One-dimensional compression-wave propagation
This benchmark studies compression-wave propagation in a water-saturated soil column subjected to a Heaviside surface load. It compares the available two-phase element formulations in numgeo and assesses the importance of relative fluid acceleration in dynamic analyses.
The column is 10 m high and consists of 100 elements. Its in-plane width is 0.1 m; the three-dimensional models additionally have a thickness of 0.1 m. The soil skeleton is linear elastic. A compressive step load of 10 kPa is applied at the top, generating a wave that travels down the column and is reflected at the base.
The principal material and analysis parameters in the supplied input files are:
| Parameter | Value |
|---|---|
| Young's modulus \(E\) | 10,000 kPa |
| Poisson's ratio \(\nu\) | 0.3 |
| Initial void ratio \(e_0\) | 1.0 |
| Solid density \(\rho^s\) | 2.7 t/m\(^3\) |
| Water density \(\rho^w\) | 1.0 t/m\(^3\) |
| Water bulk modulus \(K^w\) | \(1.1\times10^6\) kPa |
| Intrinsic permeability \(\bar{K}\) | \(1\times10^{-10}\) m\(^2\) |
| Dynamic water viscosity \(\mu^w\) | \(1\times10^{-6}\) kPa·s |
| HHT parameter \(\alpha\) | \(-0.1\) |
| Dynamic step duration | 0.05 s |
With the adopted unit weight of water, the intrinsic permeability corresponds to a saturated hydraulic conductivity of approximately \(10^{-3}\) m/s.
Figure 1: Finite-element model, boundary conditions, and Heaviside loading of the saturated soil column.
Element formulations
The archive contains two- and three-dimensional \(u\)–\(p\), \(u\)–\(U\), and \(u\)–\(p\)–\(U\) models with linear or quadratic interpolation and full or reduced integration. It also contains selected explicit-dynamic variants. The element-label convention is described on the two-phase benchmark overview.
Input files
The input files, analytical reference solution, and evaluation script can be downloaded here.
Simulation results
The analytical solution developed by Staubach and Machaček1 is used as the reference. Figure 2 compares the excess pore-water pressure at the bottom of the column and the vertical displacement at the top.
The dominant wave frequency is approximately 30 Hz. At the comparatively high hydraulic conductivity of about \(10^{-3}\) m/s, the \(u\)–\(p\) formulation increasingly deviates from the analytical solution as the wave propagates because it does not retain relative fluid acceleration as an independent kinematic contribution. The \(u\)–\(U\) and \(u\)–\(p\)–\(U\) formulations include this effect and remain in close agreement with the analytical solution throughout the analysed interval.
The linearly interpolated \(u\)–\(U\) elements also reproduce the reference response well. The reduced-integration u4u4-red element performs slightly better than its fully integrated u4u4 counterpart in this problem because reduced integration alleviates volumetric locking. Reduced-integration elements still require the stabilisation measures stated in the corresponding element reference.
Figure 2: Bottom excess pore-water pressure and top displacement for the investigated two-phase element formulations.
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P. Staubach and J. Machaček, “Influence of relative acceleration in saturated sand—Analytical approach and simulation of vibratory pile driving tests,” Computers and Geotechnics, 112, 173–184, 2019. https://doi.org/10.1016/j.compgeo.2019.03.027 ↩