Hybrid ML-FEM Viscoelastic-Viscoplastic Damage Model

A sophisticated finite element implementation combining traditional constitutive modeling with machine learning (LSTM neural networks) for modeling complex material behavior in epoxy nanocomposites under cyclic loading conditions.

Table of Contents

• Overview
• Features
• Model Description
• Installation
• Usage
• Configuration
• Output
• Examples
• Contributing
• Citation
• License

Overview

This code implements a hybrid ML-FEM approach for large deformation solid mechanics, specifically designed for epoxy nanocomposite materials with moisture content. The implementation seamlessly combines:

• Traditional physics-based constitutive models for initial simulation phases
• LSTM neural networks for accelerated computation in later phases
• Multi-network viscoelastic-viscoplastic formulation with damage evolution
• Nanoparticle and moisture effects through amplification factors

The approach is based on deal.II's step-44 tutorial and extends the Quasi_static_Finite_strain_Compressible_Elasticity example from the deal.II code gallery.

Features

🔧 Multi-Physics Material Model

• Viscoelastic behavior: Two-network model with equilibrium and non-equilibrium responses
• Viscoplastic behavior: Rate-dependent plasticity with backstress evolution
• Damage mechanics: Progressive material degradation under cyclic loading
• Environmental effects: Temperature and moisture content dependencies
• Nanoparticle reinforcement: Weight fraction-dependent amplification

🤖 Machine Learning Integration

• LSTM neural networks: Two-layer architecture with 200 hidden units
• Seamless switching: Automatic transition from physics to ML models
• Pre-trained weights: Load neural network parameters from external files
• Numerical tangent: Automatic differentiation for ML components

⚡ Advanced Numerical Methods

• Updated Lagrangian formulation: Large deformation framework
• Newton-Raphson solver: Robust nonlinear solution scheme
• Parallel processing: MPI support for large-scale computations
• Adaptive timestepping: Dynamic time step control

📊 Comprehensive Analysis

• Cyclic loading: Progressive amplitude cycling with unloading
• Multi-scale output: Element and nodal averaging options
• Real-time monitoring: Force-displacement tracking at specific points
• Paraview visualization: VTK output for post-processing

Model Description

Rheological Framework

The material model is based on a multi-network rheological representation:

Total Response = Equilibrium Network + Viscous Network + Viscoplastic Element + Damage

• Equilibrium Network (F_e, σ_eq): Instantaneous elastic response
• Viscous Network (F_v, σ_neq): Time-dependent Maxwell element behavior
• Viscoplastic Element (F_vp): Rate-dependent plastic flow with threshold
• Damage Factor (1-d): Progressive stiffness reduction

Mathematical Formulation

Multiplicative decomposition:

F_total = F_ve × F_vp^(-1)
F_ve = F_e × F_v^(-1)

Viscous flow evolution:

γ̇ = γ̇_0 × exp[(ΔG/kT) × (τ/τ_hat)^m - 1]

Viscoplastic flow:

γ̇_p = a × |ε - ε_0|^b × ε̇  (when τ > σ_0)

Environmental amplification:

X = (1 + 5φ_np + 18φ_np²) × α_Z × α_T

Hybrid ML-Physics Approach

if (timestep < switch_point) {
    // Traditional constitutive model
    update_internal_equilibrium();
} else {
    // LSTM neural network
    lstm_forward();
}

Installation

Prerequisites

• deal.II 9.0+ with MPI support
• CMake 3.10+
• C++ compiler with C++14 support
• Eigen3 linear algebra library
• Optional: Trilinos with Sacado for automatic differentiation

Build Instructions

1. Clone the repository:

git clone https://github.com/yourusername/vevp-ml-model.git
cd vevp-ml-model

2. Configure and compile:

cmake -DDEAL_II_DIR=/path/to/deal.II .
make release
make

3. Create output directory:

mkdir output

Verify Installation

./vevp_ml_model --help

Usage

Basic Execution

./vevp_ml_model

The program reads parameters from parameters.prm and generates:

• VTK files: In output/ directory for Paraview visualization
• Force-displacement data: data-for-gnuplot.sol for plotting
• Timing information: data_times.txt for performance analysis

MPI Execution

mpirun -np 4 ./vevp_ml_model

Physics-Only Mode

To run without machine learning (recommended for initial testing):

# In parameters.prm:
set Switch to ML = Off
set At which timestep switch to ML ? = 2147483646

ML-Enhanced Mode

To enable machine learning acceleration:

# In parameters.prm:
set Switch to ML = On
set At which timestep switch to ML ? = 500

Note: ML mode requires pre-trained LSTM weight files in the working directory.

