The purpose of this project is to make an accurate, robust, geometric flexible and efficient model for calculation of forces on structures from nonlinear ocean waves and breaking wave impacts. Accurate prediction of the extreme forces on wind turbine foundations, breakwaters and tidal or wave power devises are important for enhancement structural designs. The proposed model is based on an incompressible and inviscid flow approximation and the governing equations are applied in an arbitrary Lagrangian-Eulerian moving frame of reference (ALE). The Runge-Kutta method (RK) is used for time integration and mass conservation is satisfied through a pressure-corrector type calculation of the pressure. The weighted least squares method (WLS) is combined with approximate Riemann solvers to introduce numerical smoothing of the solution around steep gradients in the velocity and pressure fields. The Poisson equation is solved and the pressure boundary conditions are satisfied by a generalized finite pointset method (GFPM); This provides a geometrically flexible and stable solution for the fluid pressure. The numerical approximations of these equations are performed on unstructured point distributions and the solutions for velocity and pressure are represented by WLS approximation of multivariate polynomials. The stencils for the ALE-WLS and GFPM methods are found through a breadth first search (BFS) in a modified Delaunay graph. This graph is the discrete representation of the fluid domain and the connectivity between the calculation points. The graph is updated according to the evolving topology of the fluid domain caused by the fluid reaching or leaving a solid boundary or the free surface colliding with itself or another free-surface. After each time step the fluid domain is checked for any of these intersections and the topology is updated accordingly in its graph representation. The calculation points move in a Lagrangian way and this can cause ill-conditioning of the generalized Vandermonde matrix in the WLS and GFPM methods. To prevent this the point set is refined and coarsened by a fill-distance based adaptivity method and redistributed via a point position filtering method. The incompressible and inviscid ALE-WLS model is applied to the following standard validation test cases: deforming elliptical drop, small amplitude standing waves and the dam break problem. The deforming elliptical drop test show that the model can calculate the kinematics and dynamics of this free surface flow accurately and robustly. The small amplitude standing wave gives the same conclusions. Long time integration of this small amplitude periodic motion is possible due to accurate free surface evolution and small errors in the fluid volume. The dam break test case shows that the incompressible and inviscid ALE-WLS model can calculate nonlinear fluid motion, fluid structure impacts and overturning waves. The propagation speed of the wetting front and impact pressures are compared to experiments and the results compare reasonably well. The incompressible and inviscid ALE-WLS model is coupled with the potential flow model of Engsig-Karup et al. , to perform multiscale calculation of breaking wave impacts on a vertical breakwater. The potential flow model provides accurate calculation of the wave transformation from offshore to the vicinity of the breakwater. The wave breaking close to the breakwater and the wave impact are calculated by the incompressible ALE-WLS model. The forces calculated with the incompressible and inviscid ALE-WLS model are ˜ 1 - 2 times the corresponding compressible calculations in Bredmose et al.  for the calculations without trapped air. Among the contributions of this project are the ALE-WLS method combined with approximate Riemann solvers and the generalization of the FPM method to arbitrary order of accuracy. The WLS and GFPM stencils found using the BFS data structure, which is updated due to topology changes of the evolving fluid domain. This extension combined with ALE-WLS and approximate Riemann solvers gives a numerical model capable of calculation of forces due to breaking wave impacts. The incompressible and inviscid ALE-WLS model has been coupled with a potential flow model to provide multiscale calculation of forces from breaking wave impacts on structures.
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Walther, Jens Honore, Engsig-Karup, Allan Peter, Bingham, Harry B.