1 Topology
1.1 Unstructured topology
1.1.1 Unoriented topology
1.1.2 Oriented topology
1.1.3 Combinatorial maps as an interface
1.1.4 Unoriented map derived from oriented map
1.1.5 Tensor product topology
1.1.6 Composite topological data structures
1.1.7 Semi-structured topology
1.1.8 Refinement
1.1.9 Example application
2 Polynomials
2.1 An Overview of Polynomials
2.2 The Bernstein Polynomials
2.2.1 Definition
2.2.2 Ordering of Derivatives
2.2.3 Degree Elevation
2.2.4 Multivariate Bernstein Polynomials
2.2.4.1 Definition on Box Domains
2.2.4.2 Definition on Simplicial Domains
2.2.5 Bernstein-like Bases
2.3 The Chebyshev Polynomials
2.3.1 Definition
2.3.2 Chebyshev points
2.3.3 Evaluation
2.4 The Lagrange Polynomials
2.4.1 Definition
2.4.2 Evaluation
2.5 Change of Basis for Polynomial Forms
2.5.1 Conversion Between Chebyshev and Lagrange Forms
2.5.2 Conversion Between Chebyshev and Bernstein Forms
2.5.3 Conversion Between Bernstein and Lagrange Forms
2.5.4 Conversion Between Lagrange Bases
2.6 Construction of Polynomial Forms
2.6.1 Basic Construction
2.6.1.1 Interpolation
2.6.1.2 Projection
2.6.2 Adaptive Construction
3 Splines
3.1 The Bézier Mesh
3.1.1 Topology
3.1.1.1 Adjacencies
3.1.1.2 Cell Types
3.1.2 Cell Domains and Parameterization
3.1.3 Cell Space and Degree
3.1.4 Interface Continuity
3.1.4.1 Supersmooth Interfaces
3.2 Bernstein Representations
3.2.1 Indexing
3.2.2 Bernstein Form
3.2.3 The Trace Mapping Matrix
3.3 Continuity Constraints
3.3.1 Constraint Sets
3.3.2 Constraint Matrices
3.3.3 Constraint Construction
3.4 Splines and the Nullspace Problem
3.4.1 Basis Vectors
3.4.2 Basis Functions
3.4.3 Spline Form
3.4.4 Extracted Form
4 U-splines: Definition
4.1 Summary
4.2 Previous work
4.3 An Overview of U-splines
4.4 The U-spline Mesh
4.4.1 Ribbons
4.4.1.1 Maximum Coupling Length
4.4.1.2 Continuity Transitions
4.4.1.3 Degree Transitions
4.4.2 Admissible Layouts
4.4.3 Classification
4.5 The U-spline Space
4.5.1 Completeness and the Neighborhood of Interaction
4.5.2 Mathematical Properties
4.6 U-spline Geometry
4.6.1 Bézier Projection
4.7 Notable U-spline Examples
4.7.1 Supersmooth Interface Layout
4.7.2 Handling Degree Transitions
4.7.3 Extraordinary Vertices
4.7.4 Triangles
4.7.5 Unstructured Volumetric U-splines
5 U-splines: Construction
5.1 Bernstein basis metrics and index measurements
5.1.1 Greville points
5.1.2 Submesh domains
5.1.2.1 Indexed submesh domains
5.1.3 Equivalence relations and classes
5.1.4 Alignment
5.1.4.1 Alignment in two dimensions
5.1.4.2 Alignment in arbitrary dimensions
5.2 Basis vectors for cell nullspaces
5.2.1 Basis vectors in one dimension
5.2.2 Interface basis vectors in two dimensions
5.2.3 Cell basis vector preliminaries
5.2.3.1 Spokes and interface-element pairs
5.2.3.2 Inclusion distances
5.2.3.3 Alignment sets
5.2.4 Overview of cell basis vector construction
5.2.5 Vertex basis vectors in two dimensions
5.2.5.1 Composite vertex basis vectors
5.2.5.2 Simple vertex basis vectors
5.2.5.3 The full set of vertex basis vectors
5.2.6 Subordinate basis vectors
5.2.7 Basis vector boundaries
5.2.7.1 Basis vector boundaries in one dimension
5.2.7.2 Basis vector boundaries in two dimensions
5.2.7.3 Basis vector boundaries in arbitrary dimensions
5.3 Basis vectors in arbitrary dimensions
5.3.1 Composite cell basis vectors
5.3.2 Simple cell basis vectors
5.3.3 The full set of cell basis vectors
5.4 The U-spline basis
5.4.1 The core graph
5.4.1.1 Cores
5.4.1.2 Expansion edges
5.4.1.3 Algorithm
5.4.2 The rank one null space problem
5.4.3 Normalization
5.5 Verification of U-spline space
5.5.1 Overview of verification procedure
