Augmented Lagrangian Method
Augmented Lagrangian Method: Theoretical Foundations
What is the Augmented Lagrangian Method?
Professor, is the Augmented Lagrangian method a "best of both worlds" combination of the Penalty method and the Lagrange multiplier method?
Exactly. It's a technique that combines the simplicity of the Penalty method with the zero-penetration accuracy of the Lagrange multiplier method.
In addition to the penalty term $k_p g_n$, the Lagrange multiplier $\lambda_n$ is iteratively updated. As iterations proceed, penetration converges to zero.
So even with a small penalty stiffness $k_p$, penetration can be reduced through iteration, right?
Its dependence on $k_p$ is smaller than the Penalty method. This is its greatest advantage. Augmented Lagrangian is the default in Ansys.
Summary
Key Points:
- Penalty + Iterative Lagrange Update — Best of both worlds
- Small dependence on $k_p$ — More stable than the Penalty method
- No additional DOFs required — More efficient than the Lagrange multiplier method
- Ansys Default — The most widely recommended method
Hestenes-Powell Method 1969
The Augmented Lagrangian method is a hybrid technique combining penalty and Lagrange multipliers, independently published by M.R. Hestenes and M.J.D. Powell in 1969. By updating the multipliers with each outer iteration, it avoids the ill-conditioning of pure penalty methods while also avoiding the large-scale simultaneous equations of pure multiplier methods. Its application to contact FEM was systematized by Simo and Laursen in the late 1980s.
Computational Methods for the Augmented Lagrangian Method
Implementation of Augmented Lagrangian
Algorithm:
1. Solve initially as a Penalty method with $\lambda = 0$
2. Check penetration amount $g_n$
3. Update $\lambda$: $\lambda \leftarrow \lambda + k_p g_n$
4. Solve again with the updated $\lambda$
5. Iterate until $g_n$ becomes sufficiently small
So there's an outer loop ($\lambda$ update) and an inner loop (Newton-Raphson), right?
A double iteration loop. The inner loop satisfies equilibrium, the outer loop satisfies contact constraints. Computational cost is about 1.5 to 2 times that of the Penalty method.
Solver Settings
Summary
Multiplier Update Iteration Scheme
Augmented Lagrangian contact has a double iteration structure: an outer loop updates the Lagrange multiplier λ, and an inner loop solves the nonlinear finite element solution. In the formulation by Laursen & Simho (1993), the multiplier update formula λ_{k+1} = λ_k + ε_N g_N (g_N: penetration, ε_N: penalty) is adopted. Since ε_N can be set intuitively by back-calculating from physical gap tolerance, convergence tuning is more intuitive than with the Penalty method.
Augmented Lagrangian Method in Practice
Augmented Lagrangian in Practice
Many Ansys users unconsciously use Augmented Lagrangian. The default is the most stable.
Practical Checklist
Aircraft Engine Blade Fretting
Around 2008, GE Aviation began using ANSYS Mechanical's Augmented Lagrangian contact to analyze fretting wear in turbine blade dovetail sections. They evaluated repeated contact equivalent to 10⁷ cycles combined with a cumulative damage model, reproducing wear depth on Inconel 718 blades with an accuracy of ±15μm compared to measured values. This case was presented at an academic conference as an example that reduced prototype manufacturing costs by about 30%.
Augmented Lagrangian Method: Software & Solver Comparison
Tools for Augmented Lagrangian
Selection Guide
Evolution of ANSYS ALM Implementation
ANSYS adopted the Augmented Lagrange Method (ALM) as the default contact algorithm for CONTA174/TARGE170 elements in the late 1990s. In ANSYS 10.0 (2005), the ALM convergence criterion was revised to a contact force basis, significantly improving penetration issues in thick plates. In the current ANSYS Mechanical 2024, the "program controlled" setting, which automatically switches between ALM and the Penalty method, is the default.
Advanced Technology
Advanced Topics in Augmented Lagrangian
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