Numerical Methods & Discretization

Finite element formulations, update schemes, and multi-level operators underlying the solver implementations used in practice.

1. Updated Lagrangian vs Total Lagrangian

Notes comparing ULF and TLF for large deformation elastomers:

• Choice of configuration for evaluating F, stresses, and tangents
• Implications for stability and accuracy in tire simulations
• How this choice interacts with constitutive models and ALM constraints

2. Contact & Constraint Discretization

Discretizing contact and other constraints in a way that matches solver structure:

• Penalty vs Lagrange multiplier vs Augmented Lagrange in FE contact
• Defining and updating constraint violations structurally
• “Model problem” contact: fixed contact sets within inner solves

3. Multi-Level Operators

Restriction and prolongation viewed through the Rᵀ–L–S framework:

• Restriction operators that preserve physical structure
• Prolongation consistent with energy and stiffness
• When multi-grid is admissible—and when contact/damage breaks smoothness

4. Worked Examples (coming)

Future notes may include:

• 1D bar with ALM boundary constraints
• Structural stiffness C_eff and G_eff examples
• Toy multi-level contact problem illustrating “model problems”

These notes are intended as a bridge between textbook formulations and the actual structural and thermodynamic questions that drive tire design.