Abstract: SPH (smoothed particle hydrodnamics) method with fully variable smoothing lengths is proposed. Different from existing adaptive kernel SPH methods, fully variable smoothing lengths are considered based on an adaptive symmetrical kernel estimation. An evolution equation of density is derived which implicitly couples with a variable smoothing length equation. Based on Springel' s fully conservative formulation SPH momentum equation and energy equation are derived by using symmetrical kernel estimation instead of scatter kernel estimation. An additional iteration process is employed to solve evolution equations of density and variable smoothing lengths equation. SPH momentum equation and energy equation are solved explicitly. Computation cost added by iteration is little. The equations and algorithm are tested via three ID shock-tube problems and a 2D Sedov problem. It is showed that conservation of momentum and energy is improved substantially and variable smoothing lengths effect is corrected, especially in the 2D Sedov problem. Pressure peak position and pressure at center are more accurate than those by Springel's scheme. The method deals with large density gradient and large smoothing length gradient problems well, such as large deformation and serious distortion problems in high velocity impact and blasting.

Key words: variable smoothing length, smoothed particle hydrodynamics, meshless method, mechanics of explosion, shock wave