6. Explicit diagrams

Code_Aster allows the use of two different explicit schemes: that of centered differences and that of Tchamwa-Wielgosz [bib34], which is dissipative. These patterns, unlike the implicit ones, are conditionally stable: there is a time limit that cannot be exceeded under penalty of discrepancy.

These diagrams are very well suited to fast dynamics problems, where one seeks to analyze wave propagation phenomena. Their cost becomes prohibitive, at least on a physical basis, when you have to do numerical simulations over a long period of time, as is the case when you want to obtain a vibratory response or when the imposed load is long-lasting: from several seconds, as, for example, in an earthquake.

However, it should be noted that, compared to a resolution with an implicit schema, explicit approaches cost much less over a period of time. Indeed, while implicitly we must really solve the assembled global system of large dimensions at each step (or even at each non-linear iteration), by explicit we avoid this very expensive inversion by using a diagonal operator (based on the only mass matrix, based on the mass matrix alone, therefore, which does not require updating: with the keyword MASS_DIAG = “OUI”). With an explicit schema, you can only solve by acceleration (keyword FORMULATION = “ACCELERATION”).

6.1. The schema of centered differences

This pattern (keyword SCHEMA_TEMPS, SCHEMA =” DIFF_CENT “), the most common in fast dynamic codes, is a particular case of the Newmark family. It is built by taking \(\beta =0\) and \(\gamma =1/2\). This time integration method, which does not introduce a digital dissipation, is second order.

Its stability time step (condition CFL: Friedrichs and Levy current [bib25]) is \(2/\omega\), with \(\omega\) which is the highest natural pulsation of the structure under consideration.

Remarks:

On a physical basis, it can be shown that the critical time step is equivalent to the minimum time it takes for a wave to pass through the smallest finite element of the mesh. In practice, this value is approximated by calculating the minimum, for all finite elements, of: \(\Delta t\mathrm{=}D\mathrm{/}C\) with d which is the diameter of the largest inscribed sphere and \(c\) which is the speed of the elastic compression waves in the element in question.

This method of calculating condition CFL, which is directly applicable to massive elements, needs to be corrected for structural elements.

For conventional industrial applications, condition CFL requires very small time steps to be taken (generally of the order of \({10}^{-5}s\). for metal structures).

On the other hand, on a modal basis, due to the strong truncation generally done for industrial cases, the critical time step, directly given by \(2/\omega\) with \(\omega\) which is the cut-off pulse, will be much greater than on a physical basis.

Explicit resolution is always accelerated (the operator to be inverted is then reduced to the mass matrix alone, which is recommended to be « lumped », and therefore made diagonal).

6.2. The Chamwa-Wielgosz diagram [bib34]

This recent schema can be seen as the explicit counterpart to schema HHT. The objective is the same: to introduce high-frequency digital dissipation, which does not disturb the low-frequency solution « too much ». This Tchamwa-Wielgosz [bib34] scheme (keyword SCHEMA_TEMPS, SCHEMA =” TCHAMWA “) cannot be deduced from the Newmark family.

Just like HHT, this schema is parameterized by a real, named \(\phi\). If we take \(\phi\) =1, the pattern does not disappear. In practice it is recommended to choose a value of \(\phi\) between 1 and 1.1.

The critical time step is equal to that of the centered differences diagram divided by \(\phi\).

6.3. Explicit schema performance in Code_Aster

Code_Aster is not a code dedicated to fast dynamics, it is a generalist code oriented towards implicit approaches. It is therefore not optimized for explicit resolutions, in particular with regard to the calculation of internal forces and the structuring of the generated objects, such as the solution data structure.

It is therefore not recommended to use an explicit schema for large problems, especially since you want to calculate over a very large number of steps.

6.4. Explicit Ritz-based calculations with subdomains in Code_Aster

The comments in the previous paragraph should however be qualified insofar as it is possible to carry out explicit calculations based on Ritz with sub-domains (with « generalized » degrees of freedom) in Code_Aster .That is to say, we can use the keyword factor PROJ_MODAL to perform the calculation with an explicit time integration diagram on a modal (or Ritz) basis of projection previously calculated.

This option comes in the case where you want to dynamically condense a part of the model to linear behavior, by strictly calculating by DYNA_NON_LINE only sub-domains with non-linear behavior. This is in order to reduce the size of the calculation model. In this case, it is necessary to calculate a Ritz modal base on the whole model: the nonlinear subdomain modeled for the calculation using DYNA_NON_LINE and the other supposed linear subdomains condensed dynamically. This base must be orthogonalized with respect to the mass and linear stiffness matrices of the entire model. It should simply be representative of the movements activating the entire model.

