Computational Engineering / module 11
Nonlinear and explicit analysis
What changes when superposition fails: equilibrium paths, limit points, Newton–Raphson, arc-length, dynamic relaxation and the explicit time step.
What you will understand
- Distinguish material, geometric and directional nonlinearity, and recognise each in a model
- Explain Newton–Raphson through the residual rather than through the formula
- Say why the modified method is sometimes the better choice despite needing more iterations
- Read an equilibrium path and identify limit points, snap-through and bifurcation
- Explain why load control cannot pass a limit point and what can
- State what dynamic relaxation computes and what its intermediate states mean
- Compute an explicit stable time step and say what sets it
Lessons
Three kinds of nonlinearity
Material, geometric and directional — how to recognise each, and why the solver menu is not what decides.
Newton–Raphson and the residual
Guess, measure the out-of-balance, divide by the tangent, correct, repeat — and what each of those four words is doing.
Equilibrium paths and limit points
Where load control runs out, what displacement and arc-length control can see that it cannot, and what snap-through actually is.
Module summary
- A model can be nonlinear even when a linear solver was selected — the elements decide, not the menu
- The residual is the load the structure is not yet carrying; convergence is a decision about tolerance
- Newton–Raphson converges quadratically near the answer and can diverge far from it
- Load control cannot follow a path where the load falls; displacement or arc-length control can
- A limit point is a property of the structure; convergence failure there is not a solver setting
- Dynamic relaxation's intermediate states are fictitious — only the converged state is real
- An explicit time step is the time a stress wave takes to cross the smallest element
- One sliver element can multiply an explicit run time by a hundred
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Check yourself
A load-controlled nonlinear analysis of a shallow dome fails to converge at 96 % of the design load. What is the most useful interpretation?
Practice
A structure has tangent stiffness kT = 80 − 40u + 8u² kN/m. At what displacement is its tangent stiffness least, in metres? Then check whether it ever reaches zero.
Practice
An explicit analysis of a 60 ms impact uses 25 mm concrete elements. Concrete has E = 30 × 10⁶ kN/m² and ρ = 2.4 t/m³. Using 90 % of the critical step, how many time steps does the run need? Give the answer to the nearest hundred.