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Implementation of the BFSC plate finite element in 2D

• BFSC plate element, BFSCPlate2D: a four-node rectangular element of the classical laminated plate theory, with the nonlinear strains of von Karman
• Linear statics: [KC0]{u} = {f}
• Linear buckling: ([KC0] + lambda[KG]){u} = 0, with [KG] of the membrane forces of a static solution or of constant membrane forces
• Natural frequencies: ([KC0] - omega^2[M]){u} = 0
• Geometrically nonlinear statics: internal force vector and tangent stiffness matrix [KC0] + [KCNL] + [KG], its exact Jacobian
• Follower pressure, normal to the deformed plate: force vector and unsymmetric load stiffness matrix
• The contributions of each element are written in sparse arrays; the model, its boundary conditions and the solution of its equations are written by the user, with NumPy and SciPy

The ABD matrices of the laminates are calculated with my other repository composites, and the tests and usage examples solve the nonlinear analyses with structsolve.

Citing this library

Saullo G. P. Castro (2026). Implementation of the BFSC plate finite element in 2D (Version 0.5.0). https://github.com/saullocastro/bfscplate2d.

Documentation

The documentation is available on: https://saullocastro.github.io/bfscplate2d.

History

See CHANGELOG.md for the details of each version.

• version 0.5.0 (2026-10-26)
  • The shape functions renamed from N to S, as in bfsccylinder: the attributes Nu, ..., Nw_y and the methods update_Nu, ..., update_Nw_y of BFSCPlate2D are Su, ..., Sw_y and update_Su, ..., update_Sw_y
  • Follower pressure: update_fext_follower and update_KCfollower, the force vector and the unsymmetric load stiffness matrix of a pressure normal to the deformed mid-surface
  • BUG: BFSCPlate2D.__init__ sets init_k_KCNL, which a second call of the constructor left at its previous value
  • Sphinx documentation, with a user guide, usage examples and the solution of the analyses with structsolve.Analysis
  • The tests solve the nonlinear analyses with structsolve.Analysis and cross-check the buckling loads and natural frequencies with structsolve.lb and structsolve.freq, requiring structsolve>=0.9.3
  • Metadata in pyproject.toml; Python 3.10 to 3.14
• version 0.4.1 (2026-10-22)
  • Build flags for speed, without OpenMP
• version 0.4.0 (2026-09-21)
  • Fixed the tangent stiffness matrix and the internal force vector of the nonlinear analyses, such that KC0 + KCNL + KG is the derivative of fint
  • Python 3.13 and 3.14 support
• version 0.3.1 (2024-09-25)
  • Fixed the compatibility with integer numbers
• version 0.3.0 (2023-04-12)
  • Removed the dependency on the NumPy C API
• version 0.2.1 (2021-11-10)
  • 3-Clause BSD license
• version 0.2.0 (2021-11-01)
  • Geometrically nonlinear analyses: update_KCNL and update_fint
• version 0.1.9 (2021-09-27)
  • update_KG_cte_N, the geometric stiffness matrix of constant membrane forces
• version 0.1.7 (2020-08-21)
  • First tagged version: BFSCPlate2D, with update_KC0, update_KG and update_M

License

Distributed under the 3-Clause BSD license (https://raw.github.com/saullocastro/bfscplate2d/main/LICENSE).

Contact: S.G.P.Castro@tudelft.nl