Fully and semi-automated shape differentiation in NGSolve

Area of Science:

• Computational mechanics
• Numerical analysis
• Software engineering

Background:

• Shape optimization is crucial in engineering design.
• Automated differentiation is a powerful tool for sensitivity analysis.
• Finite element software like NGSolve offers a platform for complex simulations.

Purpose of the Study:

• To present a framework for automated shape differentiation in NGSolve.
• To enable user-defined levels of automation for shape derivative computation.
• To facilitate the development of shape optimization algorithms.

Main Methods:

• Combines the Lagrangian approach for PDE-constrained shape functions with NGSolve's automatic differentiation.
• Generates first- and second-order shape derivatives for unconstrained and constrained problems.
• Applies to linear, nonlinear, and surface-posed problems.

Main Results:

• Verified accuracy of computed derivatives using Taylor tests.
• Demonstrated the framework's applicability to diverse problems, including nonlinear elasticity and Maxwell's equations.
• Successfully implemented first- and second-order shape optimization algorithms.

Conclusions:

• The developed framework provides an efficient and flexible tool for automated shape differentiation.
• This approach significantly advances the capabilities of finite element software for shape optimization.
• The method is robust and accurate for a wide range of engineering applications.