Thermomechanical iterations

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 $ i$ has been performed and convergence is to be checked. Let us introduce the following quantities:

Now, two constants $ c_1$ and $ c_2$ are introduced: $ c_1$ is used to check convergence of the flux, $ c_2$ serves to check convergence of the solution. Their values depend on whether zero flux conditions prevail or not. Zero flux is defined by

$\displaystyle \bar{q}^{\alpha}_i \le \epsilon^\alpha \tilde{q}^{\alpha}_i.$ (596)

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

$\displaystyle r^{\alpha}_{i,max} \le c_1 \tilde{q}^{\alpha}_i$ (597)

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 $ i$, the following actions are taken: