# Boundary SPH for Robust Particle-Mesh Interaction in Three Dimensions
## 1 Introduction  
The smoothed particle hydrodynamics (SPH) approach is a widely used, mesh-free, Lagrangian representation of gaseous or fluid media that enjoys applications in a variety of simulation tasks, which range from theoretical astrophysics to material science and computational fluid dynamics (CFD) [1, 2, 3, 4]. In particular, SPH is broadly appreciated for its ability to provide numerical and computational support where model interactions between fluid particles and simulated solids, walls, and structures are called for. Nonetheless, SPH is troubled by an open, non-trivial challenge in the form of the \"Boundary Condition\" (BC) or \"Fluid-Surface Interactions\" (FSI) problem [5]. FSI arises from a conundrum in the numeric representation of fluid particles, wherein they have no direct means of interacting with 2D or 3D representations of boundary manifolds. FSI creates significantly thorny issues for applications of SPH. At a minimum, two main criteria are expected for boundaries. First, the boundary must prevent fluid particles from intersecting completely with the boundary and therefore escaping the fluid domain. Second, the boundary must minimizetruncation of the density kernel of fluid particles close to or intersecting with the boundary manifold [3; 6]. If these criteria are not met, the resulting hydrodynamics markedly deviate from real-world properties, often in ways that are visually and empirically unrealistic in simulation. One of the earliest known BC methods is [PERSON]'s \"Immersed Boundary Method\" (IBM), which approximates boundary forces via a Dirac delta function [7; 8]. The IBM has seen success in various application fields [9] but exhibits unexpected spring-like behavior in elastic boundaries [10]. Since the IBM, three broad perspectives dominate popular BC methodologies: (1) [PERSON]'s \"Boundary Particles\" method [11]; (2) fictitious \"dummy\" particles, commonly referred to as \"Ghost Particles\" or \"Mirror Particles\" [12; 13]; and (3) semi-analytical wall boundaries via boundary integrals [14; 15]. Hybrid variations of these methods are also common, such as [PERSON] et al. 's combination of boundary and ghost particles [16] or [PERSON] and [PERSON]'s \"staggered\" boundary particle rows approach [17; 18], which share resemblance with [PERSON] et al. 's \"fixed\" ghost particles variation for SPH [19; 20]. Yet, for all their successes in advancing computational representations of hydrodynamics, many of these methods are limited in one key respect: their ability to treat the complexity of the boundary manifold. Consider, for example, the case of ghost particles. They are incompatible with boundary manifolds that feature sharp edges, turns, or points. Moreover, methodologies that involve 3D boundary particles suffer in computational performance due to the difficulty of interpolating across 3D surfaces. There have been noble attempts to circumnavigate this issue. For example, [PERSON] attempted to resolve the 3D interpolation issue by using a Gaussian kernel to dictate the placement of boundary particles, but alias, this method involves the subdivision of mesh triangles and can lead to complications with bigger meshes [21]. Alongside challenges with absolute boundary properties, it is often observed that boundary particles also suffer from aberrant effects in the fluid particles as they grow close to the boundary. Invariably, this near-boundary problem manifests in a range of compromising knock-on effects that are often unsatisfactory for application domains. These issues may present as (1) the bouncing of fluid particles due to the \"bumpy\" surface formed by boundary particles; (2) unexpected pressure forces acting on fluids at the beginning of simulations; (3) tensile instability, wherein fluid particles close to the boundary sometimes clump together; and (4) the density kernel of fluid particles close to the boundary being \"truncated\" beyond the initial layer of boundary particles.