3. Finite element contact formulations

Once the contact kinematics of a bolted or frictional assembly have been defined — gap functions, normal and tangential relative displacements, and the active contact set — the next decision is how to enforce the contact constraints inside the finite element solver. Two independent choices must be made: the discretization scheme that decides which geometric entities are paired and where the constraint is evaluated (node-to-surface or surface-to-surface), and the constraint enforcement method that decides how the non-penetration and friction conditions enter the algebraic system (penalty, Lagrange multiplier, or augmented Lagrangian). These choices are orthogonal, and mixing them incorrectly is one of the most common sources of slow convergence or spurious slip in bolted-joint models.

Discretization: node-to-surface versus surface-to-surface

In a node-to-surface (NTS) formulation, the nodes of the "slave" surface are projected onto the "master" surface, and a constraint is written for each slave node. The gap is computed from the slave node to its closest point on the master facet, and the contact force is applied at that node. NTS is inexpensive and robust for small sliding, and it remains the default in many explicit codes. Its weaknesses appear when the slave mesh is coarser than the master: individual nodes may "tunnel" through a master facet, and the contact pressure is sampled only at nodes, producing a lumpy pressure distribution across a bolt head or flange.

In a surface-to-surface (STS) formulation, the constraint is integrated over the overlapping area of the two surfaces rather than at isolated nodes. The mortar method is the most rigorous realization: it introduces Lagrange multipliers on the slave surface and evaluates the weak-form contact integral with shape functions from both sides. The result is a smoother, mesh-insensitive pressure field and better performance under large sliding. For a bolted flange where the goal is to predict the pressure distribution under the washer, STS is usually worth the extra cost.

Constraint enforcement: penalty, Lagrange, augmented Lagrangian

The penalty method approximates the non-penetration condition by adding a stiff spring between the contacting surfaces. The normal contact pressure is p_n = ε_n g_n when the gap g_n is negative (penetration), where ε_n is the penalty stiffness. The method adds no unknowns, keeps the system positive definite, and is trivial to implement. Its cost is a trade-off: too small a penalty allows visible penetration and under-predicts the clamp load; too large a penalty conditions the stiffness matrix badly and stalls the Newton iteration. A common starting value is ε_n = α · E / h, with E the softer material's Young's modulus, h a characteristic element size, and α typically between 0.1 and 10.

The Lagrange multiplier method enforces the constraint exactly by introducing additional unknowns — the contact pressures themselves — and solving a saddle-point system. Penetration is zero to machine precision, and the multipliers are the physical contact pressures, which is attractive when the pressure field is the quantity of interest. The price is a larger, indefinite system, zero diagonal entries that complicate direct solvers, and the need for careful handling of the active set. For friction, the multipliers become the tangential tractions, and the stick/slip decision is made by comparing the trial traction with the friction bound μ p_n.

The augmented Lagrangian method combines both ideas. It retains the penalty term but adds a multiplier that is updated iteratively:

for each augmentation iteration k:
    solve  K u + B^T λ_k + ε B^T g(u) = f
    update λ_{k+1} = λ_k + ε g(u)
    if ||g(u)|| < tol_gap and ||λ_{k+1} - λ_k|| < tol_mult:
        converged

Here B is the constraint gradient operator, λ the multiplier vector, and ε the penalty parameter. Because the multiplier absorbs the constraint error, the penalty stiffness can be moderate, which keeps conditioning acceptable while driving penetration toward zero. This is why augmented Lagrangian is the default in many implicit contact solvers, including the general contact algorithms in Abaqus/Standard and ANSYS.

Choosing a formulation for bolted and frictional assemblies

For a bolted joint, the contact problem has three distinct zones: the bolt head or nut against the clamped part, the thread interface (if modeled), and the flange-to-flange interface. Each zone has different requirements. The bolt-head contact is a small-sliding, high-pressure patch where the pressure distribution matters for fatigue assessment — STS with augmented Lagrangian is a good fit. The flange interface is large-sliding-capable and dominated by friction; here the tangential formulation matters more than the normal one, and a penalty or augmented Lagrangian treatment with a well-chosen friction coefficient is usually sufficient.

The table below summarizes the practical trade-offs. The "typical use" column reflects common practice in implicit structural codes, not a hard rule.

Worked example: clamp load in a single-bolt joint

Consider a single M10 bolt clamping two 10 mm steel plates, modeled with a target preload of 20 kN. The bolt shank is modeled with beam or solid elements, and the head bears on the upper plate. The contact pair is defined between the bolt-head underside (master) and the upper plate top surface (slave). The plate mesh is 2 mm hexahedral elements, giving a characteristic size h = 2 mm and E = 210 GPa.

With a penalty formulation, a starting penalty stiffness is ε_n = 0.1 · 210000 / 2 = 10500 N/mm³. After the first solve, check the maximum penetration in the contact output. If it exceeds roughly 1% of the element size (0.02 mm), increase α to 1.0 and re-solve. If the Newton residual stalls or the number of iterations per increment exceeds about 15, reduce α to 0.05 and instead switch to augmented Lagrangian, which will drive the penetration down without further stiffening.

With augmented Lagrangian, the same model typically converges in 3–5 augmentation iterations per load increment, and the reported contact pressure under the bolt head should integrate to approximately the applied preload. A useful verification: sum the normal contact forces (CNORMF in Abaqus, or the equivalent in ANSYS) over the contact pair and compare with the 20 kN target. A discrepancy larger than 2% indicates either insufficient penalty stiffness, an unconverged augmentation loop, or a contact pair that does not fully cover the bearing area.

Friction enforcement and the stick/slip transition

Friction is enforced through the same three methods, but the tangential problem is path-dependent and non-associated. In the penalty approach, a tangential penalty stiffness ε_t is used, and the trial traction is compared with the Coulomb bound μ p_n. If the trial traction is below the bound, the point is in stick and the elastic tangential displacement is retained; if it exceeds the bound, the point slips and the traction is scaled back to the bound. In the Lagrange and augmented Lagrangian approaches, the tangential multiplier plays the same role, with the slip condition acting as a complementarity constraint.

For bolted assemblies, the friction coefficient is often the largest source of uncertainty. A value of μ = 0.15 is common for as-received steel, but it can range from 0.08 (lubricated) to 0.5 (clean, dry, rough). Because the clamp load retention under transverse loading depends directly on μ, a sensitivity study over μ is more valuable than a single high-fidelity run. The formulation choice should therefore be stable enough that the sensitivity study reflects physics, not solver noise — which is another argument for augmented Lagrangian in implicit bolted-joint work.

In summary, the discretization and enforcement choices should be made together. For most implicit bolted-joint analyses, surface-to-surface discretization with augmented Lagrangian enforcement provides the best balance of accuracy, robustness, and interpretability of the contact pressure field. Penalty methods remain useful for explicit and scoping analyses, while pure Lagrange multipliers are reserved for cases where exact constraint satisfaction or direct pressure output is essential.