Isogeometric Analysis and Computational Domain Parameterization

Isogeometric analysis (IGA) unites the geometric precision of computer‐aided design with the numerical rigour of finite element analysis by employing spline‐based functions for both domain representation and solution approximation. Central to this paradigm is computational domain parameterization, the process of constructing a smooth, bijective mapping between a simple parametric domain and the often complex physical geometry. Such mappings, typically realised through NURBS or B‐spline patches, enable exact geometry preservation and high‐order continuity across elements. Modern parameterization techniques address challenges of multi‐patch coupling, irregular boundaries and volumetric meshing by leveraging elliptic partial differential equations, harmonic mapping principles and advanced shape control mechanisms. The seamless integration of design and analysis afforded by IGA has broad impact across structural mechanics, fluid dynamics and optimisation, driving more accurate simulations, reducing meshing overhead and facilitating adaptive refinement strategies in industrial and research settings.

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Recent advances have employed elliptic PDE‐based frameworks to generate valid planar multipatch parameterizations suited for IGA applications. By formulating inverted Laplace or more general elliptic equations on a convex parametric domain, researchers have devised algorithms that integrate seamlessly into existing isogeometric software, offering control over cell uniformity and boundary‐layer grading. In parallel, spline‐parameterization approaches driven by physics‐informed neural networks have emerged, solving the governing elliptic problems via deep learning to produce continuous, fold‐free mappings on both convex and non‐convex domains without retraining for different mesh resolutions. Additionally, parameterization transfer algorithms have been proposed for planar B‐spline domains with similar skeleton structures. These methods extract boundary control points from a source domain and employ discrete harmonic mapping to achieve C1/G1 continuity in a target geometry, markedly reducing computational cost while maintaining high‐quality parameterizations applicable to morphing and deformation tasks.

Isogeometric Analysis and Computational Domain Parameterization publication trend

The graph below shows the total number of articles in isogeometric analysis and computational domain parameterization across all publications each year (not limited to Nature Index journals).

Technical terms

Isogeometric Analysis: A computational framework using the same spline functions for geometry representation and numerical approximation.

Computational Domain Parameterization: The construction of a smooth, bijective mapping between a simple parametric domain and a complex physical geometry.

Multipatch Domain: A composite geometric representation comprising multiple spline patches joined with continuity constraints.

Physics-Informed Neural Network (PINN): A deep learning model trained to satisfy underlying differential equations governing a physical problem.

Parameterisation Transfer: The algorithmic process of mapping an existing spline parameterization from one domain to another with similar boundary topology.

References

1. On the use of elliptic PDEs for the parameterisation of planar multipatch domains. Engineering with Computers (2024).
2. Splines Parameterization of Planar Domains by Physics-Informed Neural Networks. Mathematics (2023).
3. Parameterization Transfer for a Planar Computational Domain in Isogeometric Analysis. Computer Modeling in Engineering & Sciences (2023).