Norma

Norma is a Julia prototype for testing algorithms and ideas for coupling and multiphysics, primarily in solid mechanics and heat conduction. It is a prototyping framework built for extensibility and experimentation, supporting implicit and explicit time integration, large deformations, contact, and domain decomposition by Schwarz methods.

Impact of two bars using different time integrators and mesh types: the left bar uses hexahedral elements with an implicit integrator; the right bar uses tetrahedral elements with an explicit integrator. Each subdomain advances with its own time step.

Dynamic torsion of a solid cylinder undergoing large deformations.

Large-deformation vibration of a soft rubber ball, playing in real time.

Quick start

# self-activating CLI wrapper
bin/norma input.yaml

# multi-threaded
bin/norma input.yaml --threads 8

# or directly with Julia
julia --project=. src/Norma.jl input.yaml

Interactive:

using Pkg; Pkg.activate(".")
using Norma
Norma.run("input.yaml")

See Installation to set up the environment and Running Norma for the full set of ways to launch a simulation.

Documentation

Installation
Running Norma
Testing
Examples
Profiling and debugging
Troubleshooting
Input File Reference — every section and key of the YAML input file

Features

Capabilities currently implemented:

Material models (infinitesimal- and finite-strain kinematics)

Linear elastic
Saint-Venant–Kirchhoff
Neohookean
Seth-Hill (generalized hyperelastic with tunable exponents)
J2 plasticity (finite deformation, multiplicative split, radial return, linear isotropic hardening)

Time integration

Quasi-static (with adaptive stepping)
Newmark (implicit dynamic)
Central difference (explicit dynamic)

Solvers

Newton's method (Hessian minimizer) with optional backtracking line search
Matrix-free steepest descent
Explicit (lumped-mass) solver

Element library

1D: BAR2
2D: TRI3, TRI6, QUAD4
3D: TETRA4, TETRA10, HEX8
Gauss, Dunavant, and Keast quadrature rules

Multidomain coupling — Schwarz alternating method

Overlapping and non-overlapping (Dirichlet–Neumann) Schwarz
Robin–Robin and impedance-matching (absorbing) Schwarz, including the adjoint-paired non-overlapping impedance coupling (see docs/notes/schwarz-coupling)
Multi-time-step (subcycled) coupling with time interpolation of the partner state
Frictionless and tied contact via Schwarz
Fixed and adaptive (Aitken) relaxation with interface prediction

Boundary conditions

Dirichlet (time-dependent displacement/velocity/acceleration)
Neumann traction and follower pressure
Robin (mixed displacement–traction)
Inclined-surface (roller) support

Adaptive mesh swapping

Triggered by time, stress-recovery error, overlap displacement error, or the elastic-to-plastic transition

Nodal recovery and field transfer

L2-projection recovery (lumped and consistent) of stress, von Mises stress, deformation gradient, and internal variables
Mesh-to-mesh field transfer for multidomain and remeshing workflows

Input/output

YAML input configuration
Exodus II output (nodal, element, and global variables)
CSV output of nodal and side-set quantities
Overlap-corrected (Arlequin-blended) energy output for multidomain runs

Reduced-order models — Operator Inference and RBF kernel ROMs are available and documented with the norma-opinf companion package; they are outside the scope of this guide.

License

Norma is licensed under the BSD 3-Clause License. See LICENSE.md in the repository for details.