Numerical Methods (SEMEC0140)

This course is fully based on variational methods as introductory topic for application of CFD with the finite element method. In the current version, it is a 3-part course divided as follows:

  • Part 1 (Tools): A general overview on numerical methods, basics on Python programming, and presentation of tools like Scipy ecosystem, solvers (FENICS, OpenFOAM etc.), mesh generators (meshio, Gmsh etc.), CAE/CAD open platforms, viewers (Paraview, Mayavi etc.) and support literature.
  • Part 2 (Theory): Discussion of approximation theory, construction principles of FEM, topics on functional analysis, strong/weak form, n-dimensional IBVPs.
  • Part 3 (Practice): project-based learning oriented to student’s research (generally CFD or multiphysics simulations with Gmsh/FENICS/Paraview trio), and state-of-the-art approaches.

Resources

Remark: resources for this course can be forked from GDCOC0072.

Books
Booklets
Courses
Misc
Papers
  • Physics-informed machine learning, Karniadakis et al., Nature Reviews, 2021
    • See: i) Table 1, for a list of PINN libraries; ii) subsection “Connection to classical numerical methods”;
  • Hybrid FEM-NN models: Combining artificial neural networks with the finite element method, Mitusch et al., JCP, 2021
    • Takeaways: i) may improve convergence speed; ii) fewer optimization steps; iii) generalization for spatial variability.
    • Gaps: non-linearity solvable through training; PDE solvers not optimized for large batch training on GPU; iii) FEM weak form requires efficient quadrature integration over the NN, whose rules are unknown for now.
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