A new second-order numerical manifold method model with an efficient scheme for analyzing free surface flow with inner drains

• Development of two forms of second-order approximation for flow analysis in NMM.
• An efficient scheme to locate and iterate the free surface in second-order NMM with fixed mesh.
• Enhanced precision with linearly distributed flux for inner drains with high gradient.
• Verified high accuracy, high convergence speed with fixed, coarse mesh.

Numerical manifold method (NMM) is a numerical method known for analyzing continuous and discontinuous mechanical processes in a unified mathematical form. In this study we developed a new second-order NMM model to solve the nonlinear problem of water flow with the free surface priori unknown and the difficulty of modeling drains which could dramatically increase the meshing load. Our study consist of: (1) deriving two forms of NMM second-order approximation; (2) constructing the total potential energy for water flow by our energy-work seepage model considering Dirichlet, Neumann and material boundaries uniformly; (3) locating free surface nodes in two forms of second-order approximation; (4) tracking the free surface with an efficient iteration scheme without re-meshing; (5) deriving velocity and tunnel flux by second-order approximation. We developed a new code and demonstrate our model and code with examples including confined drainage tunnel and free surface flow through a dam. We compare the results such as tunnel flux or free surface with linear NMM, analytical or other available numerical solutions. We prove that: the two forms of second-order NMM (1) yield consistent results; (2) for modeling drains involving local intensive change, could achieve accurate result of tunnel flux calculation and dramatically save computation load with linear velocity distribution in coarse mesh; (3) for free surface iteration, are efficient with fast convergence to accurate results and with rather coarse mesh. As a result, our second-order NMM model is applicable to free surface flow with inner drains for free surface locating and flux calculation, and seepage stability analysis, laying a solid foundation for extending to coupled hydro-mechanical analysis.