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Total Lagrangian Smoothed Particle Hydrodynamics with An Improved Bond-Based Deformation Gradient for Large Strain Solid Dynamics

I. M. Wiragunarsa, L. R. Zuhal, T. Dirgantara, I. S. Putra, E. Febrianto

Abstract

Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) is one variant of SPH where the variables are described using the fixed reference configuration and a Lagrangian smoothing kernel. TLSPH elevates the computational efficiency of the standard SPH when no topological change is involved, and it alleviates the stability of SPH scheme with respect to tensile loading. However, instabilities associated with spurious mode, or hourglass/zero-energy mode, persists and often affects the simulation of solids undergoing extremely large strain. This work proposes an alternative formulation to compute deformation gradient with improved accuracy and therefore minimising the possibility of encountering the zero-energy mode. Specifically, we leverage the local discrete computation of bond-based (or pairwise) deformation gradient smoothed by the kernel. Additionally, the bond of a particle with itself is considered to preserve the polynomial reproducibility imposed by the kernel correction scheme. We showcase the convergence of the approach using a two-dimensional benchmark example. Furthermore, the accuracy, robustness, and stability of the proposed approach are assessed in various two- and three-dimensional examples, highlighting on the stability improvement that allows for solid dynamic simulations with more extreme elongation.

Total Lagrangian Smoothed Particle Hydrodynamics with An Improved Bond-Based Deformation Gradient for Large Strain Solid Dynamics

Abstract

Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) is one variant of SPH where the variables are described using the fixed reference configuration and a Lagrangian smoothing kernel. TLSPH elevates the computational efficiency of the standard SPH when no topological change is involved, and it alleviates the stability of SPH scheme with respect to tensile loading. However, instabilities associated with spurious mode, or hourglass/zero-energy mode, persists and often affects the simulation of solids undergoing extremely large strain. This work proposes an alternative formulation to compute deformation gradient with improved accuracy and therefore minimising the possibility of encountering the zero-energy mode. Specifically, we leverage the local discrete computation of bond-based (or pairwise) deformation gradient smoothed by the kernel. Additionally, the bond of a particle with itself is considered to preserve the polynomial reproducibility imposed by the kernel correction scheme. We showcase the convergence of the approach using a two-dimensional benchmark example. Furthermore, the accuracy, robustness, and stability of the proposed approach are assessed in various two- and three-dimensional examples, highlighting on the stability improvement that allows for solid dynamic simulations with more extreme elongation.
Paper Structure (19 sections, 66 equations, 25 figures, 1 table)

This paper contains 19 sections, 66 equations, 25 figures, 1 table.

Table of Contents

  1. Introduction
  2. Spatial discretisation in SPH
  3. Field approximation from point values
  4. Corrected SPH discretisation
  5. Total Lagrangian SPH for solid body
  6. Motion and deformation
  7. Total Lagrangian formulation of conservation laws
  8. Alternative formulation for computing deformation gradient
  9. One-dimensional illustration
  10. SPH implementation of the proposed deformation gradient
  11. Numerical examples
  12. Convergence analysis using swinging plate problem
  13. Momentum preservation analysis using spinning plate and cube problems
  14. Energy preservation analysis using finite deformation of a column
  15. Stability analysis using pulling test of column
  16. ...and 4 more sections

Figures (25)

  • Figure 1: Support of the kernel centred at the $i$-th particle, which is a circle in $\mathbb R^2$ with radius $R = \kappa h$. The neighbouring particles lying inside the support is coloured black.
  • Figure 2: The kernel and kernel gradient of particle $i$ due to interaction with particle $j$: (a) kernel value, and (b) gradient of kernel at $x_i=1$. (c) kernel value, and (d) gradient of kernel at $x_i=0$.
  • Figure 3: Description of the motion with respect to the initial and current configuration
  • Figure 4: Field reconstruction from point values: (a) deformation , (b) deformation gradient.
  • Figure 5: Gradient approximation of a deformation field from point values: (a) quadratic function, (b) cubic function
  • ...and 20 more figures