一种模拟节点达西渗透流速的三次样条多尺度有限单元法

Cubic-spline multiscale finite element method for simulation of nodal Darcy velocities in aquifers

提出了一种三次样条多尺度有限单元法(MSFEM-C)模拟非均质介质中的地下水流运动。该方法将三次样条法和多尺度有限单元法(MSFEM)有机结合,能够高效、精确地求解水头和达西渗透流速。MSFEM-C应用三次样条函数逼近多尺度基函数,保证了基函数的一阶导数的连续性,从而得到连续的水头一阶导数。因此,MSFEM-C通过达西定律得到的渗透流速在节点上是连续的。MSFEM-C是基于MSFEM的,它可以在局部网格单元上求解达西渗透流速,而无需在整个研究区上求解,从而可以节省很大计算量。因此,MSFEM-C在求解大尺度、长时间、非线性等高计算量问题时十分高效。通过对二维稳定流以及非线性潜水流的模拟,发现MSFEM-C在计算水头和达西渗透流速时的具有很高的效率和精度。

A cubic-spline multiscale finite element method （MSFEM-C） is proposed for the simulation of nodal Darcy velocities in the heterogeneous media. The main goal of this method is to efficiently solve the hydraulic heads and nodal Darcy velocities. It is realized by the combination of cubic-spline technique and the multiscale finite element method （MSFEM）. MSFEM-C applies cubic-spline technique to multiscale base functions so as to make their derivatives continuous. Therefore, the continuous derivatives of hydraulic head can be obtained, which ensures the continuity of the velocity field. The MSFEM-C is based on MSFEM, so that the computation of nodal Darcy velocities is decomposed from element to element. Therefore, MSFEM-C can save much computational cost, which makes it more efficinent in solving high computational problems, such as large-scale, long-term or nonlinear problems. The numerical experiments indicate that the MSFEM-C achieves accurate nodal Darcy velocities and hydraulic heads with much less computational cost.