7. Nonlinear transient dynamic calculation resolution

7.1. Introduction

In this chapter, we use the framework of a calculation in small disturbances, without shock. The only non-linearities are of material origin. This corresponds to the most frequent studies in building structure. However, geometric nonlinearities (large displacements, large deformations) or frictional contact could be modelled.

All the keywords mentioned in this chapter are attached to the DYNA_NON_LINE operator. Refer to [U4.53.01] and [U4.51.03] for more details.

7.2. Support for the convergence of computation

7.2.1. Temporal evolution of loading

The DEFI_LIST_INST command makes it possible to automatically redistribute the time step when Newton’s algorithm does not converge.

Moreover, it should be noted that, for a given calculation, it is necessary to test several time steps in order to analyze the stability of the results as a function of the chosen time step (cf. § 2.3).

7.2.2. Convergence criteria

The keyword CONVERGENCE allows you to define the value of the relative residue on the balance (RESI_GLOB_RELA). By default, this one is set to \({10}^{\mathrm{-}6}\). In order to facilitate the convergence of the calculation, this convergence criterion can be relaxed. However, this must be done with care, as the use of an excessively large convergence criterion may lead to results that are far from the real solution. We recommend that you do not use a RESI_GLOB_RELA that is greater than \({10}^{\mathrm{-}4}\).

In addition, there is an absolute residue on the balance (RESI_GLOB_MAXI). Initially, this criterion is used when the load and the support reactions become zero (for example in the case of a total discharge). In this case, the relative criterion is automatically changed to the absolute criterion.

When using a damaging behavior model, in some cases a discrepancy in the absolute residue is observed while the relative residue remains low. Therefore, it is advisable during calculations to check that RESI_GLOB_MAXI remains low. If this one reaches high values, it is recommended to resume the calculation by imposing a maximum value on the overall residue by defining a RESI_GLOB_MAXI for the calculation. It is recommended to use the default value of \({10}^{\mathrm{-}6}\).

Under ITER_GLOB_MAXI, you can change the maximum number of iterations performed to solve the global problem at each moment (10 by default). If we see that during Newton’s iterations, the convergence of the behavior model is slow, we can increase the value of this parameter.

Under ITER_GLOB_ELAS, you can change the maximum number of iterations performed with the discharge matrix when using the PAS_MINI_ELAS keyword of the factor keyword NEWTON (see § 7.2.3) to solve the global problem at each moment (25 by default).

7.2.3. Newton algorithm

The NEWTONpermet keyword is to specify the matrix used for the global iterations of Newton’s method (Figure 7.2.3-a). You can use either the elastic matrix (MATRICE =” ELASTIQUE “) or the tangent matrix (MATRICE =” TANGENTE”). Moreover, in the latter case, it is possible to switch automatically from the tangent matrix to the discharge matrix when the time step is or becomes (through redistribution) less than a minimum step (PAS_MINI_ELAS). For damage models, the discharge matrix is identified with the secant matrix (Figure 7.2.3-a). This option can be useful when automatically redividing the time step is not enough to converge a calculation. For damage models (concrete), the tangent matrix may become singular and it may be preferable to use the discharge matrix to converge.

Figure 7.2.3-a : schematic description of the Newton method resolution operators in Code_Aster (for a damage behavior model) .

Initially, it is recommended to use the tangent matrix. To optimize convergence, it is strongly recommended to update this tangent matrix as often as possible, the best strategy being, if the size of the problem allows it, to update it at all iterations (MATRICE =” TANGENTE “, =””, REAC_ITER = 1, [U4.51.03]). This is all the more advisable if you have a model using the De Borst method (see 4.2).

If convergence problems occur (singular tangent matrix), one can choose to switch to the discharge matrix (activation of the keyword PAS_MINI_ELAS). As the convergence with the discharge matrix is slower than that with the tangent matrix, the keyword ITER_GLOB_ELAS (cf. § 7.2.2) makes it possible to define a maximum number of iterations specific to the use of the discharge matrix. For damaging behavior models, the discharge matrix (which corresponds to the secant matrix) depends on the deformation state reached. It is therefore necessary to update this matrix using the REAC_ITER_ELAS keyword.

