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Implementation of the BFSC cylinder finite element

• Single-curvature BFSC (SC-BFSC) element: a four-node element of the developed mid-surface of a cylindrical shell
• Donnell-type kinematics, BFSCCylinder, and Sanders-Koiter type kinematics, BFSCCylinderSanders
• Linear statics: [KC0]{u} = {f}
• Linear buckling: ([KC0] + lambda[KG]){u} = 0, with [KG] of the membrane stress resultants of a static solution or of a constant membrane stress state
• 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 shell: 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, the tests and usage examples solve the nonlinear analyses with structsolve, and the tutorials model the stringers of stiffened cylinders with bfscplate2d.

Citing this library

Saullo G. P. Castro (2026). Implementation of the BFSC cylinder finite element (Version 0.7.1). https://github.com/saullocastro/bfsccylinder.

Documentation

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

History

See CHANGELOG.md for the details of each version.

• version 0.7.1 (2026-11-03)
  • Tutorial and tests of the Riks method deep into the post-buckling of the axially compressed cylinder of Simo, Fox and Rifai (1990)
  • Tutorial and tests of the cantilever cylinders of Schweizerhof and Ramm (1984) under a follower external pressure: linear buckling with the unsymmetric load stiffness matrix, Table 4, and Riks with the inertia count
  • Requires structsolve>=0.10.0
• version 0.7.0 (2026-10-26)
  • Follower pressure, for both kinematics: 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: update_M of the Sanders-type element uses the rotation -w,y + v/R in the rotary inertia
  • BUG: the constructors of BFSCCylinder and BFSCCylinderSanders set init_k_KCNL, which a second call of a constructor left at its previous value
  • Deterministic and verified natural frequency tests
  • Sphinx documentation, with a user guide, usage examples, the solution of the analyses with structsolve.Analysis and tutorials of stringer-stiffened cylinders
  • The tests solve the nonlinear analyses with structsolve.Analysis, requiring structsolve>=0.9.3
  • Metadata in pyproject.toml; Python 3.10 to 3.14
• version 0.6.1 (2026-09-25)
  • Cython compiler flags
• version 0.6.0 (2026-09-11)
  • Fixed the tangent stiffness matrix, now the derivative of the internal force vector, for both kinematics
  • Fixed update_KC0 of the Sanders-type element, which missed a part of the twisting curvature
• version 0.5.1 (2026-08-27), including 0.5.0
  • Fixed the bending part of the Sanders-type equations
  • Shape functions renamed from N to S
  • Python 3.14 support
• versions 0.4.1 to 0.4.8 (2024-09-25 to 2025-11-25)
  • Fixed the compatibility with integer numbers; updated continuous integration and deployment
• version 0.4.0 (2023-04-12)
  • Removed the dependency on the NumPy C API
• version 0.3.12 (2021-12-04), including 0.3.11
  • Fixed the nonlinear terms, where v was missing, and fint, which included terms of KG
• version 0.3.0 (2021-11-03)
  • Sanders-type kinematics: BFSCCylinderSanders, in bfsccylinder.sanders
• version 0.2.3 (2021-10-04), including 0.2.2
  • 3-Clause BSD license
• version 0.2.0 (2021-05-26)
  • Geometrically nonlinear analyses: update_KCNL and update_fint
• version 0.1.1 (2020-08-22)
  • First tagged version: BFSCCylinder, with update_KC0, update_KG, update_KG_constant_stress and update_M

License

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

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