A CBS-type stabilizing algorithm for the consolidation of saturated porous media,International Journal for Numerical Methods in Engineering

当前位置： X-MOL 学术 › Int. J. Numer. Meth. Eng. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

A CBS-type stabilizing algorithm for the consolidation of saturated porous media
International Journal for Numerical Methods in Engineering ( IF 2.7 ) Pub Date : 2021-06-17 , DOI: 10.1002/nme.6708
Bernhard Aribo Schrefler ₁

Affiliation

While deriving the theoretical part of paper [1] again for chapter 5 in the book “Computational Geomechanics: Theory and Applications”, Second Edition, with authors A. Chan, M. Pastor, B. Schrefler, T. Shiomi, O. C. Zienkiewicz, ISBN9781118350478, in print at Wiley, the senior author discovered errors in some equations of this paper. The errors occurred during the transcription of the mass balance equation of the fluid phase from the second edition of the textbook [2]. The procedure has not been implemented as indicated in the original Equation (24) of [1] but the component matrices have been introduced in the code PLASCON listed in [2]. Hence the errors in the procedure do not affect the results published in [1] also because the effect of the difference in the matrix (21c) is minimal as shown by the validation example in [1].¹

The correct equations are:

m^{T} L \frac{u^{n + 1} - u^{n}}{Δ t} = - ϑ_{3} {({Lm}^{T} v)}^{n + 1} + (1 - ϑ_{3}) {({Lm}^{T} v)}^{n}

(14)

m^{T} L \frac{u^{n + 1} - u^{n}}{Δ t} = - m^{T} L {\tilde{v}}^{n + 1} + Δ t m^{T} L \frac{n}{ρ^{w}} {[\tilde{H} (\nabla p)]}^{n + 1}

(14b)

\{\begin{array}{l} - L^{T} {Du}^{n + 1} + L^{T} m p^{n + 1} = ρ g^{n + 1} \\ m^{T} L \frac{u^{n + 1} - u^{n}}{Δ t} = - m^{T} L {\tilde{v}}^{n + 1} + Δ t m^{T} L \frac{n}{ρ^{w}} {[\tilde{H} (\nabla p)]}^{n + 1} \end{array}\}

(18)

\{\begin{cases} - K {\overline{u}}^{n + 1} + L_{u} {\overline{p}}^{n + 1} = F^{n + 1} \\ L_{u}^{T} {\overline{u}}^{n + 1} - {(Δ t)}^{2} C_{p} {\overline{p}}^{n + 1} = L_{u}^{T} {\overline{u}}^{n} - Δ t P_{v} {\bar{\tilde{v}}}^{n + 1} \end{cases}

(21)

[\begin{array}{c} - K & L_{u} \\ L_{u}^{T} & - {(Δ t)}^{2} C_{p} \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n + 1} = [\begin{array}{c} 0 & 0 \\ L_{u}^{T} & 0 \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n} - Δ t [\begin{array}{c} 0 & 0 \\ P_{v} & 0 \end{array}] {[\begin{array}{c} \bar{\tilde{v}} \\ 0 \end{array}]}^{n + 1} + {[\begin{array}{c} F \\ 0 \end{array}]}^{n + 1}

(21′)

C_{p} = \frac{n}{ρ^{w}} \tilde{H} \int_{Ω} {(\frac{\partial N^{p}}{\partial x_{i}})}^{T} \frac{\partial N^{p}}{\partial x_{i}} d Ω

(21c)

[\begin{array}{c} - K & L_{u} \\ L_{u}^{T} & - {(Δ t)}^{2} C_{p} \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n + 1} = [\begin{array}{c} 0 & 0 \\ L_{u}^{T} & 0 \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n} - Δ t [\begin{array}{c} 0 & 0 \\ P_{v} & 0 \end{array}] {[\begin{array}{c} \bar{\tilde{v}} \\ 0 \end{array}]}^{n + 1} + {[\begin{array}{c} F_{E} \\ 0 \end{array}]}^{n + 1}

(24)

The same errors appear also in the mass balance equations of the fluid phase in a further paper [3] on the topic and the necessary changes are as listed below:

Equation (10) should read like Equation (14b) above.

Equation (11a) should read like the first one of Equation (18) above.

Equation (13a) should read like Equation (21) above.

Equation (13d) should read like Equation (21c) above.

Equation (14a) should read like Equation (21′) above.

Equation (29) becomes

[\begin{array}{c} - K & L_{u} \\ L_{u}^{T} & - {(Δ t)}^{2} C_{p} \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n + 1} = [\begin{array}{c} 0 & 0 \\ L_{u}^{T} & 0 \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n} - Δ t [\begin{array}{c} 0 & 0 \\ P_{v} & 0 \end{array}] {[\begin{array}{c} \bar{\tilde{v}} \\ 0 \end{array}]}^{n + 1} + {[\begin{array}{c} F_{E} \\ 0 \end{array}]}^{n + 1} + {[\begin{array}{c} Δ C \\ 0 \end{array}]}^{n + 1}

and Equation (47) should read

[\begin{array}{c} - K_{1}^{n + 1} + K_{2}^{n + 1} & L_{u} \\ L_{u}^{T} & - {(Δ t)}^{2} C_{p} \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n + 1} - {[\begin{array}{c} F_{ML} \\ 0 \end{array}]}^{n + 1} = [\begin{array}{c} 0 & 0 \\ L_{u}^{T} & 0 \end{array}] {[\begin{array}{c} \overline{u} \\ \overline{p} \end{array}]}^{n} - {[\begin{array}{c} F_{ML} \\ 0 \end{array}]}^{n} - Δ t [\begin{array}{c} 0 & 0 \\ P_{v} & 0 \end{array}] {[\begin{array}{c} \bar{\tilde{v}} \\ 0 \end{array}]}^{n + 1} + {[\begin{array}{c} F_{E} \\ 0 \end{array}]}^{n + 1}

The same remarks as above for the implementation in the code PLASCON apply.

中文翻译：

一种用于饱和多孔介质固结的CBS型稳定算法

在“计算地质力学：理论与应用”一书的第 5 章中再次推导论文 [ 1 ]的理论部分，第二版，作者 A. Chan、M. Pastor、B. Schrefler、T. Shiomi、OC Zienkiewicz， ISBN9781118350478，在Wiley印刷，资深作者发现了这篇论文的一些方程中的错误。该错误发生在第二版教科书[ 2 ]中流体相质量平衡方程的转录过程中。该过程尚未按照 [ 1 ]的原始方程 (24) 中所示实施，但已在 [ 2] 中列出的代码 PLASCON 中引入了分量矩阵]。因此，程序中的错误不会影响 [ 1 ] 中公布的结果，也是因为矩阵 (21c) 中差异的影响最小，如 [ 1 ] 中的验证示例所示。¹

正确的方程是：