## 1 Introduction
However, some researches show that they are not always effective in the prevention of inner fluid particle penetration [6; 18]. This phenomenon is also found in the following 2 Dtube flow case study, as shown in Section 6.4. The semi-analytical boundary is introduced by [PERSON] et al. [19] and further improved by [PERSON] et al. [20]. A correction term is applied to the theoretical formulations to eliminate the effect of lacking neighboring particles [21; 22]. The drawback is that extra computation is needed during the simulation, as normally the correction term is not just a simple number but a matrix. Research has been carried out to find out a more efficient boundary treatment. In this paper, the ISPH method with constant fluid density is applied as it normally provides better pressure evaluations than WCSPH [21]. For the ISPH method, the application of one layer of denser wall particles is supposed to be effective in the prevention of inner particle penetration [22]. However, it is found in some situations, such as violent impact problems, that the distribution of particles could be distorted and the local particle spacing could be shortened largely. This will lead to unrealistic fluid particle exploration [23]. Hence, uniform particle distribution is preferred for a stable and accurate simulation [24; 25], where wall particles involved in calculation should be spaced the same as fluid particles. With all these considerations, a new boundary treatment, using additive particles and dynamic dummy particles is proposed in this work. Dummy particles do not really exist and are only used as neighboring particles. The solid boundary is represented by two types of boundary particles. One type is spaced just the same as fluid particles; the other is used to fill the gap of the first type with smaller intervals, called additive particles. These additive particles are not treated directly as neighboring particles, and thus the uniform particle distribution is retained. They are only used as a medium to create dummy particles. Both types of boundary particles create dummy particles, so that the kernel domain of the inner fluid particle will be complete. Five different boundary treatments shown in Section 6 are considered including the new boundary treatment, ghost particles, ghost particles plus repulsive force, uniform distributed boundary particles with dynamic dummy particles, as well as uniform distributed particles with repulsive force. Four case studies are carried out, respectively, with the five above different boundary treatment methods, including water motions in dam breaking,2D and 3D static water tank and u-tube flow. The results are displayed and compared to demonstrate the performance of the proposed boundary treatment. The exact pressure values in dam breaking, 2D, and 3D static water tanks are analyzed. The time history of the impact pressure on a specific point during dam breaking is compared with experimental values [26]. Pressure distributions of u-tube flow are displayed in different colors to provide an intuitive assessment on the performance of different boundary treatments. The results show that the new boundary treatment using additive boundary particles combined with dummy particles is effective to prevent the inner particles from penetrating and could provide accurate pressure evaluations.