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One of the historical problems appearing in SPH formulations is the inconsistencies coming from the inappropriate implementation of boundary conditions. In this work, this problem has been investigated; instead of using typical methodologies such as extended domains with ghost or dummy particles where severe inconsistencies are found, we included the boundary terms that naturally appear in the formulation. First, we proved that in the 1D smoothed continuum formulation, the inclusion of boundary integrals allows for a consistent O (h) formulation close to the boundaries. Second, we showed that the corresponding discrete version converges to a certain solution when the discretization SPH parameters tend to zero. Typical tests with the first and second derivative operators confirm that this boundary condition implementation works consistently. The 2D Poisson problem, typically used in ISPH, was also studied, obtaining consistent results. For the sake of completeness, two practical applications, namely, the duct flow and a sloshing tank, were studied with the results showing a rather good agreement with former experiments and previous results.Subject Index: 024 §1. IntroductionThe SPH scheme is a Lagrangian model based on a smoothing of the spatial differential operators of fluid-dynamics equations and on their subsequent discretization through a finite number of fluid particles. The smoothing procedure is performed at the continuum level using a compact support kernel function whose characteristic length is the smoothing length h. The resolution of the discrete SPH scheme is a function of the smoothing length h and the mean particle distance Δx. In this framework, the (continuous) equations of the fluid-dynamics should be recovered as both h and Δx/h simultaneously tend to zero. 1) Colagrossi and Landrini 2) demonstrated that when boundaries are not present, the SPH approximation of a function u (x) differs O(h 2 ) from the analytical function u (x). Quinlan et al. 3) studied the errors in the first-order derivatives of a classical SPH implementation and its dependence on the particle spacing and smoothing length when no boundaries were considered. In Ref. 4), this work was extended to 3D and a complete consistency study was performed for first-and second-order derivatives.The SPH simulations in engineering usually involve solid boundary conditions (BC) for both the velocity and pressure fields. In the SPH framework, these conditions have been implemented in the past in a number of different ways: by using boundary force-type models, 5), 6) by modifying the structure of the kernel in the Downloaded from https://academic.oup.

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A dynamical system involving a driven pendulum filled with liquid, is analyzed in the present paper series.The study of such a system is conducted in order to understand energy dissipation resulting from the shallow water sloshing and induced wave breaking. This analysis is relevant for the design of Tuned Liquid Damper devices. The complexity and violence of the flow generated by the roll motion results in the impossibility of using an analytical approach, requiring in turn the use of a suitable numerical solver. In Part I, the coupled dynamical system is thoroughly described, revealing its nonlinear features associated with the large amplitude of the forcing, both in terms of mechanical and fluid dynamical aspects. A smoothed particle hydrodynamics (SPH) model, largely validated in literature, is used to calculate the frequency behavior of the whole system. For small rotation angles, a semi-analytical model of the energy dissipated by the fluid, based on a hydraulic jump solution, is developed; the energy transfer is numerically calculated in order to extend the analysis to large oscillation angles. The experimental part of the investigation is carried out in Part II of this work.