2. User help and applications#
2.1. Tips#
In the following, we indicate some best practices that facilitate the achievement of a satisfactory energy balance.
Use an adapted formalism
All the calculations presented are part of a Lagrangian configuration. In fact, a term of average effort is used on the time step, obtained by summing a written force at the previous step with a written force at the current step. This is not a problem for operator DYNA_VIBRA. But this can lead to an incorrect energy balance for STAT_NON_LINE and DYNA_NON_LINE, when the formalism used defined under the keyword factor COMPORTEMENT is:
PETIT_REAC
SIMO_MIEHE
In these two cases, it is advisable to take small steps of time to limit the difference between the calculation configurations from one time step to the next.
Starting from a balanced state
A first precaution is to start from an initial balanced state. Balance is ensured by calculating the external force \({F}_{\mathit{ext}}\) that balances the initial state (see § 1.4.1). In the majority of cases, this corresponds to reality. However, it is strongly recommended to start from an already balanced state when possible.
Use adapted convergence criteria
Another important point to note is the influence of the convergence criterion on the quality of the energy balance. The convergence of the calculation is ensured within a tolerance, through the value of RESI_GLOB_RELA or RESI_GLOB_MAXI. When this criterion is too loose, numerical dissipation can become significant. In case of doubt, it is advisable to restart the calculation with a more stringent criterion, in order to verify that the energy dissipated by the integration decreases.
Prefer command AFFE_CHAR_MECAà over command AFFE_CHAR_CINE
The AFFE_CHAR_CINE command is special. It consists, during the resolution, in removing the degrees of freedom where a displacement is imposed. This feature is interesting because it makes it possible to reduce the size of the system to be solved. However, it can pose a problem in evaluating the energy balance under certain conditions, which are certainly quite rare. An example is given in part 2.1.2.1.
2.2. Some applications#
In this part, we choose a few test cases to decipher the energy balance. The calculations are carried out with version 11.1.12 of*Code_Aster*.
2.2.1. SSNP155A Test Case#
This test models the stamping of a sheet by a punch in a matrix (figure). The structure is modelled in plane deformations. Symmetry is taken into account to represent only one half. The punch and the matrix are modelled as edge elements, on which a movement is imposed using the AFFE_CHAR_CINE command. At the end of the calculation, we then obtain an energy balance that is obviously incorrect, since \({W}_{\mathit{ext}}\) is zero and since \({W}_{\mathit{sch}}\) should be very low compared to the other energy terms (table).
hallmark Figure 2.1.2.1-1: Geometry of the ssnop155a test case
matrix
\({W}_{\mathit{ext}}\) |
|
|
|
|
|
0 |
6,8808E+01 |
3,7883E-05 |
1,4873E-01 |
-6,8957E+01 |
-6,8957E+01 |
Table 2.1.2.1-1: Energy balance of the ssnop155a test case at the end of the calculation
The explanation is as follows: using the AFFE_CHAR_CINE command removes all the degrees of freedom of the punch from the resolution. To calculate the contribution to external work of the movements imposed via the AFFE_CHAR_CINE command, we build the vector of the movements restricted to the nodes on which a displacement is imposed, and we use the equation () with the force the vector of internal forces as force. In this test case, the external work is only that due to the imposed displacement of the punch. But since it is only modeled with edge elements, which have no stiffness, their contribution to the internal force vector is zero. We are therefore unable to recover the force that corresponds to the imposed displacement, and we obtain zero external work.
To obtain a correct balance, simply use the AFFE_CHAR_MECA command to impose the displacement. With this, in order to guarantee compliance with the imposed travel conditions, Lagrange multipliers are constructed corresponding in fact to the force to be taken into account in the calculation of external work. Another solution would have been to mesh the punch in order to be able to calculate the internal forces. In both cases, the energy balance indicated in the table below is then obtained:
\({W}_{\mathit{ext}}\) |
|
|
|
|
|
6,8979E+01 |
6,8808E+01 |
6,8808E+01 |
3,7883E-05 |
1,4873E-01 |
2,2804E-02 |
Table 2.1.2.1-2: Energy balance using AFFE_CHAR_MECA
The balance sheet has improved significantly. Nonetheless, the term \({W}_{\mathit{sch}}\) is still significant. It’s actually due to the use of a dispative HHT schema. When using a non-dissipative Newmark diagram, the term \({W}_{\mathit{sch}}\) becomes negligible compared to other energies (table):
\({W}_{\mathit{ext}}\) |
|
|
|
|
|
6,8947E+01 |
6,8872E+01 |
1,6589E-02 |
5,7774E-02 |
-1,4257E-08 |
-1,4257E-08 |
Table 2.1.2.1-3: Energy balance using AFFE_CHAR_MECA and a non-dissipative diagram
2.2.2. SDLV120A Test Case#
This test case, illustrated in the figure, models an infinite elastic bar in which a compression wave is created by imposing a displacement at one of its ends. The other end of the bar is provided with elastic paraxial elements of order 0 intended to apply absorbent conditions at the border of the mesh, in order to account for the infinite medium.
Figure 2.1.2.2-1: Geometry of the sdlv120a test case
The following evolution is then obtained represented in the figure for the various energies involved:
Figure 2.1.2.2-2: Energy evolution in the sdlv120a test case
It is observed that the energy supplied by the outside propagates in the bar in the form of total deformation energy and kinetic energy before being rapidly dissipated thanks to the absorbent boundary elements.
2.2.3. CAS test wtnv109a#
This test case models the effect of mechanics and hydraulics on thermal energy. An element is stretched by forcing it to move in the \(z\) direction, while applying constant hydraulic pressure to it, which leads to a decrease in its temperature. We chose it to illustrate the influence of the convergence criterion on the balance of the energy balance.
In the table, the energy balance is presented with the default value of criterion RESI_GLOB_RELA, namely \(1.E\mathrm{-}6\), then for a value of \(1.E\mathrm{-}12\).
RESI_GLOB_RELA |
|
|
|
|
1.E-6 |
1.1296E+02 |
1.1303E+02 |
-7.0833E-02 |
-7.0833E-02 |
1.E-12 |
1.1303E+02 |
1.1303E+02 |
1.1303E+02 |
-2.3590E-12 |
Table 2.1.2.3-1: Energy balance of the wtnv109a test case
It can be seen that the balance gap, symbolized by the value of \({W}_{\mathit{sch}}\), has been reduced by a factor of 10.