Total Lagrangian Method and Updated Lagrangian Method
Total Lagrangian Method and Updated Lagrangian: Theoretical Foundations
TL Method and UL Method
Professor, could you explain the difference between the Total Lagrangian (TL) method and the Updated Lagrangian (UL) method?
Both are formulations for large deformation, but they differ in their reference configuration.
| Characteristic | TL Method | UL Method |
|---|---|---|
| Reference Configuration | Initial configuration ($t=0$) | Last converged configuration |
| Strain Measure | Green-Lagrange Strain $E_{ij}$ | Logarithmic Strain (Modified Euler-Almansi) |
| Stress Measure | Second Piola-Kirchhoff Stress $S_{ij}$ | Cauchy Stress $\sigma_{ij}$ |
| Stiffness Matrix Formation | Integrated over initial configuration | Integrated over current configuration |
So the difference is whether we use the "initial configuration as reference" or the "current configuration as reference".
If implemented correctly, both give the same solution. They are chosen based on computational efficiency and implementation convenience. Abaqus uses UL, Nastran's SOL 106 is TL-like.
Summary
Key Points:
- TL Method — Initial configuration reference. Green-Lagrange Strain. Nastran SOL 106
- UL Method — Current configuration reference. Cauchy Stress. Abaqus default
- Same result if correctly implemented — Choice of reference configuration is a matter of numerical efficiency
- User only needs to set "NLGEOM=YES" — TL/UL selection is internal to the solver
The FEM Dichotomy of Lagrange and Euler
Finite deformation FEM formulations fall into two main categories: Total Lagrangian (TL) and Updated Lagrangian (UL). TL integrates based on the initial (reference) configuration, using the deformation gradient tensor F to accurately represent finite strain. UL uses the current configuration as reference and updates it each step. TL is suitable for hyperelastic materials (rubber), while UL is better for large metal deformation (with contact). The establishment of this distinction is the achievement of Malvern, Ogden, and Simo (1980s-90s).
Computational Methods for Total Lagrangian Method and Updated Lagrangian
Formulation Details
Principle of Virtual Work for TL method (Weak Form):
UL method:
So the TL method integrates over the initial volume $V_0$, and the UL method over the current volume $V$.
The difference in integration domain affects computational cost. TL method has complex Green-Lagrange strain calculation, UL method requires configuration updates.
Summary
How to Use Second Piola-Kirchhoff Stress
The Total Lagrangian formulation uses the conjugate pair of "Second Piola-Kirchhoff stress S" and "Green-Lagrange strain ε". Unlike Cauchy stress, S is a "nominal stress based on the reference configuration" that maps the deformed area back to the pre-deformation configuration. Because it's based on the initial configuration, the material constant tensor C (elastic tensor) remains constant regardless of deformation, which naturally fits the FEM implementation of hyperelastic models like Mooney-Rivlin.
Total Lagrangian Method and Updated Lagrangian in Practice
Practical Usage Distinction
Users rarely need to be aware of TL/UL. It's best to rely on the solver's default.
Practical Checklist
Application of TL to Rolling Tire FEM Analysis
The contact deformation of a radial tire (meridian cross-section) involves hyperelastic large deformation where rubber stretches over 100%, making Total Lagrangian formulation essential. Since the 1980s, Bridgestone has used TL-formulation FEM to predict tire contact patch shape, contact pressure, and rolling resistance, achieving 90% accuracy (compared to measurements) by the 2000s. Today, multifunctional analysis combining TL formulation with Mullins effect (softening) and creep is standard.
Total Lagrangian Method and Updated Lagrangian: Software & Solver Comparison
Tools
Selection Guide
There's no need to choose a solver based on TL/UL selection. All solvers support large deformation.
Marc Special Hyperelastic Elements and Large Deformation
MSC Marc was originally developed as an FEM code specialized for Total Lagrangian analysis of rubber, with built-in support for Ogden, Mooney-Rivlin, Neo-Hookean...
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