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### Nonlinear calculations

For nonlinear calculations the solution is found by iteration. Because a step is possibly too large to obtain convergence, the option exists to subdivide the step into a finite number of increments. The size of the initial increment in a step is defined by the user (line beneath *STATIC, *DYNAMIC, *VISCO, *HEAT TRANSFER or *COUPLED TEMPERATURE-DISPLACEMENTS) and also the number of increments can be controlled by the user (parameter DIRECT). However, in most cases it is advisable to let the program determine the size of the increments, based on the convergence rate of the previous increments. The solution in each increment is obtained by iteration until the residual forces are small enough.

Therefore, the structure of nonlingeo corresponds to the flow diagram in Figure 126. It lists all subroutines, each line is a subroutine. On the upper right ``preliminary'' is an abbreviation for five subroutines which recur often. If a subroutine or a group of subroutines is enclose by square brackets, it means that it is only run under certain conditions. In detail, the structure of nonlingeo looks like:

- before the first increment
- determine the number of advective degrees of freedom and the number of radiation degrees of freedom (envtemp.f)
- expanding the radiation degrees of freedom in case of cyclic symmetry (radcyc.c)
- initialization of contact fields and triangulation of the independent contact surfaces (inicont.c)
- take into account time point amplitudes, if any (checktime.f).
- calculate the initial acceleration and the mass matrix (specific heat
matrix for transient heat transfer calculations) for dynamic calculations.
(initialaccel.c). This includes:
- determine the load at the start of the increment (tempload.f)
- for thermal analyses: determine the sink temperature for forced convection and cavity radiation boundary conditions (radflowload.f)
- update the location of contact and redefine the nonlinear contact spring elements (contact.f)
- update the coefficients of nonlinear MPC's, if any.
- if the topology of the MPC's changed (dependence of nonlinear MPC's on other linear or nonlinear ones) or contact is involved: call remastruct
- determine the internal forces (results.f).
- construction of the stiffness and mass matrix and determination of the external forces (mafillsm.f); This is also done for explicit calculations in order to get the mass matrix.
- subtract the internal from the external forces to obtain the residual forces;
- solving the system of equations with in spooles.c, preiter.c or any other available sparse matrix solver. For explicit dynamic calculations explicit calculation of the solution (no system needs to be solved). The solution is the acceleration at the start of the step.

- for each increment
- before the first iteration
- determine the load at the end of the increment (tempload.f)
- for thermal analyses: determine the sink temperature for forced convection and cavity radiation boundary conditions (radflowload.f)
- update the location of contact and redefine the nonlinear contact spring elements (contact.f)
- update the coefficients of nonlinear MPC's, if any.
- if the topology of the MPC's changed (dependence of nonlinear MPC's on other linear or nonlinear ones) or contact is involved: call remastruct.
- prediction of the kinematic vectors
- determination of the internal forces (results.f). The difference between the internal and the external forces are the residual forces. If the residual forces are small enough, the solution is found. If they are not, iteration goes on until convergence is reached. The residual forces are the driving forces for the next iteration.

- in each iteration
- determine the load at the end of the increment (tempload.f)
- for thermal analyses: determine the sink temperature for forced convection and cavity radiation boundary conditions (radflowload.f)
- update the location of contact and redefine the nonlinear contact spring elements (contact.f)
- update the coefficients of nonlinear MPC's, if any.
- if the topology of the MPC's changed (dependence of nonlinear MPC's on other linear or nonlinear ones) or contact is involved: call remastruct and redetermine the internal forces (results.f).
- construct the system of equations and determination of the external forces (mafillsm.f); for explicit dynamic calculations no system has to be set up, only the external forces are determined (rhs.f).
- subtract the internal from the external forces to obtain the residual forces (calcresidual.c);
- solving the system of equations with in spooles.c, preiter.c or any other available sparse matrix solver. For explicit dynamic calculations explicit calculation of the solution (no system needs to be solved).
- calculating the internal forces and material stiffness matrix in each integration point in results.f
- deriving the new residual by subtracting the updated internal forces from the external forces (calcresidual.c).
- If the residual is small enough iteration ends (checkconvergence.c). The convergence criteria are closely related to those used in ABAQUS.

- after the final iteration, if output was not suppressed by user input control:
- determining the required results for all degrees of freedom, starting from the displacement solution for the active degrees of freedom. This is done in subroutine results.f, including any storage in the .dat file.
- storing the results in the .frd file. For structures not exhibiting cyclic symmetry this is performed in routine out.f, for cyclic symmetric structures routine frdcyc.c is called before calling out. If an error occurred during the matrix fill the output is reduced to the pure geometry.

- before the first iteration
- after the final increment (only if no output resulted in this final
increment due to user input control)
- determining the required results for all degrees of freedom, starting from the displacement solution for the active degrees of freedom. This is done in subroutine results.f, including any storage in the .dat file.
- storing the results in the .frd file. For structures not exhibiting cyclic symmetry this is performed in routine out.f, for cyclic symmetric structures routine frdcyc.c is called before calling out. If an error occurred during the matrix fill the output is reduced to the pure geometry.

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**Contents**guido dhondt 2012-10-06