Advanced examples#

Example	Highlight
Arc-length methods	\({\color{green}Advanced\ solvers:}\) Arc-length continuation for tracing nonlinear equilibrium paths through limit points and buckling instabilities.
Dynamic relaxation	\({\color{green}Advanced\ solvers:}\) Fictitious inertia and adaptive damping for nonlinear static equilibrium and buckling problems.
Explicit dynamics	\({\color{green}Time\ integration:}\) Lumped-mass central-difference integration of a 3D elastic waveguide, verified against the analytical P-wave solution and conservation of total energy.
Periodic boundary conditions	\({\color{brown}Boundary\ conditions:}\) Periodic constraints enforced through a linear multipoint constraint.
Nonlinear Robin boundary conditions	\({\color{brown}Boundary\ conditions:}\) Nonlinear Robin boundary terms implemented with surface residual kernels.
Quiet-element method	\({\color{orange}Domain\ control:}\) Progressive element activation for a transient directed energy deposition (DED) thermal simulation.
Stokes flow	\({\color{red}Mixed\ formulations:}\) Multiple finite-element variables and a mixed velocity-pressure formulation for incompressible Stokes flow.
Phase-field fracture	\({\color{red}Multiphysics:}\) Brittle fracture simulation with staggered displacement-phase-field solves, an irreversible history field, and custom derivatives for repeated eigenvalues.
Dendrite growth	\({\color{red}Multiphysics:}\) Coupled phase-field and thermal-diffusion simulation of dendritic solidification with implicit and explicit finite-element schemes.
Finite-strain plasticity	\({\color{blue}Constitutive\ models:}\) Finite-strain \(J_2\) plasticity with linear isotropic hardening.
Crystal plasticity	\({\color{blue}Constitutive\ models:}\) Finite-strain FCC crystal plasticity with sensitivity examples and a 100-grain polycrystal simulation.
Differentiable mesh	\({\color{teal}Differentiability:}\) Implicit-adjoint differentiation of a finite-element objective with respect to nodal coordinates, verified by central finite differences.
Neural-network surrogate model	\({\color{purple}Machine\ learning:}\) An energy-based constitutive surrogate for a cellular metamaterial, trained on RVE data and deployed in a macroscale simulation.