Immersed boundary-conformal isogeometric method for linear elliptic problems.

Area of Science:

• Computational Mechanics
• Numerical Analysis
• Geometric Modeling

Background:

• Traditional immersed boundary methods offer geometric flexibility but lack conformal discretization advantages.
• Conformal discretizations provide higher accuracy and better control over mesh resolution near boundaries.
• Existing methods struggle with complex geometries and automatic satisfaction of interface conditions.

Purpose of the Study:

• To introduce a novel Immersed Boundary-Conformal Method (IBCM).
• To combine the geometric flexibility of immersed boundaries with the accuracy of conformal discretizations.
• To demonstrate IBCM's effectiveness on complex geometries and boundary-layer phenomena.

Main Methods:

• Extruding a boundary representation to create a conformal layer.
• Cutting a background B-spline mesh with the conformal layer using Boolean operations.
• Coupling the conformal layer with selected regions (interior/exterior) via Nitsche's method.
• Employing minimal stabilization for arbitrarily cut elements.

Main Results:

• IBCM demonstrates improved accuracy and expected convergence in 2D benchmark problems.
• Successful application to complex geometries, including a spanner model and fiber-reinforced composites.
• Effectiveness shown in simulations exhibiting boundary-layer phenomena.

Conclusions:

• IBCM successfully integrates the benefits of immersed boundary and conformal discretization methods.
• The method offers intuitive mesh control, higher accuracy, and automatic satisfaction of interface conditions.
• IBCM shows significant potential for simulating problems with complex geometries and boundary layers.