439 Progress of Theoretical Physics, Vol. 128, No. 3, September 2012 A Boundary Integral SPH Formulation Consistency and Applications to ISPH and WCSPH Fabricio Maci` a, Leo M. Gonz´ alez, Jose L. Cercos-Pita and Antonio Souto-Iglesias Naval Architecture Department (ETSIN), Technical University of Madrid (UPM), 28040 Madrid, Spain (Received May 11, 2012; Revised July 17, 2012) 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. Introduction The SPH scheme is a Lagrangian model based on a smoothing of the spatial dif- ferential 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 frame- work, 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 con- ditions 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.com/ptp/article/128/3/439/1841294 by guest on 24 June 2022