Insights on using the boundary integral SPH formulations to calculate Laplacians with Dirichlet boundaries

Handling boundary conditions in Smooth Particle Hydrodynamics (SPH) has been known as a challenging problem for a long time, especially when particle approximation is implemented for truncated kernels on the boundary. This work develops a novel particle consistent gradient formulation to impose the Dirichlet boundary condition on a Laplacian with boundary integrals. Boundary integral formulation allows the accurate imposition of boundary conditions without creating virtual particles to extend boundaries, as used in conventional SPH modeling practices. Here, boundary integral formulations are presented for Laplacian operators and the imposition of Dirichlet boundary conditions on arbitrary boundary shapes. We also explore using this particle consistent first-order derivative twice to evaluate the entire Laplacian. Further, the convergence characteristics of different Laplacian formulations with boundary integrals are discussed. Finally, numerical experiments solving explicit thermal and fluid problems are presented to demonstrate the robustness of various Laplacian operators for practical use.