Configuration

Key Parameter Categories

Loading Conditions

subsection Boundary conditions
  set Load type = cyclic_to_zero        # Cyclic loading with unloading
  set First stretch = 0.5               # Progressive amplitudes (mm)
  set Second stretch = 0.7
  # ... up to Seventh stretch
  set Total cycles = 7
end

Material Properties

subsection Material properties
  # Hyperelastic networks
  set mu1 = 760.0                       # Equilibrium network modulus (Pa)
  set mu2 = 790.0                       # Viscous network modulus (Pa)
  set nu1 = 0.23                        # Poisson's ratio
  
  # Viscous flow
  set gamma_dot_0 = 9.7746e11           # Reference strain rate
  set dG = 1.9761e-19                   # Activation energy
  set m = 0.657                         # Rate sensitivity
  
  # Viscoplasticity
  set sigma0 = 5.5                      # Yield threshold (Pa)
  set a = 0.179                         # Flow parameter
  set b = 0.910                         # Flow exponent
  
  # Environmental effects
  set temp = 296.0                      # Temperature (K)
  set zita = 0.0                        # Water content
  set wnp = 0.0                         # Nanoparticle weight fraction
end

Numerical Settings

subsection Finite element system
  set Polynomial degree = 1             # Linear elements
  set Quadrature order = 2              # 2×2×2 Gauss points
end

subsection Time
  set Delta de = 5e-4                   # Displacement increment
  set load_rate = 0.0165                # Loading rate (mm/s)
end

Parameter Sensitivity

Parameter	Physical Meaning	Typical Range	Effect
mu1, mu2	Network stiffness	100-10000 Pa	Overall stiffness
gamma_dot_0	Viscous rate	1e8-1e12 s⁻¹	Relaxation speed
sigma0	Yield threshold	1-50 Pa	Plastic onset
wnp	Nanoparticle fraction	0-0.1	Reinforcement
zita	Moisture content	0-0.05	Softening effect

Output

Generated Files

1. solution-*.vtk: Paraview visualization files

   • Displacement fields
   • Stress components (Cauchy stress)
   • Strain measures (Green-Lagrange)
   • Damage variables

2. data-for-gnuplot.sol: Time history data

   • Force-displacement curves
   • Volume evolution
   • Reaction forces at boundaries

3. data_times.txt: Performance metrics

   • Assembly times
   • Solution times
   • MPI communication overhead

Visualization

Paraview workflow:

1. Load solution-*.vtk files
2. Apply "Warp by Vector" filter for deformed configuration
3. Color by stress components or damage
4. Create animations for loading cycles

Gnuplot example:

plot 'data-for-gnuplot.sol' using 6:9 with lines title 'Force-Displacement'

Examples

Example 1: Pure Epoxy Matrix

# In parameters.prm
set wnp = 0.0          # No nanoparticles
set zita = 0.0         # Dry conditions
set mu1 = 800.0        # Base matrix properties

Example 2: Nanocomposite (5% CNT)

set wnp = 0.05         # 5% nanoparticle weight fraction
set zita = 0.0         # Dry conditions
# Networks automatically amplified by factor X

Example 3: Moisture Effects

set wnp = 0.0          # Pure matrix
set zita = 0.02        # 2% moisture content
# Softening through α_Z factor

Example 4: ML Acceleration

set Switch to ML = On
set At which timestep switch to ML ? = 200
# Hybrid physics→ML transition

Contributing

We welcome contributions! Please follow these guidelines:

1. Fork the repository
2. Create a feature branch
3. Test your changes thoroughly
4. Submit a pull request with detailed description

Development Setup

# Debug build for development
cmake -DDEAL_II_DIR=/path/to/deal.II -DCMAKE_BUILD_TYPE=Debug .
make

Testing

# Run with small problem for testing
# Modify parameters.prm for quick convergence
./vevp_ml_model

Citation

If you use this code in your research, please cite:

@article{bahtiri2023machine,
  title={A machine learning-based viscoelastic--viscoplastic model for epoxy nanocomposites with moisture content},
  author={Bahtiri, Betim and Arash, Behrouz and Scheffler, Sven and Jux, Maximilian and Rolfes, Raimund},
  journal={Computer Methods in Applied Mechanics and Engineering},
  volume={415},
  pages={116293},
  year={2023},
  publisher={Elsevier},
  doi={10.1016/j.cma.2023.116293}
}

License

This project is licensed under the GNU General Public License v2.1

Acknowledgments

• deal.II Development Team for the finite element library
• Leibniz Universität Hannover for research support
• Original step-44 tutorial authors for the foundation code

Contact

For questions or support:

• Lead Developer: Betim Bahtiri
• Institution: Leibniz Universität Hannover
• Email: [email protected]
• Issues: Use GitHub issue tracker for bug reports and feature requests

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Note: This implementation is designed for research purposes. For production use, please validate results against experimental data and consider additional verification steps.