Isogeometric Analysis (IGA) 等几何分析（IGA）

Course overview

Isogeometric Analysis (IGA) is a computational methodology that tightly integrates Computer-Aided Design (CAD) and Finite Element Analysis (FEA) by using spline-based functions—such as B-splines and NURBS—for both geometry representation and numerical simulation.

By enabling higher-order smoothness and exact geometry representation, IGA provides significant advantages over classical finite element methods, particularly for applications involving:

• fluid and solid mechanics,
• multi-physics coupling,
• shape and topology optimization.

Originally motivated by practical engineering challenges, IGA has developed a strong theoretical foundation over the past two decades. This course aims to strike a balance between mathematical foundations, computational techniques, and engineering applications of IGA.

Learning objectives

After completing this course, students will be able to:

• understand spline-based geometry representations and their role in numerical analysis,
• formulate and solve linear and nonlinear PDEs using IGA,
• implement collocation- and Galerkin-based IGA methods,
• apply IGA to physics and optimization problems,
• critically assess the strengths and limitations of IGA in practical applications.

Teaching team

The course is taught by a lecturer team with complementary expertise:

• Stefanie Elgeti – Computer-aided design optimization
  (DCSE visiting professor, TU Wien)
• Matthias Möller – Computational simulation and numerical methods
  (TU Delft)
• Ye Ji – Computer-aided geometric modeling and parameterization
  (TU Delft)
• Jingya Li – Fluid–structure interaction and multi-physics coupling
  (TU Delft)

Course format and prerequisites

The course combines:

• lectures,
• hands-on programming exercises,
• practical examples,
• and a final project.

Prerequisites:

• basic knowledge of numerical methods and numerical linear algebra,
• prior programming experience (MATLAB, Python, or C++).

Course materials

All course materials—including slides, exercises, and code examples—will be made available via Brightspace:

👉 https://brightspace.tudelft.nl/d2l/home/776010

Syllabus – WI4450

Special Topics in Computational Science and Engineering

Quarter 3 – Core lectures (7 weeks)

Lecture 1 – 10 February
Introduction to Computer-Aided Engineering (CAE) and design optimization with IGA;
overview of key components (geometry representation, objective functions, optimizers).

Lecture 2 – 17 February
Geometry modeling with B-splines and NURBS;
practical introduction to G+Smo and SplinePy.

Lecture 3 – 24 February
Surface and volume parameterization for analysis-suitable IGA
(Coons patches, geometry creation, and limitations).

Lecture 4 – 3 March
Collocation-based IGA for Poisson’s problem;
Greville abscissae, error analysis, and convergence rates.

Lecture 5 – 10 March
Galerkin-based IGA for Poisson’s problem;
variational formulation and numerical quadrature.

Lecture 6 – 17 March
Linear elasticity problems in IGA.

Lecture 7 – 24 March
Optimization algorithms and project kickoff.

Quarter 4 – Advanced topics and projects (7 weeks)

Lecture 8 – 22 April
Nonlinear Poisson’s equation;
defect correction, Newton–Raphson methods, derivative computation, and convergence enhancement.

Lecture 9 – 29 April
Fast matrix assembly techniques for Galerkin IGA
(e.g., weighted quadrature).

Lecture 10 – 6 May
Topology optimization.

Lecture 11 – 13 May
Immersed methods and solvers.

Lecture 12 – 20 May
Multi-patch IGA and coupling techniques
(e.g., Nitsche’s method).

Lecture 13 – 27 May
Advanced parameterization techniques.

Lecture 14 – 3 June
Robust optimization under uncertainty.

Final project presentations

18–19 June (all lecturers)