Nonlinearity
When linear FEM is not enough — geometric, material, and contact nonlinearities explained.
Linear static analysis assumes that the structure behaves proportionally: double the load, double the stress, double the deformation. This holds for the vast majority of engineering problems. Nonlinear analysis is needed when that assumption breaks down.
The three sources of nonlinearity
1. Geometric nonlinearity (large deformations)
Linear analysis assumes the geometry doesn't change during loading. If the structure deforms significantly — more than roughly 5–10% of its smallest dimension — this assumption fails. The stiffness matrix needs to be updated at each load step.
Typical cases: thin flexible structures, snap-through buckling, rubber-like materials, slender beams under large bending.
2. Material nonlinearity (plasticity)
Linear analysis assumes the material stays elastic — stress and strain remain proportional (Hooke's law). Once the material yields, the relationship becomes nonlinear. Permanent deformation remains after unloading.
Typical cases: overload scenarios, crash analysis, forming simulations, safety factor calculations beyond yield.
3. Contact nonlinearity
Contact between parts changes during loading — surfaces that were touching may separate, surfaces that were apart may come into contact. The solver doesn't know in advance which nodes are in contact, so it must iterate.
Contact is covered in more detail in the Contact Mechanics article.
When to use nonlinear analysis
Nonlinear analysis is significantly more expensive than linear. Solve times are 10–100× longer and convergence is not guaranteed. Only use it when there is a specific, identified reason.
| Situation | Use nonlinear? | | --- | --- | | Small deformations, elastic material | No — linear is correct | | Thin flexible part, large deflection | Yes — geometric nonlinearity | | Stress exceeds yield strength | Yes — material nonlinearity | | Parts touching and separating | Yes — contact nonlinearity | | Rubber or elastomer | Yes — material and geometric |
Load stepping
Nonlinear solvers apply the load in steps, iterating to equilibrium at each step. The number of steps and the convergence tolerance affect both accuracy and solve time. Too few steps can cause the solver to diverge; too many add unnecessary computation.
Dr.Q sets reasonable defaults. For difficult problems, reduce the load step size and tighten the convergence tolerance.
Always start with linear
Even for problems that require nonlinear analysis, always run a linear analysis first:
- Verify the model setup (boundary conditions, loads, units)
- Check that reaction forces balance applied loads
- Identify which regions exceed yield — this tells you where nonlinearity matters
- Use the linear result as a sanity check for the nonlinear one
A nonlinear result that differs wildly from the linear result is either physically correct (large plastic zone, buckling) or a modelling error. Start from the linear case to tell the difference.