Simplified Adaptive Triangulation of the Contact Boundaries of the Dam Model

Abstract

To numerically solve the system of integral equations, it is customary to establish a discrete grid within each integration area. In the context of 3D modeling, these areas correspond to surfaces situated in space. The standard discretization technique employed for the computational domain is triangulation. This study addresses the integral equation system pertinent to the electrical tomography of dams. The structural model encompasses an embankment dam, the upstream and downstream water bodies, the dam's base, and a potential leakage region on the upstream side. An alternative configuration may be encountered in specific scenarios with no water downstream. Consequently, the model may incorporate up to nine distinct contact boundaries. Accordingly, the system of integral equations comprises an equivalent number of equations. Effectively resolving this system through numerical methods necessitates applying triangulation techniques to these diverse surfaces. While mathematical packages like Matlab offer triangulation functions, they may not fully address the specific demands of the problem. Additionally, the grid resolution should be heightened in proximity to key elements such as the sounding line, the supply electrode, and the various contact lines within the medium. These considerations transform the triangulation task into a distinct subtask within the numerical simulation of the resistivity tomography problem. In this paper, we provide our specific approach to this problem. The simplification of the triangulation algorithm is rooted in the predominant utilization of the two-dimensional geometric properties inherent to the object under study. For most contact boundaries, the triangulation is constructed layer by layer with a gradual modulation in triangle dimensions as one progresses from one layer to the next, orthogonal to the axis of the dam. Concerning the surface corresponding to the leakage area, cylindrical coordinates are used for surface parameterization. This approach enables partitioning the surface into discrete strata, facilitating a systematic, layer-by-layer grid construction. Additionally, points at the intersections of contact boundaries are integrated into the pre-existing triangulation by applying a standard function within the Matlab package. In the future, the mathematical modeling based on the Integral Equation Method with adaptive discretization will help incorporate real-time computations into information systems related to monitoring hydraulic structures.