A differential quadrature algorithm based on staggered grids for coupled analysis of saturated soils

Abstract

Biot’s theory of consolidation is always analyzed numerically for the complexity of the actual problems. The differential quadrature method (DQ) is a high-order numerical algorithm which is popular for its easy implementation and high accuracy and has already been applied successfully in geotechnical engineering. However, spurious pressure oscillations will be observed when strong pressure gradients appear if it is employed directly for coupled consolidation analysis of saturated soils. In the present study, a staggered differential quadrature (SGDQ) algorithm is developed and a non-uniform staggered grid of Chebyshev-Gauss-Lobatto points is proposed to enhance the numerical stability of DQ. Different numbers of grid points are employed to discretize the displacement and the pore pressure. The equations of equilibrium are approximated at the displacement points and the condition of continuity is established at the pressure points. The derivatives of the pore pressure at the displacement points and the derivatives of the displacement at the pressure points are dealt with through a pre-process of polynomial interpolation. Detailed derivations of the formulations are given and one- and two-dimensional numerical tests are provided. It can be seen that non-physical pressure oscillations observed in the differential quadrature method are removed by the present formulation and therefore the stability and numerical accuracy is greatly improved.

The equilibrium equation for a general elastoplastic soil can be given as

$$ \operatorname{div}\boldsymbol{\upsigma} =\mathbf{0} $$

(A.1)

where σ is the total stress. An incremental strategy is usually adopted for elastoplastic analysis and the equilibrium equation can be written as:

$$ \operatorname{div}{\boldsymbol{\upsigma}}^{n+1}=\operatorname{div}\left({\boldsymbol{\upsigma}}^n+\Delta {\boldsymbol{\upsigma}}^n\right)=\mathbf{0} $$

(A.2)

where the superscripts denote the time increments. Introduce the principle of effective stress:

$$ \Delta {\boldsymbol{\upsigma}}^n=\Delta {\boldsymbol{\upsigma}}^{\hbox{'}n}+\Delta {p}^n\boldsymbol{\updelta} $$

(A.3)

where σ^'n is the effective stress and δ is the identity vector and Eq. (A.2) turns into:

$$ \operatorname{div}\left(\Delta {\boldsymbol{\upsigma}}^{\hbox{'}n}\right)+\nabla \left(\Delta {p}^n\right)=-\operatorname{div}{\boldsymbol{\upsigma}}^n=\mathbf{0} $$

(A.4)

The incremental effective stress can be related to the incremental strain Δεⁿ by a elastoplastic constitutive matrix D^ep:

$$ \Delta {\boldsymbol{\upsigma}}^{\hbox{'}n}={\mathbf{D}}^{ep}\Delta {\boldsymbol{\upvarepsilon}}^n $$

(A.5)

and the incremental strain can be expressed in terms of the incremental displacement Δuⁿ as

$$ \Delta {\boldsymbol{\upvarepsilon}}^n=\mathbf{L}\left(\Delta {\mathbf{u}}^n\right) $$

(A.6)

where L is a differential operator. The combination of Eqs. (A.4)–(A.6) gives

$$ \operatorname{div}\left({\mathbf{D}}^{ep}\mathbf{L}\left(\Delta {\mathbf{u}}^n\right)\right)+\nabla \left(\Delta {p}^n\right)=\mathbf{0} $$

(A.7)

Eqs. (2) and (A.7) are the control equations for elastoplastic consolidation analysis which can be approximated by the differential quadrature analog.