Mathematical Foundations of Adaptive Isogeometric Analysis.

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Computer-Aided Engineering

Background:

• Isogeometric analysis (IGA) integrates computer-aided design and analysis.
• Adaptive methods enhance computational efficiency and accuracy in IGA.
• Mathematical foundations of adaptive IGA are crucial for its advancement.

Purpose of the Study:

• To provide a comprehensive review of the state-of-the-art in adaptive isogeometric analysis.
• To discuss recent mathematical developments, particularly concerning spline technologies and adaptive algorithms.
• To analyze the convergence and quasi-optimality of adaptive methods within the IGA framework.

Main Methods:

• Review of existing literature on adaptive isogeometric analysis.
• Analysis of spline technologies for singularity resolution in IGA.
• Formulation and discussion of convergence and quasi-optimality for adaptive finite element and boundary element methods in IGA.

Main Results:

• Overview of advanced spline technologies applicable to adaptive IGA.
• Detailed examination of the mathematical theory underpinning adaptive IGA algorithms.
• State-of-the-art formulations for convergence and quasi-optimality in adaptive IGA.

Conclusions:

• Adaptive isogeometric analysis is a rapidly evolving field with significant theoretical underpinnings.
• Spline technologies play a key role in addressing singularities within adaptive IGA.
• The convergence and quasi-optimality of adaptive algorithms are well-established for finite and boundary element methods in IGA.