An adaptive fifth-degree kernel for suppressing stress instability in SPH for compressible flows

Abstract

The phenomenon of stress instability is frequently observed in smooth particle hydrodynamics (SPH), which manifests as unphysical clustering or separation of particles, and constrains the application of SPH. In this paper, we propose an adaptive fifth-degree kernel function for alleviating stress instability in compressible flows. The shape of kernel function can be adaptively modified according to the particle states, circumventing the conditions associated with instability and thus alleviating both compressive and tensile instability. For the case where two particles in a particle pair use different kernels, discrete formulations of the momentum and thermal equations within the Conservative Reproducing Kernel Smoothed Particle Hydrodynamics (CRKSPH) framework are employed to ensure the conservation. Several benchmark cases of compressible flows are simulated utilizing the adaptive fifth-degree kernel, and results indicate that the adaptive fifth-degree kernel can sustain homogeneous particle distributions and clear shock wave fronts after multiple bounces of shock and rarefaction waves, effectively alleviating the stress instability inherent in classical SPH methods. In addition, the adaptive fifth-degree kernel can ensure reasonable spacing of particle pairs even when negative pressure is encountered, and avoid particle clustering or voids in the computational domain.