2. Benchmark solution#
For each modeling, this test carries out an inter-comparison between the reference solution (obtained with a very fine time step), the solution with a moderately coarse discretization, the solution with the effect of temperature (or another control variable), the solution by changing the system of units (\(\mathrm{Pa}\) in \(\mathrm{MPa}\)), and the solution obtained after rotation or symmetry.
2.1. Definition of robustness test cases#
We propose 3 analysis angles to test the robustness of the integration of laws of behavior:
studies of equivalent problems
tangent matrix check
study of the discretization of the time step
For each of them, we study the evolution of the relative differences between several calculations using the same law but presenting different parameters or calculation options. The exploitation focuses on the invariants of the stress tensor: tensor trace, Von-Mises constraint and scalar internal variables: generally it is the cumulative plasticity.
The global convergence criteria are the values provided by default by Code_Aster. (RESI_GLOB_RELA =10-6, ITER_GLOB_MAXI =10). A usual Newton diagram was adopted for the updating of the tangent matrix:
calculation of the tangent prediction matrix at each converged increment (REAC_INC =1)
calculation of the coherent tangent matrix at each Newton’s iteration (REAC_ITER =1).
2.2. Studies of equivalent problems#
For a rough discretization of the paths: 1 time step for each segment of the journey, the solution obtained for each law is compared to 3 strictly equivalent problems for the state of the material point:
\(\mathrm{Tpa}\), same path with a change of unit, we substitute the \(\mathrm{Pa}\) for the \(\mathrm{MPa}\) in the material data and the possible parameters of the law,
Trot, path by imposing the same \(\stackrel{ˉ}{\varepsilon }\) tensor after a rotation: \(R\stackrel{ˉ}{\varepsilon }{R}^{T}\) where \(R\) is a rotation matrix. For case \(\mathrm{2D}\), the angle of rotation will be \(\alpha =0.9\) radian, for the configuration \(\mathrm{3D}\), we chose the Euler angles with the arbitrary values {\(\psi =0.9\) radian \(\theta =0.7\) radian and \(\phi =0.4\) radian},
Tsym, path by imposing the tensor \(\stackrel{ˉ}{\varepsilon }\) after a symmetry: permutation of the axes \(x\) and \(y\) in \(\mathrm{2D}\), permutation of \(x\) in \(y\), \(y\) in \(z\) and \(z\) in and in \(x\) in in 3D.
For each of these problems, the solution (stress invariants, cumulative equivalent plastic deformation) must be identical to the basic solution, obtained with the same time discretization. The reference value of the difference is therefore 0. In practice, this means that the difference found must be of the order of machine precision, i.e. approximately 1.E-15.
2.3. Tangent matrix test#
For each behavior, the tangent matrix is also tested, by difference with the matrix obtained by disturbance. Again, the reference value is 0.
2.4. Study of the discretization of the time step#
We study the behavior of law integration as a function of discretization. For the same modeling, and therefore a given behavior, several discretizations are studied here at different times, by multiplying the number of steps in the loading path by 5. In the reference [1], discretization is extended up to 3125 increments per segment on the same principle. Here, in order to limit the duration of the tests, we limit ourselves to 3 successive refinements. This leads to the following discretization:
Number of intervals per load segment |
5 |
25 |
Total number of steps over the entire trip |
40 |
200 |
Calculation |
T1 |
Tref reference solution |
The reference solution, \({T}_{\mathrm{réf}}\), is the one obtained for \(\text{N}=25\), i.e. 200 steps for the entire journey. These different solutions make it possible to judge the sensitivity to large time steps and the robustness of the integration.
