A Rust-based parametric Finite Element Analysis (FEA) tool for Direct Parameterisation of Invariant Manifolds (e.g., Spectral Submanifolds, SSM) in geometrically nonlinear structural dynamics. Enables high-dimensional model order reduction (MOR) by assembling parametric matrices dependent on mesh geometry and material properties.

• Parametric FEA: Assembles stiffness/mass/damping matrices as polynomial functions of design parameters (geometry, material).
• Geometric Nonlinearity: Supports large deformations and nonlinear constitutive laws.
• SSM/MOR Integration: Compatible with spectral submanifold reduction techniques for efficient dynamics analysis.
• Rust-Powered: Memory-safe, parallelizable, and high-performance numerics via ndarray, nalgebra, or petgraph.
• Extensible: Modular design for adding new elements, materials, or solvers.

✅ Reduced-Order Modeling (ROM): Generate low-dimensional SSMs for high-DOF systems.
✅ Structural Optimization: Sensitivity analysis via parametric matrix derivatives.
✅ Nonlinear Dynamics: Study bifurcations, instabilities, or forced responses.

git clone https://github.com/yourusername/ParametricFEA-SSM.git  
cd ParametricFEA-SSM  
cargo build --release  

use parametricfea::prelude::*;

let mesh = Mesh::from_gmsh("beam.geo");  // Load parametric mesh  
let material = Steel::new(E=210e9, nu=0.3);  
let fe = NonlinearBeamElement::new(quadrature=3);  

// Assemble parametric stiffness matrix K(p)  
let K = fe.assemble_param_matrix(&mesh, &material, |p| p[0] * p[1]^3);  

The tool implements:

1. Parametric Assembly: Matrices are expressed as K(p) = K₀ + ∑ p� Kᵢ where p are design parameters.
2. Hyper-Reduction: Integrates with SSM theory (Jain & Haller, 2022) for nonlinear MOR.
3. Automatic Differentiation: For analytic sensitivity computation (optional dfdx feature).

src/  
├── elements/         # Finite elements (beam, shell, solid)  
├── materials/        # Constitutive laws  
├── assembly/        # Parametric matrix assembly  
├── reduction/       # SSM/MOR interfaces  
└── io/              # Mesh/result handling  

PRs welcome! Key needs:

• More element types (e.g., piezoelectric, composites)
• GPU acceleration via rust-cuda.

For collaborations/questions:

• Email: your.email@example.com
• Twitter: @yourhandle

This README balances technical depth with accessibility. Adjust:

• Add a "Benchmarks" section if performance is a highlight.
• Link to papers/thesis if applicable.
• Include GIFs of nonlinear deformations/SSM projections (if visual).

Would you like me to emphasize any specific aspect (e.g., GPU support, validation cases)?