NNFEM is a

• lightweight educational 2D finite element library with truss and 2D quadrilateral elements. Different constitutive relations are supported, including plane stress/strain, hyperelasticity, elasto-plasticity, etc. It supports unstructured grid.
• neural network-enabled finite element library, which supports learning a neural network-based constitutive relations with both direct data (i.e, strain-stress pairs) and indirect data (i.e. full displacement field) via automatic differentiation, and solving finite element problems with network-based constitutive relations. In principle, it allows you to insert and learn a neural network anywhere in your finite element codes.

⚠️ NNFEM.jl is now superseded by AdFem.jl, a computational-graph-based finite element library for inverse modeling. NNFEM.jl will be no longer actively developed.

Documentation

Install NNFEM

Install via Julia registery

using Pkg; Pkg.add("NNFEM")

If you intend to develop the package (add new features, modify current functions, etc.), we suggest developing the package (in the current directory (NNFEM.jl))

julia> ]
pkg> dev .

When necessary, you can delete the package (in the current directory (NNFEM.jl))

julia> ]
pkg> rm NNFEM

If you only want to use the package and do not want to install the dependencies manually, do

julia> ]
pkg> activate .
(NNFEM) pkg> instantiate

Code structure

Basic finite element library

• elements are in /src/elements, including finite/small strain 2D quad and 1D truss elements.

• constitutive relations are in /src/materials, including plane stress/strain, hyperelasticity, elasto-plasticity, etc.

• solvers are in /src/solvers/Solver.jl, including generalized-alpha solver, etc.

• finite element domain, and core functions are in /src/fem.

Neural network based constitutive relations

• neural network based constitutive relations are in /src/materials/NeuralNetwork1D.jl and src/materials/NeuralNetwork2D.jl.

• neural network based finite element solvers are in /src/solvers/NNSolver.jl, which compute the loss for indirect data training.

• different customized neural networks are in /deps/CustomOp, which enables designing constraint-embedded neural networks.

Applications

There are several applications in research/ConstitutiveRelations/Plate and research/ConstitutiveRelations/Truss/Case1D

• Data_* runs the finite element solver to generate the test data and produces Data/1.dat and Data/domain.jld2

• NNLearn.jl learns an ANN with strain-to-stress data generated previously (extracted from each Gaussian quadrature points of the train sets). It produces learned_nn.mat. This is refered as direct training.

• Train_NN* learns an ANN from displacement data and associated loading condition. This is refered as indirect training.

• Test_NN* substitutes the constitutive law with the learned NN and test the hybrid model (NN + FEM) on the test sets.

• NN_Test_All* substitutes the constitutive law with the learned NN and test the hybrid model (NN + FEM) on the all test cases, and visualize the time-histories of the displacement and von-Mises stress fields.

Troubleshooting

Python dependencies

NNFEM is based on ADCME, you need to first install ADCME.jl, which will install a private Python environment for you. Take it easy, it will NOT mess your local environment!

A bit more about what is under the hood: PyCall relies on the python version installed in $HOME/.julia/conda/3/bin/python, you can check the path with

julia> using PyCall
julia> PyCall.python

If you want to use Python packages via PyCall, install python packages, e.g., tikzplotlib, via

$HOME/.julia/conda/3/bin/python -m pip install tikzplotlib

Build customized operators

NNFEM includes some custom operators. Those operators are implemented in C++ and will be compiled automatically when you invoke Pkg.build("NNFEM"). However, in the case you encounter any compilation issue, you can go into the deps directory, and run build.jl

cd deps
julia build.jl

Submit an issue

You are welcome to submit an issue for any questions related to NNFEM.

Research

1. Huang, Daniel Z., Kailai Xu, Charbel Farhat, and Eric Darve. "Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks"

2. Kailai Xu, Huang, Daniel Z., and Eric Darve. "Learning Constitutive Relations using Symmetric Positive Definite Neural Networks"