[Formula: see text] regularity properties of singular parameterizations in isogeometric analysis.

Area of Science:

• Numerical analysis
• Computer-aided design (CAD)
• Computational mathematics

Background:

• Isogeometric analysis (IGA) integrates CAD and finite element analysis using NURBS.
• IGA leverages NURBS for geometric representation, simplifying the analysis process.
• Singular parameterizations in NURBS can lead to test functions lacking necessary regularity.

Purpose of the Study:

• To investigate the regularity properties of test functions in isogeometric analysis with singular NURBS parameterizations.
• To identify conditions ensuring regularity for accurate numerical solutions.
• To propose methods for handling insufficient regularity in IGA.

Main Methods:

• Analysis of NURBS basis functions and domain parameterization.
• Investigation of test function regularity under singular parameterizations.
• Derivation of conditions for guaranteed regularity and development of modification schemes.

Main Results:

• Identified specific classes of singularities in 1D and 2D NURBS parameterizations.
• Derived conditions to ensure the regularity of test functions in IGA.
• Developed a modification scheme for the discretized function space to address insufficient regularity.

Conclusions:

• Regularity issues in IGA test functions due to singular NURBS parameterizations are addressed.
• The study provides conditions and methods to ensure accurate numerical solutions for complex geometries in IGA.
• The findings are applicable to higher-dimensional computational domains parameterized via sweeping.