Stabilize Solution for Mortar Contact

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Mortar Method

In previous article, mortar method which is one of the solution for mechanical contact, is shown. As it mentions, solution is not stable because of the characteristics of matrix. So some stabilizers for the solution should be introduced. In this article, I will summarize these stabilizers, Dual Mortar, Coarse Mortar and Augmented Lagrangian as my note.

The exact form of mortar is as follows:

\underbrace{
\begin{bmatrix}
\mathbf{K}_1 & \mathbf{0} & \mathbf{B}_1^{\mathrm{T}} \\
\mathbf{0}   & \mathbf{K}_2 & \mathbf{B}_2^{\mathrm{T}} \\
\mathbf{B}_1 & \mathbf{B}_2 & \mathbf{0}
\end{bmatrix}
}_{\text{global stiffness matrix } \mathbf{A}_{\mathrm{aug}}}
\begin{bmatrix}
\mathbf{u}_1 \\[4pt]
\mathbf{u}_2 \\[4pt]
\boldsymbol{\lambda}
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{f}_1 \\[4pt]
\mathbf{f}_2 \\[4pt]
\mathbf{0}
\end{bmatrix}

It turns out that

\mathbf{B}_1 \mathbf{u}_1 + \mathbf{B}_2 \mathbf{u}_2 = 0

This extended system is a saddle-point problem where the discontinuity of displacement at the contact interface Γ is constrained by the Lagrange multiplier λ.

It seems that this is just formulation with a constraint with Lagrange multipliers, but it is actually not in Dual Mortar.

Dual Mortar

In the dual mortar formulation, the Lagrange multiplier space is constructed using a biorthogonal (dual) basis. In the dual mortar formulation, the Lagrange multiplier space is constructed using a biorthogonal (dual) basis defined on the contact interface (\Gamma_c). Specifically, the Lagrange multiplier basis functions (\mu_i) are chosen to be biorthogonal to the primal trace basis functions (\phi_j), i.e.,

\int_{\Gamma_c} \mu_i  \phi_j  d\Gamma_c = \delta_{ij}

Here, ({\phi_j}) denotes the trace of the primal finite element basis on the slave side of the interface.

This biorthogonality property leads to a local (diagonal or block-diagonal) structure of the coupling matrices, enabling efficient elimination of the Lagrange multipliers via static condensation.

In the standard mortar formulation, the coupling matrix between multipliers and primal trace functions is

M_{ij} = \int_{\Gamma_c} \mu_i  \phi_j  d\Gamma

This matrix is generally dense, even if the original finite element matrices are sparse.
As a result:

• The saddle-point system becomes globally coupled
• Static condensation is expensive
• Memory cost increases

With the biorthogonality condition,

M=I\\
(B=I)

the coupling matrix becomes the identity matrix.

This leads to:

• The Lagrange multipliers becoming locally supported
• Possibility of static condensation
• Improved sparsity
• Better computational efficiency
• Reduced memory footprint

Coarse Mortar

In the mortar method, the Lagrange multiplier space is intentionally chosen to be coarser than the displacement space. For example, the displacement field may use P2 elements, while the multiplier space uses P1 or even P0 elements, i.e., a lower polynomial order.

The reason is stability and efficiency. If the multiplier space has the same richness as the displacement trace space, the number of degrees of freedom increases significantly, the saddle-point system becomes larger, and the formulation is more prone to violating the inf–sup condition. This often leads to instability and deteriorated convergence.

By reducing the order of the multiplier space, we effectively restrict it to only the physically necessary constraint modes. Mathematically, the constraint is written as

Bu=0

but choosing a coarser multiplier space reduces the rank of B. In other words, we deliberately weaken the constraint.

The essential idea is to stabilize the formulation by lowering the dimensionality of the interface constraint space. Physically, this means ignoring small-scale oscillations along the interface and enforcing only macroscopic compatibility.

Augmented Lagrangian

In the augmented Lagrangian (AL) approach, a penalty term is added to the energy functional in addition to the classical Lagrange multiplier term. The interface contribution becomes

\int_{\Gamma_c}
\lambda  (u_1 - u_2)
+
\frac{\rho}{2}
| u_1 - u_2 |^2
 d\Gamma

The first term enforces the constraint through a Lagrange multiplier, while the second term penalizes the constraint violation with a parameter ( \rho > 0 ).

In the discrete system, this modifies the stiffness structure. Instead of a purely saddle-point system, the primal block gains an additional term of the form

K_1 + \rho B_1^T B_1

and similarly for the coupled interface contributions. The matrix is no longer purely indefinite in the same way as the classical Lagrange multiplier formulation.

The motivation is numerical robustness. A pure Lagrange multiplier method leads to a saddle-point system that is indefinite and sensitive to inf–sup stability. The multiplier field can exhibit oscillations, and the linear system can be difficult to solve.

By introducing the augmented term, the constraint is softened. This improves numerical stability, enhances conditioning, and makes the multiplier field behave more physically (less oscillatory and better controlled).

The essential idea is to stabilize the constraint enforcement by combining Lagrange multipliers with a penalty mechanism — a hybrid between exact constraint enforcement and soft penalization.