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Lee, H., 2021. Propagation of solitary wave around conical island in level-set finite element framework. In: Lee, J.L.; Suh, K.-S.; Lee, B.; Shin, S., and Lee, J. (eds.), Crisis and Integrated Management for Coastal and Marine Safety. Journal of Coastal Research, Special Issue No. 114, pp. 16–20. Coconut Creek (Florida), ISSN 0749-0208. Though the solitary wave is a single wave, it consists of a complex spectrum of frequencies, which enables in-depth analysis and reliable generation both in the laboratory and in the numerical model. For these reasons, solitary waves are known to be a good candidate for the description of waves including tsunamis since they effectively model the important effects of the long wave on the coasts very well. In addition, it is supposed to propagate over constant depth without appreciable changes, allowing for consistent referencing of its offshore or incident wave height. In this study, the level-set scheme, which is combined with the incompressible Navier-Stokes solver based on the fractional step algorithm in the framework of the finite element method, was applied to the modeling of wave propagation and runup on a circular conical island. Unstructured hexahedral meshes were generated for this purpose and MPI (Message Passing Interface) based parallel algorithms were developed to accelerate the computation. The physical behavior of waves are discussed in detail and the numerical results (e.g., runup heights) are compared with the experimental data. The good agreements were observed.

Lee, H. and Lee, N., 2018. Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 106–110. Coconut Creek (Florida), ISSN 0749-0208.The characteristic Galerkin finite element model for the discrete Boltzmann equation is developed for the depth-integrated shallow water equations. The lattice Boltzmann method (LBM) simulates microscopic collision and streaming of fluid particles and, then, evaluates the macroscopic velocity, pressure, etc. from the computed particle distribution. Due to the inherent geometric flexibility of the unstructured mesh in finite element method, the mesh clustering near large gradients and the numerical treatments of complex geometries has become possible. Some benchmark problems of practical importance (subcritical flow over hump, flow around cylinder, curved channel, etc.) are tested and the numerical results are in good agreement with previous experimental or analytic solutions.

Recent development of computing power and theoretical advances in computational fluid dynamics have made possible numerical simulations of water waves with full three-dimensional Navier-Stokes equations. In this study, an internal wave maker using the mass source function approach was combined with the level set finite element method for generation and propagation of water waves. The model is first applied to the two-dimensional linear wave generation and propagation. Then, it is applied to the three-dimensional simulation of the wave generation and the problem of wave force evaluation on the vertical wall. To effectively utilize computational resources and enhance the speed of execution, parallel algorithms are developed and applied for the three-dimensional problems. The results of numerical simulations are compared with theoretical values and good agreements are observed.

Numerical solutions of 1D and 2D shallow water equations are presented with Runge-Kutta Discontinuous Galerkin (RKDG) finite element method. For 1D problems, the transcritical flows such as dam-break flows and a flow over a bump with hydraulic jump were simulated and the numerical solutions were compared with the exact solutions. As a formulation of approximate Riemann solver, the Local Lax-Wendroff (LLF) fluxes were employed and minmod slope limiter was used for 1D flows. For 2D applications, the classical problems of lateral transition were simulated. The HLL (Harten — Lax — van Leer) fluxes were adopted and a van Albada type gradient-reconstruction type slope limiter was applied. For the time integration, 3rd-order and 2nd-order TVD Runge-Kutta schemes were used for 1D and 2D simulations, respectively. In all case studies, good agreement was observed.

With the increase of the frequency in large-scale floods and natural disasters, the demands for highly accurate numerical river models are also rapidly growing. Generally, flows in rivers are modeled with previously developed and well-established numerical models based on shallow water equations. However, the so-far-developed models reveal a lot of limitations in the analysis of discontinuous flow or flow which needs accurate modeling. In this study, the numerical shallow water model based on the discontinuous Galerkin method was applied to the simulation of one-dimensional transcritical flow, including dam break flows and a flow over a hump. The favorable agreement was observed between numerical solutions and analytical solutions.

Recent development of computing power and theoretical advances in computational fluid dynamics have made possible numerical simulations of water waves with full Navier-Stokes equations. In this study, an internal wave maker using the mass source function approach was combined with the level set finite element method for generation of waves. The model is first applied to the two-dimensional linear wave generation and propagation. Then, it is applied to the three-dimensional simulation of the same problem. To effectively utilize computational resources and enhance the speed of execution, parallel algorithms are developed and applied for the three-dimensional problem. The results of numerical simulations are compared with theoretical values and good agreements are observed.

We present a numerical method for free surface flows that couples the incompressible Navier-Stokes equations with the level set method in the finite element framework. The implicit characteristic-Galerkin approximation together with the fractional four-step algorithm is employed to discretize the governing equations. The schemes for solving the level set evolution and reinitialization equations are verified with several benchmark cases, including stationary circle, rotation of a slotted disk and stretching of a circular fluid element. The results are compared with those calculated from the level set finite volume method of Yue et al. , which employed the third-order essentially non-oscillatory (ENO) schemes for advection of the level set function in a generalized curvilinear coordinate system