To show the convergence speed as a function of the time step, we report here the solutions presented in [1], up to 3125 time steps for each of the 8 segments of the loading path.
2.4.1. Law LEMAITRE#
Gaps |
\(\text{N1}\) |
|
|
|
|
|
||
V1_N |
3.15e-02 |
3.00e-02 |
3.00e-02 |
1.35e-02 |
3.25e-03 |
5.74e-04 |
0.00e+00 |
|
VMIS |
1.64e-02 |
1.33e-02 |
1.33e-02 |
3.58e-03 |
7.95e-04 |
1.38e-04 |
0.00e+00 |
|
TRAC |
2.25e-14 |
2.22e-14 |
2.22e-14 |
2.18e-14 |
2.39e-14 |
3.36e-14 |
0.00e+00 |
|
SIXX |
4.70e-02 |
4.09e-02 |
4.09e-02 |
1.05e-02 |
2.16e-03 |
3.64e-04 |
0.00e+00 |
|
SIYY |
2.30e-01 |
1.87e-01 |
1.87e-01 |
4.64e-02 |
9.71e-03 |
1.65e-03 |
0.00e+00 |
|
SIZZ |
9.71e-02 |
7.43e-02 |
7.43e-02 |
1.78e-02 |
3.79e-03 |
6.47e-04 |
0.00e+00 |
|
SIXY |
4.70e-02 |
7.04e-02 |
7.04e-02 |
2.74e-02 |
5.40e-03 |
9.05e-04 |
0.00e+00 |
|
SIXZ |
2.45e-01 |
2.23e-01 |
2.23e-01 |
5.76e-02 |
1.19e-02 |
2.01e-03 |
0.00e+00 |
|
SIYZ |
1.92e-01 |
1.36e-01 |
1.36e-01 |
4.41e-02 |
9.03e-03 |
1.53e-03 |
0.00e+00 |
2.4.2. Law VISC_CIN1_CHAB v=10-5#
Gaps (A2) |
\(\text{N1}\) |
|
|
|
|
|
|
V1_N |
3.53e+00 |
1.14e+00 |
1.14e+00 |
2.45e-01 |
4.78e-02 |
7.98e-03 |
0.00e+00 |
VMIS |
7.83e-02 |
5.64e-02 |
5.64e-02 |
2.35e-02 |
5.52e-03 |
9.60e-04 |
0.00e+00 |
TRAC |
1.33e-14 |
1.37e-14 |
1.37e-14 |
1.33e-14 |
2.25e-14 |
2.25e-14 |
0.00e+00 |
SIXX |
1.27e-01 |
6.25e-02 |
2.89e-02 |
2.89e-02 |
6.93e-03 |
1.21e-03 |
0.00e+00 |
SIYY |
2.51e-01 |
9.65e-02 |
5.05e-02 |
1.26e-02 |
2.23e-03 |
0.00e+00 |
|
SIZZ |
2.51e-01 |
4.71e-02 |
2.04e-02 |
2.04e-02 |
5.53e-03 |
9.91e-04 |
0.00e+00 |
SIXY |
1.32e-01 |
6.54e-01 |
6.54e-01 |
2.32e-01 |
5.35e-02 |
9.32e-03 |
0.00e+00 |
SIXZ |
9.85e-02 |
7.60e-02 |
7.60e-02 |
3.21e-02 |
7.63e-03 |
1.34e-03 |
0.00e+00 |
SIYZ |
6.24e+00 |
1.62e+00 |
1.62e+00 |
9.91e-02 |
1.71e-02 |
3.05e-03 |
0.00e+00 |
2.4.3. Law VISC_CIN2_CHAB v=10-5#
Gaps (A2) |
\(\text{N1}\) |
|
|
|
|
|
|
V1_N |
3.53e+00 |
1.14e+00 |
1.14e+00 |
2.45e-01 |
4.78e-02 |
7.98e-03 |
0.00e+00 |
VMIS |
7.83e-02 |
5.64e-02 |
2.35e-02 |
2.35e-02 |
5.52e-03 |
9.60e-04 |
0.00e+00 |
TRAC |
1.33e-14 |
1.37e-14 |
1.37e-14 |
1.33e-14 |
2.25e-14 |
2.25e-14 |
0.00e+00 |
SIXX |
1.27e-01 |
6.25e-02 |
2.89e-02 |
2.89e-02 |
6.93e-03 |
1.21e-03 |
0.00e+00 |
SIYY |
2.51e-01 |
9.65e-02 |
5.05e-02 |
1.26e-02 |
2.23e-03 |
0.00e+00 |
|
SIZZ |
2.51e-01 |
4.71e-02 |
2.04e-02 |
2.04e-02 |
5.53e-03 |
9.91e-04 |
0.00e+00 |
SIXY |
1.32e-01 |
6.54e-01 |
6.54e-01 |
2.32e-01 |
5.35e-02 |
9.32e-03 |
0.00e+00 |
SIXZ |
9.85e-02 |
7.60e-02 |
7.60e-02 |
3.21e-02 |
7.63e-03 |
1.34e-03 |
0.00e+00 |
SIYZ |
6.24e+00 |
1.62e+00 |
1.62e+00 |
9.91e-02 |
1.71e-02 |
3.05e-03 |
0.00e+00 |
2.4.4. Law VISC_TAHERI v=10-5#
Gaps (A2) |
\(\text{N1}\) |
|
|
|
|
|
|
V1_N |
3.30e-02 |
4.17e-02 |
4.17e-02 |
2.04e-02 |
5.14e-03 |
9.19e-04 |
0.00e+00 |
VMIS |
8.29e-02 |
2.87e-02 |
2.87e-02 |
7.27e-03 |
1.52e-03 |
2.59e-04 |
0.00e+00 |
TRAC |
6.79e-14 |
6.78e-14 |
6.78e-14 |
6.80e-14 |
6.70e-14 |
8.73e-14 |
0.00e+00 |
SIXX |
8.71e-02 |
3.26e-02 |
3.26e-02 |
8.47e-03 |
1.78e-03 |
3.03e-04 |
0.00e+00 |
SIYY |
1.36e-01 |
5.39e-02 |
1.82e-02 |
1.82e-02 |
4.69e-03 |
8.40e-04 |
0.00e+00 |
SIZZ |
6.04e-02 |
3.26e-02 |
3.26e-02 |
1.47e-02 |
3.85e-03 |
6.92e-04 |
0.00e+00 |
SIXY |
1.68e+00 |
7.74e-01 |
7.74e-01 |
2.08e-01 |
4.43e-02 |
7.57e-03 |
0.00e+00 |
SIXZ |
5.37e-01 |
2.97e-01 |
2.97e-01 |
9.61e-02 |
2.17e-02 |
3.77e-03 |
0.00e+00 |
SIYZ |
2.43e-01 |
5.68e-01 |
5.68e-01 |
3.11e-01 |
7.77e-02 |
1.38e-02 |
0.00e+00 |
2.5. Bibliographical references#
LEVASSEUR: « Third-party Application Maintenance of the _Aster code » Verification of the robustness and reliability of the integration of behavioral laws in ASTER. Report PRINCIPIA RET .693.127.01 December 2006.