Angular-momentum enhanced non-hourglass formulation for SPH solid dynamics

Updated Lagrangian smoothed particle hydrodynamics (SPH) for solid dynamics is often plagued by numerical instabilities, particularly hourglass modes that produce unphysical zigzag patterns. While recent essentially non-hourglass (SPH-ENOG) and generalized non-hourglass (SPH-GNOG) formulations have improved stability, they suffer from poor angular momentum conservation, limiting their accuracy in rotational problems. To overcome this, this paper presents two angular-momentum enhanced non-hourglass formulations. First, we enhance the SPH-ENOG method with rotation matrices derived from Rodrigues’ formula, creating SPH-ENOG-A for elastic materials, which explicitly accounts for rigid rotations during time integration, thereby significantly enhancing angular momentum conservation. To strictly enforce linear momentum conservation, the average of the rotation matrices is computed and applied to each particle. We then extend this approach to reformulate the corrective term in SPH-GNOG, yielding SPH-GNOG-A—a unified method for both elastic and plastic materials that not only improves angular momentum conservation but also eliminates prior dependencies on material-specific coefficients. Validated against elastic (oscillating plates, spinning solids) and plastic (Taylor bars, high-velocity impacts) benchmarks, our methods retain the hourglass-free stability, convergence, and accuracy of their predecessors while achieving a significant leap in angular momentum conservation. © 2025 Elsevier Inc.     «

Updated Lagrangian smoothed particle hydrodynamics (SPH) for solid dynamics is often plagued by numerical instabilities, particularly hourglass modes that produce unphysical zigzag patterns. While recent essentially non-hourglass (SPH-ENOG) and generalized non-hourglass (SPH-GNOG) formulations have improved stability, they suffer from poor angular momentum conservation, limiting their accuracy in rotational problems. To overcome this, this paper presents two angular-momentum enhanced non-hourgla...     »