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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.

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 |

### Keywords:

saturated porous media; drained extremes; undrained; dynamic loading; transient pore fluid motion; finite element procedures; initial accuracy studies; initial boundary value problem; spatial discretization; one- dimensional problem; time integration schemes; errors; temporal discretization; linear problems; accuracy analyses
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\textit{B. R. Simon} et al., Int. J. Numer. Anal. Methods Geomech. 10, 461--482 (1986; Zbl 0597.73108)

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### References:

[1] | Biot, J. Appl. Phys. 12 pp 155– (1941) |

[2] | Biot, J. Appl. Phys. 33 pp 1482– (1962) |

[3] | Prevost, Int. J. Eng. Sci. 8 pp 787– (1980) |

[4] | Ghaboussi, J. Eng. Mech. Div. ASCE EM4 pp 947– (1972) |

[5] | Ghaboussi, J. Soil Mech. Found. Div. ASCE SM10 pp 849– (1973) |

[6] | Ghaboussi, J. Geotech. Div. ASCE GT3 pp 341– (1978) |

[7] | Prevost, Comp. Meth. Appl. Mech. Eng. (1982) |

[8] | and , ’A unified approach to soil mechanics problems’, in Finite Elements in Geomechanics, Wiley, New York, 1977, pp. 151-178. |

[9] | ’Nonlinear problems of soil statics and dynamics’, Proc. Europe-U. S. Symp. on Nonlinear Finite Element Analysis in Structural Mechanics, Ruhr Universitat, Bochum. Springer-Verlag, 1980. |

[10] | Zienkiewicz, Int. J. Numer. Anal. Methods Geomech. 8 pp 71– (1984) |

[11] | Simon, Int. J. Numer. Anal. Methods Geomech. 8 pp 381– (1984) |

[12] | The Finite Element Method, McGraw-Hill, London, 1977. |

[13] | Simon, Int. J. Numer. Anal. Methods Geomech. 10 pp 483– (1986) |

[14] | Sandhu, Int. J. Eng. Sci. 8 pp 989– (1970) |

[15] | and , Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1976. |

[16] | Belytschko, Comp. Meth. Appl. Mechs. Eng. 17/18 pp 259– (1979) |

[17] | Fellippa, Comp. Meth. Mechs. Eng. 24 pp 61– (1980) |

[18] | Park, Int. J. Numer. Methods Eng. 19 pp 1669– (1983) |

[19] | Hughes, Comp. Meth. Appl. Mech. Eng. 17/18 pp 159– (1979) |

[20] | and , Finite Elements in Plasticity–Theory and Practice, Pineridge Press, Swansea, U. K. 1980. · Zbl 0482.73051 |

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