A particle finite element method applied to long wave run-up

Abstract

SUMMARYThis paper presents a Lagrangian-Eulerian ÿnite element formulation for solving uid dynamics problems with moving boundaries and employs the method to long wave run-up. The method is based on a set of Lagrangian particles which serve as moving nodes for the ÿnite element mesh. Nodes at the moving shoreline are identiÿed by the alpha shape concept which utilizes the distance from neighbouring nodes in di erent directions. An e cient triangulation technique is then used for the mesh generation at each time step.In order to validate the numerical method the code has been compared with analytical solutions and a preexisting ÿnite di erence model.The main focus of our investigation is to assess the numerical method through simulations of threedimensional dam break and long wave run-up on curved beaches. Particularly the method is put to test for cases where di erent shoreline segments connect and produce a computational domain surrounding dry regions.

Some improvements on free surface simulation by the particle finite element method

“…We found that the mass increased about 0.2% when the circular drop impacted the water. Afterwards, the mass became small, which was very similar to the results in [21] and the error was about −2.5% at time 4 s. Figure 11 illustrates the error of the mass when the mesh-push method was used. First, the order of magnitude was about 10 −5 most of the time but there were some special points with large negative errors.…”

“…As illustrated in Figure 5(a), assume that all the angles after deformation are larger than half of their initial values, respectively. Take 1 as an example; its original value is 1 = atan2(|r 21 ×r 31 |, r 21 ·r 31 ), and the angle after deformation is set to 1 = 1 2 . An equation like Equation (19) can be derived and solved to get a suitable time step.…”

“…The PFEM does not conserve the volume exactly [21]. After updating the nodes, minute errors in the volume of the fluid may occur.…”

“…The PFEM does not conserve the volume exactly [21]. The error of volume can be understood according to two aspects.…”

The natural volume method (NVM): Presentation and application to shallow water inviscid flows

“…The rapid distortion of used mesh, and thus the frequent re-meshing, represents the biggest challenge to overcome for scientists. The paper by Birknes and Pedersen [33] could be taken as a typical work. It introduces the use of the 'particle FEM (PFEM)' to the SWE to simulate the long wave run-up.…”