5.5.1.1 One dimension
5.5.1.2 Two dimensions
5.5.1.3 Three dimensions
5.6 U-spline test cases with Bézier extraction coefficients
5.6.1 U-spline extraction coefficients near a supersmooth interface
5.6.2 U-spline extraction coefficients with non-rectangular support
5.6.3 U-spline extraction coefficients on mesh equivalent to analysis-suitable T-spline with non-crossing edge extensions
5.6.4 U-spline extraction coefficients near an extraordinary vertex
5.6.5 U-spline extraction coefficients near a triangle
5.7 Ribbon processing
5.8 Interface Continuity Constraints in Two Dimensions
5.8.1 Quadrilateral-Quadrilateral Interface
5.8.2 Quadrilateral-Triangle Interface
5.8.3 Triangle-Triangle Interface
6 U-splines: Tutorial
6.1 Building intuition: Constraints
6.2 Building intuition: Splines
6.3 Building intuition: Basis vectors
6.4 Building intuition: The U-spline mesh
6.5 Building intuition: The U-spline basis
7 Inverse Mapping
7.1 An Overview of Inverse Mapping
7.2 Bounding Volume Hierarchy
7.2.1 Nearest Cells Search
7.2.1.1 AABB Bounds
7.3 Point Inversion
7.3.1 The General Approach
7.3.1.1 Point inversion over a simple simplex
7.3.1.2 Point inversion over a tensor product of simplices
7.3.2 Optimizations
7.3.2.1 Full Rank Mapping
7.3.2.2 Affine Mapping
7.3.2.3 Composite Mapping
7.4 Algorithms
8 Fitting
8.1 An Overview of CAD Fitting
8.2 CAD Modification and Meshing for U-spline Surfaces
8.3 CAD Modification and Meshing for U-spline Volumes
8.4 Unstructured U-spline CAD Fitting
8.4.1 Submanifold Fitting
8.5 Structured U-spline CAD Fitting
8.5.1 Tensor Product Fitting
8.5.1.1 Affine Fitting
8.5.1.2 -linear Fitting
8.5.1.3 Coons Patch Fitting
8.5.2 Composite Fitting
8.5.2.1 Constructing the Composite Bézier Mesh
8.5.2.2 Constructing the Prolongation Operator
8.5.2.3 Constructing the Restriction Operator
8.6 Improving CAD Fitting Through Local Smoothing
8.7 U-spline CAD Fitting Examples
8.7.1 Surface Fitting
8.7.2 Volume Fitting
9 Coreform IGA Mesh
9.1 Rectilinear Background Mesh Construction
9.1.1 Setup
9.1.2 Axis Layout
9.1.3 Resolution
9.1.3.1 Exact Element Size
9.1.3.2 Exact Interval Count
9.1.4 Padding and Final Mesh
9.1.5 Numerical Robustness
10 Trimming
10.1 Trimmed U-splines
10.1.1 Embedded Atlas
10.1.2 Skin Atlas
10.1.3 Trimmed Atlas
10.1.4 Chart Representation
10.1.5 Atlas Representation
10.1.6 Skin and Trimmed Atlas Construction
10.1.6.1 Processing Edges
10.1.6.2 Processing Faces
10.1.6.3 Processing Volumes
10.1.6.4 Construction from ACIS Intersection Graph
10.1.7 The Trimmed U-spline Mesh
10.2 Derivatives of Degenerate Charts
10.3 Approximate Extracted Form
10.3.1 Extracted Tessellations
11 Basis Conditioning
11.1 An Overview of Basis Extension
11.2 Element Classification
11.3 Element Associations
11.3.1 Association Generations
11.4 Association Extension Operator
11.5 Initial conditions with basis extension
11.6 Effects of basis conditioning on stress
11.6.1 Simple cube patch test
11.6.1.1 Tension - zero poisson ratio
11.6.1.2 Tension - non-zero poisson ratio
11.6.1.3 Shear - zero poisson ratio
11.6.2 Spherical pressure vessel
12 Quadrature
12.1 An Overview of Quadrature
12.2 Gauss Quadrature
12.3 Tensor-product Rules
12.4 Mapped Quadrature Rules
12.5 Composite Quadrature Rules
12.6 Matching Moments
12.7 Nonnegative Least Squares Quadrature
12.8 Pruned Quadrature Rules
13 Assembly
13.1 Vectorization
13.2 Scattering
13.3 Bases
13.4 Bilinear Forms
13.4.1 Active Scatter Matrices
13.5 Basis Extension
13.5.1 Section heading
13.5.2 Section heading
13.6 Partitioning
13.7 Incorporating Constraints
13.8 Representing Symmetric Tensors
