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Fully and semi-automated shape differentiation in NGSolve.

Peter Gangl1, Kevin Sturm2, Michael Neunteufel2

  • 1TU Graz, Steyrergasse 30, 8010 Graz, Austria.

Structural and Multidisciplinary Optimization : Journal of the International Society for Structural and Multidisciplinary Optimization
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This study introduces a framework for automated shape differentiation in finite element software, enabling efficient shape optimization. The method accurately computes derivatives for various complex problems, enhancing engineering design processes.

Keywords:
Automated differentiationShape Newton methodShape derivativeShape optimisation

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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.