A Lagrangian Reproducing Kernel Kirchhoff--Love Shell Formulation with Damage-Enhanced Kernel Update

  • Wang, Jiarui (Southern University of Science and Technology)
  • Ma, Ke (Vanderbilt University)
  • Bazilevs, Yuri (Vanderbilt University)

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In the Lagrangian reproducing kernel particle method (RKPM) [1], the basis functions are constructed with reference to the material configuration, and this formulation offers well-known advantages over traditional Lagrangian mesh-based approaches in large-deformation problems. While the approximation-function values are stored and merely accessed during the simulation, which provides efficiency, the necessary assumption of one-to-one mapping fails when the deformation becomes extreme. Moreover, the topology of material points remains unchanged when the separation happens in the case of fracture and fragmentation simulations. In this work, a damaged-kernel enhanced general-purpose RKPM thin shell formulation is introduced. The degenerated shell approach is employed in the present formulation, where the structure is considered as a 3D solid that is subjected to the constraints of the Kirchhoff--Love shell kinematics [2]. A Lagrangian formulation is used. Principal component analysis (PCA) is adopted as a local parameterization technique to address the challenge of surface geometry representation of meshfree methods, and the isoparametric RK approximation is employed for the approximation of the kinematic variables. An in-plane Taylor-series-expansion-based enrichment [2] is developed to stabilize nodal quadrature. The damage-based kernel degeneration algorithm is proposed to effectively simulate material separation. The effectiveness of the proposed RKPM KL shell formulation is demonstrated using an extensive set of linear-elastic and finite-deformation inelastic test cases. REFERENCES [1] Chen, J. S., Pan, C., Roque, C. M. O. L., Wang, H. P. (1998). A Lagrangian reproducing kernel particle method for metal forming analysis. Computational Mechanics, 22(3), 289-307. [2] Wang, J., Bazilevs, Y. (2025). A general-purpose meshfree Kirchhoff–Love shell formulation. Engineering with Computers, 41(3), 1379-1410.