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Delta Voronoi Smoothed Particle Hydrodynamics, ẟ-VSPH


A Lagrangian scheme that combines Voronoi diagrams with smoothed particle hydrodynamics (SPH) for incompressible flows has been developed. Within the Voronoi tessellation, the differential operators are discretized by adding contributions from neighboring cells following the Voronoi particle hydrodynamics (VPH) method, (Serrano and Español, Phys. Rev. E, 64, 2001; Hess and Springel, Month. Not. of the Royal Astronomical Society, 406(4), 2010). This approach is structurally similar to the one used by SPH, although the neighbor set is larger in the latter. Due to this similitude, and the feasibility to approximate the fluid as weakly compressible in both SPH and VPH, coupling the two methods is a natural choice explored in detail in this thesis.

The Voronoi cells allow enforcing the boundary conditions directly on the portion of their perimeter in contact with the boundary. Two sub-domains are then defined based on the proximity to the boundaries. The VPH formulation is used in those particles close to solid boundaries, where SPH consistency and boundary conditions implementation become problematic. The SPH formulation is limited to the region such that no boundary falls within the kernel domain. Some overlapping of both sub-domains is allowed to provide a buffer zone to progressively transition from one method to the other. In this zone, fields are calculated using both methods and then coupled by a weighting function that depends on the distance to the closest boundary. As particles move, they can cross the buffer zone and change sub-domains.

The unbounded nature of Voronoi tessellations requires further treatment close to boundaries and free-surfaces to ensure that the Voronoi cells associated with the particles do not extend beyond the fluid domain. Boundaries are discretized as segments, or D-dimensional triangles in higher dimensions, and virtual particles are added mirroring the particles next to the boundaries. Since the boundary becomes a symmetry plane, no Voronoi cell can cross it. However, Voronoi cells resulting from the tessellation are always convex, so an additional trimming algorithm has been implemented to deal with concave boundaries. In the case of free-surfaces, a new detection algorithm determines if a particle is next to the free-surface and estimates the free-surface normal direction, which is used to bound the corresponding Voronoi cell. Periodic and inlet/outlet boundary conditions are also included in the coupled scheme.

The ẟ-SPH correction proposed by Cercos-Pita et al., Applied Math. Modelling, 40(19), 2016, is extended to the VPH formulation. In addition, the density field is reinitialized every certain number of time steps following the scheme proposed by Colagrossi and Landrini, J. Comp. Phys, 191, 2003. A shifting algorithm inspired by Lloyd, IEEE Trans. of Inform. Theory, 28(2), 1982, is also included to avoid excessive deformation of the Voronoi cells.

The code key subroutines are written in C++, and the main time-stepping program is implemented in Matlab. A leap-frog numerical scheme is used to iterate forward in time. The program architecture is summarized graphically, with a detailed analysis of the neighbor boundaries and particles search algorithm included afterwards. A linear damping term is used during the initialization process to mitigate possible inconsistencies from the user-defined initial conditions. Finally, the energy balance of the coupled scheme is discussed.

The method presented in this thesis is suitable for free-surface flows with moving solid boundaries within the fluid domain. However, prior to extending it to such complicate problems, the accuracy of the coupled scheme is discussed by means of a set of well-known verification benchmarks.

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