Advances in particle distribution for SPH numerical schemes: from explicit to implicit shifting techniques

In meshless numerical methods such as Smoothed Particle Hydrodynamics (SPH), the lack of uniformity in particle distribution, which is manifested as the presence of voids or clusters, affects negatively the accuracy. In these models, the particles follow the Lagrangian trajectories, and for highly distorted flows, their distribution is severely perturbed, generating numerical issues and compromising the quality of the simulations. For this reason, methodologies, called Particle Shifting Technique (PST), have been introduced to reduce these phenomena. The PSTs presented in the literature have an explicit approach meaning that it is not possible imposing a maximum predefined level of perturbation in the particle distribution. In the present thesis, an explicit shifting technique has been extended and optimized in the framework of Arbitrarily Lagrangian-Eulerian SPH (ALE-SPH) schemes, increasing the accuracy without extra computational overheads. Then, a novel approach for particle shifting, which can be adopted in meshless numerical methods, has been developed and analyzed. The proposed methodology, called Implicit Iterative Particle Shifting (IIPS), uses an iterative procedure to reduce the spatial particle anisotropy, which is associated with the discretisation error. Through the implicit iterative minimization problem, which is based on the particle concentration gradient, the algorithm is able to control the particle spatial distribution and therefore, the anisotropy of the particles. The implicit method has been implemented in the software ASPHODEL of the ANDRITZ group, which adopts an SPH-ALE solver. In order to demonstrate its effectiveness, the IIPS performances have been compared to the explicit shifting technique. Due to the characteristics of ALE-SPH models, in order to keep the scheme consistency, two different methodologies to update the physical quantities, named "Implicit iterative particle shifting with fictitious time step" and "Implicit iterative particle shifting with MLS reconstruction", have been proposed and tested in two-dimensional test cases: the Taylor-Green vortex, the moving box inside a rectangular box and the jet impacting a flat surface. With these applications, it has been shown that for affordable computational overheads, the IIPS maintains isotropic particle distribution, significantly increasing the accuracy, confirming its superiority in comparison to existing explicit shifting approaches.