An Enhanced Hourglass Control Scheme in Variationally Consistent Total Lagrangian SPH Solid Mechanics
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This paper presents a novel variationally consistent hourglass control scheme, referred to as cHCS (complete Hourglass Control Scheme), incorporated within the framework of a variationally consistent Total Lagrangian SPH for non-linear finite-strain structural mechanics. The cHCS scheme can be considered an extended and enhanced version of the HCS (Hourglass Control Scheme) scheme by Ganzenmüller et al. [1]. The HCS scheme will be shown to have several limitations: (i) it has a tuning parameter which is problem- and resolution-dependent, (ii) it is based on a 1D approximation of compression wave speed, (iii) it is based on linear elasticity, (iv) the material acoustic model is not properly considered. The cHCS overcomes these challenges by (i) incorporating a dynamic coefficient which accounts for the time history of error at each calculation point, (ii) employing a multi-dimensional wave speed and thus properly reflecting material multi-dimensionality, (iii) considering non-linear elasticity via principal stretches, (iv) modelling the material acoustic behaviour more accurately through consideration of both compression and shear waves. The cHCS stabilises hourglass modes based on deformation and relative displacement fields. The cHCS force is shown to be consistent with a penalty-based energy potential and thus preserves the structure of the Euler-Lagrange equation in the Lagrangian mechanics framework. Implementation of cHCS together with cR (complete Riemann stabilization term) [2], which is based on velocity jumps and corresponds to a dissipation potential, results in effective suppression of hourglass modes in a variationally consistent and conservative manner, yielding smooth and accurate kinematics and dynamics. Validations are made through simulations of several classical test cases in 3D, including torsional deformation of a T-shaped column, a manufactured torsional deformation and bending of a slender column.