In the case where convergence problems remain, it may be interesting to perform a calculation with the elastic matrix to see how Newton’s iterations take place in this case.

Note

Linear research theoretically makes it possible to improve the convergence of Newton’s method [R5.03.01]. However, it is not operational with the DYNA_NON_LINE operator.

7.2.4. Material nonlinearities

The COMPORTEMENT keyword is used to define the nonlinear behavior models implemented in DYNA_NON_LINE. In case of non-convergence of the calculation, it may be necessary to modify the parameters associated with the De Borst algorithm (cf. § 4.2, for a multi-layer shell calculation).

7.2.5. Temporal integration diagram

In DYNA_NON_LINE, we have various implicit (NEWMARK and HHT) and explicit (DIFF_CENT and TCHAMWA) time schemes described briefly in § 2.2.

In seismic studies, it is recommended to use the implicit mean acceleration method because it does not introduce artificial numerical damping: SCHEMA_TEMPS =_F (SCHEMA =” NEWMARK “, FORMULATION =” DEPLACEMENT”, =” “, ALPHA =0.25, DELTA =0.5,).

If high frequency parasitic oscillations are observed in the response of the structure, it is recommended to use a diagram HHT which makes it possible to introduce significant digital damping for high frequencies without almost impacting low frequencies: SCHEMA_TEMPS =_F (SCHEMA =” HHT “(=””, FORMULATION =” “, =” DEPLACEMENT”, ALPHA = alph, MODI_EQUI =” OUI “,). By default, we recommend using ALPHA = -0.3. It should be noted that digital amortization is proportional to the time step, cf. [R5.05.05]; the implementation of a time step division therefore reduces digital amortization.

In case of persistent non-convergence and if all the other methods for aiding convergence have been tested, we can try to continue the calculation using the explicit diagram of centered differences: SCHEMA_TEMPS =_F (SCHEMA =” DIFF_CENT “, =””, FORMULATION =” “,). ACCELERATION

However, it is recommended to opt for this explicit pattern with the greatest caution. Indeed, such a calculation has important specificities, in particular with regard to the time step of the calculation, which must respect the condition CFL [5] of time is generally of the order of 10-15 to 10-6 seconds.

This calculation strategy will be more relevant when Code_Aster has a method for switching from implicit to explicit time schema (and vice versa). This method will eventually make it possible to change the integration diagram in time during the resolution of a non-linear transient dynamic problem. When the structure is in a relatively regular phase of evolution, an implicit scheme is used, which makes it possible to have relatively large time steps. Then, during phases very disturbed by concrete damage, we switch to an explicit diagram that could make it possible to overcome the difficulties (snap-back). We can then switch back to an implicit pattern if the rest of the response becomes more regular.

7.3. Synchronization of calculations

From the static solution under its own weight, seismic calculations are carried out one after the other. The final state of the previous calculation \((n)\) serves as the initial state for the following calculation \((n+1)\):

ETAT_INIT =_F (CRITERE = “RELATIF”, EVOL_NOLI = gravity load,).

Before continuing with calculation \(n+1\), it is necessary to verify that the initial state provided by the calculation n corresponds to the structure at rest: the acceleration field must be very weak throughout the structure. When the seismic load applied ends with a weak phase allowing the structure to return to rest (non-truncated signal), the calculations can be carried out directly.

In the opposite case, in order to make the acceleration field zero throughout the structure at the end of the calculation n (elimination of the parasitic acceleration field), it is recommended to bring the structure back to the state of complete rest by applying high fictional damping to the structure over a short time interval (a few tenths of a second, for example). For this, it is necessary to create fictional materials that have a high elementary depreciation. After the seismic calculation, another dynamic calculation (DYNA_NON_LINE) is carried out with static loading only and using materials with high damping. At the end of this calculation the acceleration field is necessarily almost zero and it can be used as the initial state of the following calculation \((n+1)\).