinePDF
Subdivision and G-Spline Hybrid Constructions for High-Quality Geometric and Analysis-Suitable Surfaces
1 Introduction
2 Categories of and Requirements for Surfaces with Irregular Configurations
2.1 Requirements on the Geometry
2.2 The Isogeometric Approach
2.3 Requirements for Analysis: Flexibility-Increasing Refinability
3 Subdivision Surfaces
4 G-Splines
5 Guided Surfaces
6 Hybrid Surfaces
7 Conclusion
References
521119_1_En_9_Chapter_OnlinePDF
Meshing as the Choice of Basis Functions for Finite Element Analysis
1 Content
2 What is Analysis About, and What Are the Real Criteria?
3 Generality
3.1 The Patch Test
3.2 Computation
3.3 Competitors
4 Evaluating the Criteria
4.1 Computing Cost
4.2 Error
4.3 Timeliness
5 Quad and Hex Meshing
5.1 Issues
5.2 Known Approaches to Quad and Hex Meshing
6 BUT There Exist Solid Shapes for Which No Good Hex Mesh Exists
6.1 The Pyramid
6.2 The Varying Dihedral Angle Along an Edge
6.3 The Leadscrew
6.4 The Slab with Different Features on Opposite Faces
6.5 The Block with a Tangential Cylinder on Top
7 MAGIC!
7.1 Transition to the Boundary
7.2 Fairly Thin Regions with Different Patterns on the Opposite Faces
7.3 Other Opportunities
8 Conclusions
References
521119_1_En_10_Chapter_OnlinePDF
Scattered Data Approximation by LR B-Spline Surfaces: A Study on Refinement Strategies for Efficient Approximation
1 Introduction
2 Background
2.1 Locally Refined Splines
2.2 Scattered Data Approximation
3 Refinement Strategies and Success Criteria
4 Applied Refinement Strategies
4.1 Main Categories
4.2 Restrictions to the Introduction of New Knots
4.3 Modifications to Particularly Restricted Strategies
4.4 Components for Refinement Strategies
4.5 Composition of Refinement Strategies for the Tests
5 Refinement Study
5.1 Banc du Four
5.2 Gaustatoppen
5.3 Scan Lines
5.4 Søre Sunnmøre, Sea Bed
5.5 Sea Bed, Large Point Cloud
6 Summary of Results
7 Conclusion
References
521119_1_En_11_Chapter_OnlinePDF
A Block ILUT Smoother for Multipatch Geometries in Isogeometric Analysis
1 Introduction
2 Model Problem
3 p-Multigrid Method
4 (Block) ILUT Smoother
5 Numerical Results
5.1 Benchmarks
5.2 Spectral Analysis
5.3 Iteration Numbers
6 Application: Yeti Footprint
7 Conclusions
References
521119_1_En_12_Chapter_OnlinePDF
Completeness Characterization of Type-I Box Splines
1 Introduction
2 Preliminaries
3 Box Splines on Type-I Triangulations
4 Characterization of Box Spline Spaces
5 Hierarchical Type-I Box Splines
6 Concluding Remarks
References
521119_1_En_13_Chapter_OnlinePDF
THU-Splines: Highly Localized Refinement on Smooth Unstructured Splines
1 Introduction
2 THB-Splines
2.1 Initialization
2.2 Selection
2.3 Truncation
3 Local D-Patch Construction
3.1 Bézier Extraction
3.2 D-Patch Method
3.3 Smooth Transition
3.4 Global Refinement
4 THU-Splines
4.1 Initialization
4.2 Selection
4.3 Truncation
4.4 Consistent Refinement with THU-Splines
5 Numerical Examples
6 Conclusion and Future Work
References
521119_1_En_14_Chapter_OnlinePDF
HexGen and Hex2Spline: Polycube-Based Hexahedral Mesh Generation and Spline Modeling for Isogeometric Analysis Applications in LS-DYNA
1 Introduction
2 Pipeline Design
3 HexGen: Polycube-Based Hex Mesh Generation
3.1 Surface Segmentation
3.2 Polycube Construction
3.3 Parametric Mapping and Octree Subdivision
3.4 Quality Improvement
4 Hex2Spline: Unstructured Spline Construction
4.1 Blending Functions on an Unstructured Hex Mesh
4.2 TH-Spline3D for Local Refinement
4.3 Bézier Extraction for LS-DYNA
4.4 Applying Hex2Spline to the Rod Model
5 Applications Using HexGen and Hex2Spline
6 Conclusion and Future Work
References
521119_1_En_15_Chapter_OnlinePDF
C1 Triangular Isogeometric Analysis of the von Karman Equations
1 Introduction
2 Rational Triangular Bézier Splines
3 Formulations and Methods
3.1 Von Karman Equations
3.2 Mixed Form of the von Karman Equations
3.3 Finite Element Discretisations
4 Numerical Results
4.1 Convergence Study
4.2 Efficiency
4.3 Accuracy and Reliability in Complex Geometries
5 Conclusion
References
