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Evaluation of u-w and u-\(\pi\) finite element methods for the dynamic response of saturated porous media using one-dimensional models. (English) Zbl 0597.73108

The class of problems considered here lies between the ’undrained’ and ’drained’ extremes where dynamic loading is applied and transient pore fluid motion is significant.
Various finite element methods have been described in the literature, yet no comparison of numerical and analytical results has been presented for these dynamic problems since no analytical solution was available. An earlier paper by B. R. Simon and the last two authors describes an exact, one-dimensional solution for this class of problems [ibid. 8, 381- 398 (1984; Zbl 0539.73128)].
In the present paper, we present some of the available finite element procedures and carry out initial accuracy studies for each procedure by comparing corresponding numerical and exact solutions. We begin with a brief summary of the general theoretical formulation of the initial boundary value problem and then describe several finite element approaches for spatial discretization. These approaches are applied to the one-dimensional problem, and analytical and finite element solutions are compared. Various time integration schemes are then implemented, and errors associated with the combined spatial and temporal discretization are assessed. Only linear problems are considered; however, the results should provide insight into the application of finite element procedures to nonlinear problems. All time integration schemes are chosen to be stable, so only accuracy analyses are carried out here.

MSC:

74L10 Soil and rock mechanics
74S99 Numerical and other methods in solid mechanics
76S05 Flows in porous media; filtration; seepage
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