岩土体渗流自由面问题的重心插值无网格方法

Barycentric interpolation meshless method for solving rock and soil seepage free surface problems

岩土体的渗透破坏、地下工程的防渗设计等无不与渗流计算有关。针对渗流自由面问题,提出一种重心拉格朗日插值的配点型无网格方法。由于渗流自由面问题的求解区域是不规则区域,该方法通过将不规则求解区域嵌入一个正则矩形区域,在正则区域上采用重心拉格朗日插值近似未知函数,利用配点法离散渗流问题的控制方程,将重心拉格朗日插值的微分矩阵离散成代数方程表达的矩阵形式。将自由面上的边界条件通过重心拉格朗日插值离散,通过置换方程法和附加方程法施加边界条件,利用正则区域上的重心插值配点法,通过迭代确定最终自由面的位置。数值算例表明所提出的无网格方法对于求解渗流自由面问题的正确性和高精度。

The seepage failure of rock and soil and the seepage design for underground engineering are closely related to the seepage calculation.A meshless collocation method based on barycentric Lagrange interpolation for solving seepage free surface problems in 2D is presented.Embedded the irregular seepage domain into a regular rectangular domain,the seepage governing equation is extended into the regular rectangular domain.The unknown function of seepage in seepage domain is approximated by barycentric Lagrange interpolation in regular rectangular domain.Then,the barycentric Lagrange interpolation collocation method（BLICM） can be used to solve seepage problems in the regular rectangular domain.The governing equation of seepage problem is discretized by using BLICM into algebraic equations.Using notations of differentiation matrix and Kronecker product of matrices,the system of algebraic equations can be rewritten as matrix form.The boundary conditions are discretized by barycentric Lagrange interpolation and applied by using replacement equation method or/and additional equation method.The final location of free surface is obtained by iteration computation of BLICM in regular rectangular domain.The numerical results compared with others numerical methods indicate the correctness and high accuracy of the proposed method.