ABSTRACT A steady state Eulerian smoothed particle hydrodynamics (SPH) solver is proposed by integrating the Semi‐Implicit Method for Pressure‐Linked Equations (SIMPLE) algorithm. By removing time dependent terms from the governing equations, the proposed approach directly solves for steady‐state velocity, pressure, and temperature fields for incompressible flows using a matrix‐based formulation. To efficiently handle the resulting large, sparse linear systems, a matrix‐free Bi‐CGSTAB iterative solver is employed and the overall computational algorithm is fully parallelized on GPUs to accelerate performance. The accuracy and efficiency of the proposed steady solver are validated through several benchmark tests, including pipe flow with and without obstacles, 2D/3D lid‐driven cavity flow with and without heat transfer, and natural convection flow. Compared to conventional transient Eulerian SPH solvers, the proposed method achieves 8.97–17.36 times speedup while maintaining high accuracy, making it a promising tool for steady state CFD analysis and as a precursor to transient simulations.
Kim et al. (2026) studied this question.