Lagrange Multiplier Method for Contact
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind." In static analysis, this term is set to zero, based on the assumption that "forces are applied slowly enough that acceleration can be ignored." It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness = strong." Not true. Stiffness is "resistance to deformation," strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but modeling "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they deliberately absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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.
Linear Elements (1st-order Elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix each iteration. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying the full load at once, applies it in small increments. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs. Iterative Method
The direct method is like "solving a system of equations accurately with pen and paper"—reliable but can take too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—in cooking, which step is most prone to failure? Actually, it's the "prep work." If mesh quality is poor, no matter how good the solver is, the results will be a mess.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer can be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct."
How to Think About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling the real constraint conditions is...
Related Topics
なった
詳しく
報告