Lagrange Multiplier Method for Contact
Lagrange Multiplier Method for Contact: Theoretical Foundations
What is the Lagrange Multiplier Method?
Professor, how is the Lagrange multiplier method different from the penalty method?
The penalty method is like "pushing back penetration with a spring" (allowing slight penetration). The Lagrange multiplier method strictly satisfies the constraint condition (makes penetration completely zero).
Introduces additional unknowns (Lagrange multipliers $\lambda$). $\lambda$ is the contact pressure itself.
Penetration is completely zero... That's more accurate than the penalty method.
Penetration accuracy is high, but the system of equations becomes larger due to additional DOFs (Lagrange multipliers), leading to increased computational cost. Furthermore, the matrix may contain zero diagonals, which can worsen the condition number.
Penalty Method vs. Lagrange Multiplier Method
| Characteristic | Penalty Method | Lagrange Multiplier Method |
|---|---|---|
| Penetration | Small but non-zero | Completely zero |
| Additional DOFs | None | Lagrange multipliers ($\lambda$) |
| Condition Number | Worsens with penalty stiffness | Worsens with zero diagonals |
| Parameter Dependence | Depends on $k_p$ | No parameters |
| Computational Cost | Low | High |
| Compatibility with Explicit Methods | Good | Poor (Implicit methods only) |
High accuracy but also high cost.
In Abaqus, the KINEMATIC option corresponds to the Lagrange multiplier method. The contact surfaces do not slip and "penetration" is zero. Suitable for precise bolt fastening or pressure vessel contact.
Summary
Key Points:
- Penetration is completely zero — More rigorous than the penalty method
- Requires additional DOFs (Lagrange multipliers) — Increases computational cost
- Abaqus KINEMATIC — Lagrange multiplier method
- Suitable for precise contact (bolt fastening, pressure vessels)
- Cannot be used with explicit methods — Implicit methods only
Lagrange's Analytical Mechanics, 1788
In his 1788 publication "Mécanique analytique," Joseph-Louis Lagrange systematized a method for incorporating constraint conditions into variational principles using multipliers (later called Lagrange multipliers). It took about 180 years for this mathematical framework to be applied to contact problems. In the 1970s, B.M. Irons and others introduced multipliers into the Galerkin weak form for contact constraints, completing the FEM formulation that accurately satisfies the non-penetration condition.
Computational Methods for Lagrange Multiplier Method for Contact
Implementation of the Lagrange Multiplier Method
Extended system of equations:
$[C]$ is the constraint matrix, $\{\lambda\}$ are the Lagrange multipliers (contact pressure).
The bottom-right is a zero matrix... that seems bad for the condition number.
It's a saddle point problem. Not an issue for direct solvers, but difficult to converge with iterative solvers.
Solver Settings
Summary
Discretization of Mixed Variational Methods
In contact FEM using the Lagrange multiplier method, a mixed variational method is used that couples displacement degrees of freedom with contact forces (Lagrange multipliers) as unknowns. The stiffness matrix after discretization becomes a saddle point problem, losing positive definiteness in the diagonal blocks. In the 1980s, the Brezzi-Babuška condition (inf-sup condition) was established as a criterion for appropriate selection of contact multiplier spaces, providing design guidelines for stable element pairs.
Lagrange Multiplier Method for Contact in Practice
Practical Use of the Lagrange Multiplier Method
Use only for problems where zero penetration is mandatory. For most problems, the penalty method is sufficient.
Appropriate Use Cases
| Situation | Reason |
|---|---|
| Sealing surfaces of pressure vessels | Penetration = leakage. Zero is mandatory. |
| Precision bolt fastening | Accurate evaluation of seating pressure. |
| Press-fit (interference fit) | Precise control of interference amount. |
Practical Checklist
Press Die Contact Analysis
In deep drawing press die analysis using ABAQUS Standard, introduced by Toyota around 2003, the Lagrange multiplier method was adopted for contact between the blank holder and the material. By strictly maintaining zero penetration, the prediction accuracy for sheet thickness reduction rate improved to within ±3%. Previously, with the penalty method, tuning contact stiffness took several days, but with the multiplier method, the number of trials was halved.