2. Benchmark solution#

For each modeling, this test carries out an inter-comparison between the reference solution (obtained with a 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 (\(\mathit{Pa}\) in \(\mathit{MPa}\)), and the solution obtained after rotation.

2.1. Definition of robustness test cases#

Two analysis angles are proposed to test the robustness of the integration of laws of behavior:

  • studies of equivalent problems

  • 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.

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. Equivalent problem studies#

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 2 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 \(\mathit{MPa}\) in the material data and the possible parameters of the law,

  • \(\mathrm{Trot}\), path by imposing the same \(\stackrel{ˉ}{\varepsilon }\) tensor after a rotation: \({t}^{}R\mathrm{\cdot }\stackrel{ˉ}{\varepsilon }\mathrm{\cdot }R\) where \(R\) is a rotation matrix, corresponding to a rotation of 30 degrees around the \(\mathit{Oz}\) axis.

For each of these problems, the solution (constraint invariants, scalar internal variable) 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 in the order of machine precision, i.e. around \(1.E\mathrm{-}15\).

2.3. 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, we study here two discretizations at different times, by multiplying the number of steps in the loading path by 5. This leads to the following discretization:

Calculation

\({T}_{1}\)

\({T}_{\mathrm{réf}}\) reference solution

Number of intervals per load segment

5

25

Total number of steps over the entire trip

40

200

The reference solution, \({T}_{\mathrm{réf}}\), is the one obtained for \(N=25\), i.e. 200 steps for the entire journey. These solutions make it possible to judge the sensitivity to large time steps and the robustness of the integration.

The maxima of the differences between the two solutions for the entire loading path are reported in §3.3.