13.8.1 Symmetric Forms of Tensors and Arrays
13.8.2 Product Rule
13.8.3 Quotient Rule
13.8.4 Chain Rule
13.8.4.1 Scalar-valued Function of Multivariate Scalar Function
13.8.4.2 Vector-valued Multivariate Function of a Vector-valued Multivariate Function
13.8.5 Derivative of One Mapping with Respect to Another
13.9 Sum Factorization
13.9.1 Symmetrization
13.9.2 Spatial Derivatives
13.9.3 Tensor Product Spaces
13.9.4 An Illustrative Example
13.9.5 Prospects for applying sum-factorization in practical trimmed analysis
14 The Flex Representation Method
14.1 Flex Modeling and Spectrum
14.2 An Illustrative Example
15 Solids
15.1 Kinematics
15.2 Local Mesh Size
15.3 Weak Form
15.4 Semi-Discrete Form
15.5 Penalty Enforcement of Dirichlet Boundary Conditions
15.5.1 Displacement Dirichlet Conditions
15.5.2 Penalty Parameter Estimation
15.5.3 Interpretation of the Penalty Parameter
15.5.4 Velocity and Acceleration Dirichlet Conditions
15.6 Time Stepping Algorithms
15.6.1 Initial Acceleration
15.6.2 Implicit Dynamics
15.6.2.1 Implicit Dynamics With High-Frequency Damping
15.6.2.2 Backward Euler
15.6.2.3 Adaptive Time Stepping
15.6.3 Explicit Dynamics
15.6.3.1 Explicit Central Difference
15.6.3.2 Explicit Dynamics With High-Frequency Damping
15.6.3.3 Maximum Frequency Estimation
15.6.3.4 Mass-Proportional Damping
15.7 Near-incompressibility
15.7.1 Stabilized Static Pressure Equation
15.7.2 Stabilized Pressure Rate Equation
15.7.2.1 Explicit Time Integration
16 Contact
16.1 Mechanical Contact
16.1.1 Kinematics
16.1.2 Equilibrium
16.1.3 Frictionless residual construction
16.1.4 Frictional residual construction
16.1.5 Collision Detection
16.1.6 Contact force modifications
16.1.6.1 Effective signed distance modification
16.1.6.2 Contact force scaling
16.1.7 Interference fits
16.2 Tied Contact
16.2.1 Residual
17 Abaqus Plugin
17.1 Abaqus Plugin
17.1.1 Bulk Element Formulation
17.1.1.1 Kinematics and constitutive update
17.1.1.2 Bulk residual
17.1.1.3 Bulk tangent
17.1.2 Penalty Boundary Enforcement
17.1.2.1 DOF partition and projection
17.1.2.2 Variational penalty form
17.1.2.3 Penalty residual and tangent
18 Adaptivity
18.1 An overview of refinement
18.1.1 Refinement as embedding
18.1.1.1 Topology and parameterization
18.1.1.2 Local basis and interface continuity requirements
18.1.1.3 Projection of fields
18.1.2 Local refinement
18.2 Hierarchical U-splines
18.2.1 Basis construction
18.2.1.1 Core algorithm
18.2.1.2 Truncation
18.2.1.3 Algorithms
18.3 Structured Hierarchical Splines
18.3.1 Mass matrix assembly
18.4 Simulations on hierarchical U-splines
18.4.1 Body-fitted plate with a hole
18.4.2 Immersed plate with a hole
19 Constitutive Models
19.1 Abstract Interface for Materials
19.1.1 Scalar Stiffness Estimation
19.2 Isotropic neo-Hookean material model
19.3 Elastoplasticity
19.3.1 Elastoplastic theory
19.3.2 Radial return algorithm
19.3.2.1 Precision improvement
19.3.3 Consistent spatial elastoplastic moduli
19.3.4 Material failure
19.4 Thermal expansion
19.4.1 Thermal stresses in linear elasticity
19.4.2 Approximate thermal pressure in finite strain theory
19.5 Viscoelasticity
19.5.1 Viscoelastic theory
19.5.2 Time discretization
20 Structural Elements
20.1 Summary
20.2 Previous work
20.3 Geometric description
20.3.1 Reference configuration
20.3.2 Current configuration
20.3.3 Deformation gradient and local bases
20.4 Partial differential equations
20.5 Rotations
20.6 Weak form
20.6.1 Variation of the internal force
20.7 Material models
20.8 An FRM shell
20.8.1 The matrix form
20.8.2 Body fitted verification
20.8.3 Aspects of the FRM for shells
21 References