Solving Boundary Handling Analytically in Two Dimensions for Smoothed Particle Hydrodynamics

TL;DR

This paper introduces a fully analytic method for evaluating boundary integrals in 2D Smoothed Particle Hydrodynamics, enabling higher-order boundary conditions and coupling with mesh-based solvers, with significant accuracy improvements.

Contribution

The authors develop a closed-form solution for boundary integrals over triangles, using Chebyshev polynomials and hypergeometric functions, enhancing SPH boundary handling.

Findings

Outperforms numerical quadrature by up to five orders of magnitude

Works with arbitrary triangle geometries and kernel functions

Enables flexible coupling of meshes with SPH

Abstract

We present a fully analytic approach for evaluating boundary integrals in two dimensions for Smoothed Particle Hydrodynamics (SPH). Conventional methods often rely on boundary particles or wall re-normalization approaches derived from applying the divergence theorem, whereas our method directly evaluates the area integrals for SPH kernels and gradients over triangular boundaries. This direct integration strategy inherently accommodates higher-order boundary conditions, such as piecewise cubic fields defined via Finite Element stencils, enabling analytic and flexible coupling with mesh-based solvers. At the core of our approach is a general solution for compact polynomials of arbitrary degree over triangles by decomposing the boundary elements into elementary integrals that can be solved with closed-form solutions. We provide a complete, closed-form solution for these generalized integrals, derived by relating the angular components to Chebyshev polynomials and solving the resulting radial integral via a numerically stable evaluation of the Gaussian hypergeometric function $_2F_1$. Our solution is robust and adaptable and works regardless of triangle geometries and kernel functions. We validate the accuracy against high-precision numerical quadrature rules, as well as in problems with known exact solutions. We provide an open-source implementation of our general solution using differentiable programming to facilitate the adoption of our approach to SPH and other contexts that require analytic integration over polygonal domains. Our analytic solution outperforms existing numerical quadrature rules for this problem by up to five orders of magnitude, for integrals and their gradients, while providing a flexible framework to couple arbitrary triangular meshes analytically to Lagrangian schemes, building a strong foundation for addressing several grand challenges in SPH and beyond.