To find the solution at the end of a given increment a set of nonlinear equations has to be solved. In order to do so, the Newton-Raphson method is used, i.e. the set of equations is locally linearized and solved. If the solution does not satisfy the original nonlinear equations, the latter are again linearized at the new solution. This procedure is repeated until the solution satisfies the original nonlinear equations within a certain margin. Suppose iteration has been performed and convergence is to be checked. Let us introduce the following quantities:
where represents all elements, all nodes belonging to a given element, all degrees of freedom for field belonging to a given node and is the flux for a given degree of freedom of field in a given node belonging to a given element at the end of iteration . Right now, there are two kind of fluxes in CalculiX: the force for mechanical calculations and the concentrated heat flux for thermal calculations.
where denotes the change due to iteration .
where denotes the change due to the present increment. In mechanical calculations the solution is the displacement, in thermal calculations it is the temperature.
Now, two constants and are introduced: is used to check convergence of the flux, serves to check convergence of the solution. Their values depend on whether zero flux conditions prevail or not. Zero flux is defined by
The following rules apply:
The values in square brackets are the default values. They can be changed by using the keyword card *CONTROLS. Now, convergence is obtained if
AND if, for thermal or thermomechanical calculations (*HEAT TRANSFER, *COUPLED TEMPERATURE-DISPLACEMENT or *UNCOUPLED TEMPERATURE-DISPLACEMENT), the temperature change does not exceed DELTMX,
AND at least one of the following conditions is satisfied:
If convergence is reached, and the size of the increments is not fixed by the user (no parameter DIRECT on the *STATIC, *DYNAMIC or *HEAT TRANSFER card) the size of the next increment is changed under certain circumstances:
If no convergence is reached in iteration , the following actions are taken:
from which can be determined. Now, if
(which means that the estimated number of iterations needed to reach convergence exceeds ) OR , the increment size is adapted according to and the iteration of the increment is restarted unless the parameter DIRECT was selected. In the latter case the increment is not restarted and the iterations continue.