C3. Implementation

The correct method consists of always proceeding step by step so that non-linearities are not introduced all at once:

Performing an initial calculation with an elastic constitutive law before using non-linear ones,
Performing a calculation without contact before performing one with it, a calculation with no friction before including it, …

Every step should be verified properly, and it is necessary to make sure that the solution converges when the mesh and the time-step are refined.

Finally, it should be clear that in non-linear cases, the problem to be solved may have no unique solution (instability, bifurcation, etc.).

Choosing the discretization

The discretization must be chosen focusing on the fact that it must be consistent with the loadings that one wishes to apply, especially if those are not monotonic.

The more important the non-linearities are, the smaller the load-increments must be. Also, if the behavior of the model is complex, the integral of the constitutive law is not completely implicit: in that case, it should be verified that the increments are sufficiently small for the model to converge.

According to the standards, the management of the time-steps can be automated to a certain extend. This allows us to reduce the time-steps if there is a convergence problem, or even to increase it if the non-linearity is weak and the convergence is fast. For the sake of diligence, it is advised to verify that the convergence is reached by performing a second calculation with smaller load increments.

Choosing the boundary conditions

It should be clear that in the case of softening-types constitutive laws or if a limit load exists, the loading with imposed forces might become not admissible, as shown in the figure below.

In the first case, the closer the imposed forces get to the limit load, the harder it will be for the model to converge until it becomes impossible. In the second case, it will be impossible to exceed the effort corresponding to the first peak and obtain a solution after that.

Examples of response with limit load or loss of stiffness

There are other cases, where there is no unique solution for a given effort or displacement. It happens, for example, in the case of buckling of thin shells or when the problem has bifurcations, for instance, dissipative branch and elastic branch (see figure below).

Response (a) with multiple branches (b) non-monotonic loading and displacement

If the model requires imposed forces, the problem can be solved by continuation or arc length methods (see Code_Aster [R5.03.80] documentation for instance).

For an algorithm to work effectively, it is then necessary to conceive its model and to prepare its data, so that one looks for a unique solution by introducing multiple restrictions, such as:

The need for equilibrium stability (in a specific way to be explored!)
Equilibrium with a “minimization of plastic deformation energy”,
Equilibrium with a “minimization of average curvature”,
Equilibrium with a “minimization of the difference between reaction forces”

Evaluation of the convergence of the global non-linear algorithm

The stopping criterion for the Newton algorithm must be considered carefully. It is advised to verify what limit criterion is programmed by default in the software. For example, certain algorithms test the convergence using the norm of the residual (in that case, the criterion depends on the unit used for the forces in the calculation, the precision is not the same if the required residual is 1N or 1MN). Other software use a criterion where the norm of the residual relates to the norm of the loading (in this case, the convergence does not depend on the chosen physical unit). Special precautions must be taken when different elements are mixed (solid, structural elements) or when compound elements are used.

In general, one must be careful when the convergence criterion is modified from the recommended value to reach certain results.