2. Benchmark solutions

2.1. Calculation method

The multi-model method is used to perform the calculation. For each step \(n+1\) of the installation, a model containing strictly the \(n+1\) layers applied is associated. The states of stress, displacement and internal variables at the end of the previous step are transferred to the next step by field transfer operations. The initial field of displacement of the laid soil layer must vary linearly from bottom to top. It varies in fact between the value of the settlement of the lower layers and the geometric dimension to be respected, associated with an initial displacement of zero.

2.2. Reference quantities and results

Solutions are post-treated in terms of compaction. However, the use of the multi-model method does not give direct access to the settlement of the column: the vertical displacement of the last layer laid is the sum of the settlement actually suffered by it and of the settlement already carried out when it was not there: it is this last component that must be removed.

For example, consider layer 4 (see figure below). This one is asked at moment \(n=4\). The setback only makes sense for \(n\ge 4\). Let \(\delta {u}_{4}^{n}\) be the packing increment between the times \(n\) and \(n+1\) above layer 4. The compaction of layer 4 is defined at time \(n\): \(\Delta {u}_{4}^{n}\), as the accumulation of compaction increments undergone by the layer during the overall process of constructing the soil column, i.e. by the installation of successive layers located above it (\(n\ge 5\)).

So: \(\Delta {u}^{n\ge 4}=\sum _{i=4}^{n}\delta {u}_{4}^{i}\) with \(\delta {u}_{4}^{n}={u}_{4}^{n}-{u}_{4}^{n-1}\)

Thus, by decomposing the iterative process, we easily rewrite the packing as being: \(\Delta {u}_{4}^{n\ge 4}={u}_{4}^{n}-{u}_{4}^{0}\). i.e. the displacement above the 4th layer at the moment \(n\) (under the action of the layers located above it), minus its displacement during its installation (\(n=4\)). Validation is carried out by comparison with GEFDYN solutions provided by École Centrale Paris.		At moment \(n\ge 4\): \(\begin{array}{c}\Delta {u}_{4}^{n\ge 4}=\sum _{i=4}^{n}\delta {u}_{4}^{i}\\ \text{=}\sum _{i=4}^{n}{u}_{4}^{i}-{u}_{4}^{i-1}\\ \text{=}{u}_{4}^{n}-{u}_{4}^{4}={u}_{4}^{n}-{u}_{\mathrm{4,0}}\end{array}\)

For C modeling, the elementary option calculation PDIL_ELGA is also carried out in order to validate numerical developments for HM models. The values obtained are tested in non-regression.

2.3. Uncertainties about the solution

The results established during modeling with the Elements Finis GEFDyn software from Ecole Centrale Paris are accurate according to the levels of convergence criteria used in this software. The definition of convergence criteria is specified in the software user manual []. The value of the criteria relating to displacements and hydraulic pressure is equal to \({10}^{-3}\) and the value of the criteria relating to mechanical (forces) and hydraulic (flow) imbalances is equal to \({10}^{-2}\).

2.4. Bibliographical references

[1] D. Aubry, A. Modaressi. GEFDyn, Science Manual. Ecole Centrale Paris, LMSS -Mat, 1996.