The modelled calculation domain thus includes a non-linear calculation sub-domain \(I\) and a series of linear sub-domains \(E\) « external » to the calculation which will be dynamically condensed.

The matrices \(M\), \(C\), \(K\), the forces \(F\) can be broken down into a part associated with the field \(I\) and a part associated with the sub-domain (s) \(E\). The balance of the structure solved by DYNA_NON_LINE can then be written in terms of degrees of freedom of global finite elements \(U\):

\(({M}_{I}+{M}_{E})\mathrm{.}\ddot{U}={F}_{I}^{\text{ext}}+{F}_{E}^{\text{ext}}–{F}_{I}^{\text{int}}(U,\dot{U})–{K}_{E}U-({C}_{I}+{C}_{E})\mathrm{.}\dot{U}\) eq 8.4-1

Let \(\{\Phi \}\) be a Ritz modal base orthogonalized by elementary substructuring across all sub-domains \(I\) and \(E\). We then use the Ritz transform: \(U=\Phi q\) and so by projecting eq 7.4-1 by \({}^{t}\Phi\) to the left we get:

\(({\tilde{M}}_{I}+{\tilde{M}}_{E})\mathrm{.}\ddot{q}={}^{t}\Phi \mathrm{.}{F}_{I}^{\text{ext}}+{\tilde{F}}_{E}^{\text{ext}}{–}^{t}\Phi \mathrm{.}({F}_{I}^{\text{int}}+{C}_{I}\mathrm{.}\dot{U})-{\tilde{K}}_{E}\mathrm{.}q-{\tilde{C}}_{E}\mathrm{.}\dot{q}\) eq 8.4-2

Modes \(\Phi\) reduced to \(I\) are rated \(\phi\) and degrees of freedom \(U\) reduced to \(I\) are rated \(u\) so eq 7.4-2 becomes for each time step:

\(\tilde{M}\mathrm{.}\ddot{q}={\phi }^{t}\mathrm{.}{F}_{I}^{\text{ext}}–{\phi }^{t}\mathrm{.}({F}_{I}^{\text{int}}(u,\dot{u})+{C}_{I}\mathrm{.}\dot{u})+{\tilde{F}}_{E}^{\text{ext}}-{\tilde{K}}_{E}\mathrm{.}q-{\tilde{C}}_{\mathrm{E.}}\dot{q}\) eq 8.4-3

We therefore need generalized matrices of mass \(\tilde{M}=({\tilde{M}}_{I}+{\tilde{M}}_{E})\) diagonal as well as the matrices of the subdomains \(E\): \({\tilde{K}}_{E}\) and \({\tilde{C}}_{E}\) full as well as the generalized vector \({\tilde{F}}_{E}^{\text{ext}}\) which participate in terms complementary to the standard dynamic problem reduced to the standard dynamic problem reduced to the sub-domain \(I\), which would be solved without dynamic condensation in the absence of sub-domains. \(E\) by DYNA_NON_LINE:

\(\tilde{M}\mathrm{.}\ddot{q}={\phi }^{t}\mathrm{.}{F}_{I}^{\text{ext}}–{\phi }^{t}\mathrm{.}({F}_{I}^{\text{int}}(u,\dot{u})+{C}_{I}\mathrm{.}\dot{u})\)

At each time step, we evaluate and add the complementary terms resulting from dynamic condensation, cf. eq 7.4-3, to those of the standard dynamic problem without dynamic condensation and we solve with the explicit time integration scheme, cf. [§7.1] and [§7.2], see [] and [], in generalized coordinates \(q\), having expressed with the explicit diagram the terms in \(\mathrm{u},\dot{\mathrm{u}}\) on the sub-domain \(I\).

Knowledge of the orthogonalized Ritz modal base \(\{\phi \}\) then makes it possible to obtain both the degrees of freedom \(u\) on the resolution sub-domain \(I\) and \(U\) on the whole model consisting of its sub-domains.

We will only enter behind the keyword MODE_MECA the \(\{\phi \}\) modes obtained by reducing the Ritz base to the nonlinear domain of calculation \(I\) treated by DYNA_NON_LINE, behind the keyword MASSE_GENE the generalized mass matrix \(\tilde{M}\), as well as behind the key words RIGI_GENE AMOR_GENE the generalized operators of the sub- domains \(E\): \({\tilde{K}}_{E}\), \({\tilde{C}}_{E}\) and finally with the keyword factor EXCIT_GENE \({\tilde{F}}_{E}\). An example of calculation is provided by test case SDNV107A [V5.03.107].

The main nuance compared to the restriction of the use of an explicit schema for large problems consists both in the possibility of reducing the size of the calculation model and in the use of a CFL condition on the time step linked to the maximum frequency of the projection base, which makes it possible to greatly reduce the number of calculation time steps.