A Modified Algorithm to Generate Flow Nets from the Nodal Potential and Stream Values of Eight-Node Quadrilateral Elements

Eight-node quadrilateral isoparametric elements of the serendipity type have frequently been used in finite-element analyses of two-dimensional seepage problems. The shape functions for these elements are quadratic. Hence, nonlinear variation in the potential and stream function values across each element could be approximated to a high degree of accuracy. This also necessitates a commensurate high-order interpolation function to locate, in a straightforward way, equipotential lines and streamlines. In this paper, a quadratic interpolation algorithm for locating deformation contours is modified to suit flow net generation. The modification lies in the procedure for identifying the pairs of the points of intersection to be joined when there are four, six, or eight points of intersection of the contour segments of the same level and the edges of an element. The original algorithm finds the pairs of intersection points in a local coordinate system by testing all possible cases that may be encountered. The modified algorithm considers that in most, if not all, scenarios, equipotential lines and streamlines extend monotonically from one impervious boundary of the flow domain to another and from an inflow boundary to an outflow boundary, respectively. The intersection points are rapidly paired by converting their local coordinates to global coordinates and sorting the order of the intersection points according to their global coordinates. The modified algorithm eliminates the need for an exhaustive search and complex matching process, enhancing computational efficiency. The modified algorithm is verified against an exact analytical solution to the flow net for a levee under-seepage flow. Excellent agreement is obtained. Two additional illustrative examples are analyzed. One is unconfined seepage through a rectangular dam, and the other is confined seepage beneath unsymmetrical cofferdams. The equipotential lines and streamlines obtained from the modified algorithm are shown to be smoother and more accurate than those obtained using popular commercial software (GeoStudio 24.2.0), especially when a coarse finite-element mesh is adopted.