Quadrature-free immersed isogeometric analysis.

Area of Science:

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

Background:

• Solving partial differential equations (PDEs) on complex geometries is computationally intensive.
• Traditional finite element methods often require complex mesh generation and numerical integration (quadrature).
• Isogeometric analysis (IGA) offers a promising alternative by using CAD representations directly.

Purpose of the Study:

• To develop a novel, efficient, and accurate method for solving PDEs on 3D CAD geometries.
• To eliminate the need for quadrature schemes in isogeometric analysis.
• To demonstrate the method's applicability to complex industrial CAD models.

Main Methods:

• Employs immersed isogeometric discretizations on boundary representations (B-Reps).
• Utilizes a new analytical technique for evaluating polynomial integrals over spline boundary representations.
• Transforms finite element operators into polynomial integrals, then surface and line integrals via the divergence theorem.
• Performs analytical evaluation of line integrals for machine precision accuracy.

Main Results:

• Achieves optimal error convergence orders for 2D and 3D elliptic problems.
• Demonstrates high accuracy through numerical experiments.
• Successfully applies the methodology to complex, industrial-level 3D CAD models.

Conclusions:

• The proposed immersed isogeometric method provides an accurate and efficient alternative for solving PDEs on complex CAD geometries.
• Eliminating quadrature schemes simplifies the analysis and improves computational performance.
• The method is robust and scalable for real-world engineering applications.