水－土連成FEM解析手法における動水勾配の空間離散化

u-p定式化における有限体積法に基づく連続式の離散化形式の精度向上

宇野 浩樹^＊１・池上 浩樹^＊１

飽和地盤の挙動を表現するための支配方程式は，飽和土を土骨格と間隙水の二相系からなると仮定したBiotの飽和多孔質体理論（例えば，Biot, 1962）によって導かれる。支配方程式に対するいくつかの定式化のうち，本研究では，土骨格変位uと間隙水圧pを未知数とし，飽和土のつり合い式と連続式を保存式とするu-p定式化に着目する。連続式を有限体積法で離散化する場合，従来，要素境界面上の動水勾配は隣接する要素間の水頭差を要素重心間の距離で除して表現するが，メッシュ分割方法によっては解析精度が低下するという課題がある。本研究では，境界面上の動水勾配を有限要素法で離散化した間隙水のつり合い式に基づく節点での値で表現し，Mandel-Cryer効果を伴う2次元圧密のシミュレーション解析に適用した。その結果，過剰間隙水圧などの数値解は，メッシュ分割方法に依存せず，理論解を精度良く再現することが確認された。

キーワード：有効応力解析，u-p定式化，連続式，有限体積法，動水勾配，圧密

＊１ 技術センター 社会基盤技術研究部 地盤研究室

Spatial Discretization of Hydraulic Gradient for Soil-Water Coupled FEM Analysis Method

Improvement in Accuracy of Discretized Form of Continuity Equation Based on Finite Volume Method for the u-p Formulation

Hiroki UNO^*1 and Hiroki IKEGAMI^*1

The governing equation for the behavior of saturated ground is derived from the theory of saturated porous media (e.g. Biot, 1962). Among several formulations for the governing equation, this study focuses on the u-p formulation, in which the soil skeleton displacement and pore water pressure are unknowns and the equilibrium equation for saturated soil and the continuity equation are conservation equations. When discretizing the continuity equation with the finite volume method, hydraulic gradients on element boundary surfaces are conventionally evaluated by dividing the difference in hydraulic head between adjacent elements by the distance between the element centers of gravity. However, previous studies have pointed out that the accuracy in the numerical analyses might be reduced, dependent on mesh partitioning methods. In this study, the hydraulic gradients on the boundary surfaces were expressed by response values at nodes obtained from the equilibrium equation for pore water discretized by the finite element method, and its validity was examined through numerical simulations on a two-dimensional consolidation problem accompanied by the Mandel-Cryer effect. As a result, it was confirmed that the numerical solutions, such as excess pore water pressure, accurately reproduced the theoretical solutions without depending on mesh partitioning methods.

Keywords: effective stress analysis, u-p formulation, continuity equation, finite volume method, hydraulic gradient, consolidation

*1 Geotechnical Research Section, Infrastructure Technology Research Department, Taisei Advanced Center of Technology

34 - TAISEI ADVANCED CENTER OF TECHNOLOGY TECHNICAL REPORT No.58 2025