Incremental Solution Methods & Convergence

What Is It?

Incremental solution methods solve a non-linear problem by applying the load in small increments and iterating within each increment to find equilibrium. At each increment, the solver applies a portion of the load, computes the structural response, checks whether equilibrium is satisfied, and iterates if it is not. The process continues until the full load is applied. Convergence — the condition where the residual force is within tolerance — must be achieved at each increment before proceeding to the next.

Why It Matters

The incremental solution process is the engine of implicit non-linear analysis. Understanding how it works — how increments are sized, how iterations are performed, what convergence means and what happens when convergence fails — is essential for diagnosing convergence problems, choosing appropriate solver settings and interpreting results. A non-linear analysis that fails to converge produces no useful result. A non-linear analysis that converges but with inappropriate settings may converge to the wrong solution.

Convergence is a numerical requirement, not proof of physical correctness. A converged solution means the solver found equilibrium — not that the material model, contact definition or boundary conditions are correct.

The Incremental Process

The load is applied in increments from zero to the final value. Within each increment, the solver iterates to find the equilibrium state. The iteration process uses the tangent stiffness to predict a displacement correction, applies the correction, recomputes the internal forces and checks the residual. If the residual is within tolerance, the increment has converged. If not, another iteration is performed. If the iteration fails to converge after a maximum number of attempts, the increment is cut back — the load step is reduced and the solver tries again with a smaller increment.

Load: 0 → ΔF₁ → ΔF₂ → ... → F_final

Within each increment:
  Iteration 1: solve Kt·Δu = R → update u → compute F_int → check R
  Iteration 2: solve Kt·Δu = R → update u → compute F_int → check R
  ...
  Converged? R < tolerance → proceed to next increment
  Not converged after max iterations? → cutback (reduce ΔF) and retry

Load Increment

The load increment is the portion of the total load applied in one step. The size of the increment affects both convergence and computational cost. Large increments may fail to converge if the non-linearity is strong — the response changes too much in one step for the iterations to find equilibrium. Small increments converge more reliably but require more steps and more computation. Automatic stepping — where the solver adjusts the increment size based on convergence behaviour — is the standard approach. The solver increases the step if convergence is easy and decreases it if convergence is difficult.

Iteration and Residual

Within each increment, the solver iterates. At each iteration, the residual — the difference between external and internal forces — is computed. If the residual is zero (or within tolerance), equilibrium is achieved. If not, the solver uses the tangent stiffness to compute a displacement correction that reduces the residual, applies the correction and checks again. The number of iterations per increment depends on the non-linearity — typically 2–5 for mild non-linearity, more for strong non-linearity or contact changes.

Iteration process:

1. Compute residual:  R = F_ext − F_int(u)
2. If |R| < tolerance:  converged → proceed
3. Solve:  K_t · Δu  =  R
4. Update:  u  =  u + Δu
5. Recompute F_int(u)
6. Go to step 1

where:
K_t  =  tangent stiffness matrix
Δu   =  displacement correction
R    =  residual force vector

Convergence Tolerance

The convergence tolerance determines when the residual is considered small enough to accept equilibrium. The tolerance is typically expressed as a norm — a scalar measure of the residual vector. Common norms include the force norm (the magnitude of the residual force), the displacement norm (the magnitude of the displacement correction) and the energy norm (the work done by the residual through the correction). The tolerance should be tight enough to ensure adequate equilibrium but not so tight that excessive iterations are required for no engineering benefit. Typical values are 0.1–1% of the reference force or displacement.

Cutback and Automatic Stepping

When an increment fails to converge after the maximum number of iterations, the solver cuts back — reduces the increment size and retries. This is an automatic mechanism that allows the solver to navigate difficult non-linear regions. The cutback factor (typically 0.25–0.5) determines how much the increment is reduced. If the reduced increment also fails, the solver may cut back again or terminate. Automatic stepping adjusts the increment size based on convergence history — larger steps when convergence is easy, smaller steps when it is difficult. This optimises the balance between computational cost and convergence reliability.

Why Reducing Step Size Improves Convergence

A smaller load increment means the structural response changes less per step. The non-linearity is milder within a smaller step — the tangent stiffness is closer to the actual stiffness, the contact changes are less abrupt, the material state change is smaller. This makes it easier for the Newton–Raphson iterations to find equilibrium. The cost is that more increments are needed to reach the final load, increasing computation time. The trade-off between step size and computation time is managed by automatic stepping.

Common Convergence Problems

• Contact chattering — contact nodes oscillate between open and closed, preventing equilibrium
• Plasticity convergence — material yielding can cause sudden stiffness changes that destabilise iterations
• Buckling or snap-through — the structure passes through a limit point where the tangent stiffness becomes singular
• Excessive distortion — elements deform so severely that their stiffness becomes ill-conditioned
• Inappropriate tolerance — too tight causes excessive iterations; too loose accepts inaccurate equilibrium
• Too few iterations allowed — the maximum iteration count is too low for the non-linearity level

Key Takeaways

• Non-linear problems are solved in load increments with iterations within each increment
• Convergence means the residual force is within tolerance — equilibrium is achieved
• Automatic stepping adjusts increment size based on convergence behaviour
• Cutback reduces the increment when convergence fails, trading computation for reliability
• Convergence is a numerical requirement — not proof of correct physics or material model