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Advanced Tutorials

This section covers advanced topics in FEAX for specialized applications.

Available Tutorials​

  • Periodic Boundary Conditions - Learn how to apply periodic boundary conditions using prolongation matrices for unit cell analysis, homogenization problems, and repeating structures.
  • Lattice Structure Homogenization - Computational homogenization of lattice structures using FEAX's flat toolkit with graph-based structure definition and periodic boundary conditions.
  • Adaptive Topology Optimization - 3D topology optimization with Gmsh adaptive remeshing, Heaviside continuation, and gradient-based mesh refinement.
  • Cohesive Fracture with Matrix-Free Newton Solver - Quasi-static fracture simulation using the matrix-free Newton solver and cohesive zone model with energy-based formulation and automatic differentiation.
  • Third Medium Contact - Frictionless contact via the third-medium method with HuHu-LuLu biharmonic regularization, incremental loading, and non-symmetric BC elimination.
  • Cahn-Hilliard Phase Separation - Transient spinodal decomposition using mixed formulation, backward Euler time integration, and the ImplicitPipeline time-stepping interface.
  • Laminated Plates (Shell) - First-order shear-deformation (Mindlin) plates and classical lamination theory with feax.mechanics.shell: lamina/laminate stiffness, thermal (N_T, M_T) resultants, von Kármán nonlinearity, and a load-stepped thermal-warping example.
  • Batched Topology Optimization with Surface Loads - Multi-load topology optimization using jax.vmap to vectorize FE solves, SIREN density fields, and gradient-based updates across 10 load cases in parallel.
  • Differentiable Linear Buckling - Eigenvalue buckling analysis with fe.create_linear_buckling_solver: shift-invert backends ("sparse" ARPACK default, "dense", "matfree", GPU "cudss"), analytic eigenvalue sensitivities, and end-to-end design gradients.
  • Narrow-Band & Giga-Voxel Topology Optimization - Implicit StructuredGrid domains, SparseDesign storage, NarrowBand sub-mesh solves, the matrix-free NarrowBandCMG geometric multigrid, and the moving-band OC + multires drivers that scale SIMP compliance to giga-voxel resolutions.
  • Automatic Sparse Differentiation (feax.asd) - Coloring-based sparse Jacobians/Hessians, CSR pattern algebra, and the assembled solver paths they unlock: direct factorization with extra_residual_fn couplings and assembled PᵀJP (direct/AMG) for periodic problems.