2. Transient nonlinear dynamic calculation methods#

The general advice for carrying out a nonlinear dynamic transient calculation is not detailed in this document. However, it is important to emphasize the specificity of such calculations. The use of the dynamic operator DYNA_NON_LINE ([U4.53.01], [U2.06.13], and [R5.05.05]) requires additional precautions compared to a static nonlinear calculation STAT_NON_LINE ([U4.51.03], [U2.04.01], and [R5.03.01]).

In particular, it is important to note that, as the temporal evolution to be represented is large in size, we have a problem that is often time-consuming CPU. We are therefore required to set up models that are not too bulky.

Moreover, some options available in nonlinear statics are not legal in dynamics. This is the case in particular with load control, which helps the convergence of a static calculation. In dynamics, the loading history being « real », you can no longer use this type of method as it is.

2.1. Operator DYNA_NON_LINE#

The operator DYNA_NON_LINE ([U4.53.01], [U2.06.13] and [R5.05.05]) makes it possible to calculate the dynamic evolution of a structure whose material and/or geometry has a non-linear behavior.

We solve the equations of dynamics with:

  • discretization by finite elements on the mesh on a « physical » basis,

  • implicit or explicit time integration methods for solving the temporal problem,

  • Newton-Raphson integration methods for solving the nonlinear incremental problem associated with mechanical equilibrium. This method consists in iterating over the resolution with a tangent operator.

In dynamics, unlike statics, there can be no resolution operator that is strictly non-invertible. This is due to the presence of the mass matrix in this operator. However, this does not in any way guarantee the convergence of the calculation. The most common problems involve resolving behavior.

In paragraph § 7.2, we give advice on using the DYNA_NON_LINE operator to improve the convergence of algorithms.

2.2. Choice of the temporal integration scheme#

The user can currently choose between four time patterns in DYNA_NON_LINE:

Implicit schemas:

  • an implicit non-dissipative diagram: Newmark diagram of mean acceleration (or trapezium rule),

  • an implicit dissipative diagram: diagram HHT (which introduces high frequency digital dissipation),

Explicit diagrams:

  • an explicit non-dissipative diagram: diagram of centered differences,

  • an explicit dissipative scheme: the Tchamwa-Wielgosz scheme (which introduces high frequency digital dissipation).

We will refer to [14], [U2.06.13] and [R5.05.05] to know in detail the rules for using these different methods of temporal integration.

In the context of building structure studies under seismic loading, it is recommended to use the classical implicit scheme of average acceleration, which does not provide numerical dissipation.

We will see in paragraph 7.2 that, in the event of numerical instability of the calculations (high frequency oscillations), it may be necessary to use a HHT type scheme.

In addition, in case of severe non-convergence, it is possible to try to continue the calculation using an explicit diagram. However, the DYNA_NON_LINE operator is oriented towards implicit approaches. It is not optimized (vectorized) for explicit resolutions. Therefore, it is currently recommended to use an explicit schema with the utmost caution.

2.3. Choice of the calculation time step#

2.3.1. Implicit schemas#

The time step to choose must meet a number of conditions [R5.05.05], [U2.06.13]:

  • the time step must be small enough to correctly represent the time sampling of the load;

  • it is recommended to choose for reasons of precision (« Shannon » type criterion on the cutoff frequency), a time step such as: \(\Delta t\mathrm{\le }\frac{1}{(10\mathrm{\ast }{f}_{\mathit{max}})}\), with \({f}_{\mathrm{max}}\) the highest frequency to be represented;

  • in addition, it may be interesting to determine an approximation of condition CFL (Courant-Friedrichs-Lewy, [5]) of the problem in order to have a lower bound of the time step to use. The stability time step for condition CFL is given by: \({t}_{c}=L/c\) with with \(L\) the characteristic length of the smallest element of the mesh and \(c\) the speed of the one-dimensional elastic compression waves given by \(c=\sqrt{\frac{E}{\rho }}\). The use of a much shorter time step (more than one order of magnitude) than condition CFL does not make physical sense and can be a source of high-frequency digital oscillations. In particular, when redividing the time step, care will be taken to ensure that the time step used remains close to condition CFL (it should be noted that the stability time step is generally very low).

In practice, it is necessary in a non-linear regime to ensure the low sensitivity of the response obtained for calculations with different time steps.

When we have a converged result for a time step \(\Delta {t}_{1}\), we will ensure the stability of this result by taking a time step less than or equal to \(\mathrm{0,1}\mathrm{\times }\Delta {t}_{1}\) . If the answer is identical, the time step \(\Delta {t}_{1}\) is satisfactory.

2.3.2. Explicit diagrams#

These patterns, in contrast to the implicit ones, are conditionally stable. It is imperative to use a time step less than the stability time step (condition CFL). Otherwise, the calculation may diverge (for example, abnormally high accelerations are observed). The stability time step is determined at the start of operator DYNA_NON_LINE. The option “STOP_CFL “= OUI ensures that the condition CFL is never exceeded.

Note

The calculation of condition CFL is not programmed for all the elements (in particular the discrete elements are ignored); the condition CFL estimated by Code_Aster may therefore be larger (less penalizing) than the real condition CFL, with the risks of sudden discrepancy that result.