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Semi-analytical methods for plates, shells and stiffened panels

• Linear and non-linear static analyses, with field outputs of displacements, strains and stresses
• Linear buckling
• Vibration, with or without pre-stress
• Panel flutter using the piston theory
• Geometrically non-linear static analyses (postbuckling) using the Newton-Raphson method or the arc-length methods of Riks and Crisfield, with the product of the tangent stiffness matrix by a vector available without assembly
• Dead or follower (hydrostatic) pressure loads, with the unsymmetric load stiffness of the follower pressure
• Multi-domain assemblies of plates, cylindrical shells, cylinders, stiffened panels, with the domains connected exactly with the null-space method, the default, or by penalty stiffnesses. Debonding defects or cohesive zone can be simulated
• Prescribed displacements and rotations imposed exactly with the null-space method, MultiDomain(..., pd_method='null-space'), for displacement control in linear static, linear buckling and geometrically non-linear (Newton-Raphson, Riks, Crisfield) analyses, with the reaction forces; or by penalty stiffnesses, the default
• CLPT, FSDT and TSDT kinematics for plates and cylindrical shells
• Donnell and Sanders-Koiter kinematics for cylindrical shells

The structural matrices are solved with structsolve.

Citing this library

Saullo G. P. Castro, Nathan D'Souza. (2026). Semi-analytical methods for plates, shells and stiffened panels (Version 0.14.0). Zenodo. DOI: https://doi.org/10.5281/zenodo.2541522.

Documentation

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

Installation

To get the latest version:

python -m pip install panels

Running in the browser with Pyodide

A WebAssembly wheel is published on PyPI for Pyodide 314 (CPython 3.14), such that panels runs in the browser, e.g. in JupyterLite or in a web page with Pyodide. NumPy, SciPy and matplotlib come with Pyodide, and micropip installs composites and structsolve from PyPI:

import micropip
await micropip.install("panels")

import io
from structsolve import lb
from panels.shell import Shell, load

s = Shell(a=0.6, b=0.4, stack=[0, 45, -45, 90], plyt=0.125e-3,
          laminaprop=(142.5e9, 8.7e9, 0.28, 5.1e9, 5.1e9, 3.0e9), m=10, n=10)
s.Nxx = -1.
eigvals, eigvecs = lb(s.calc_kC(), s.calc_kG(), silent=True)
print('critical load: Nxx = %1.2f N/m' % -eigvals[0])
s.results['eigvals'] = eigvals
s.results['eigvecs'] = eigvecs

# saving to memory, since a web page has no file system
buf = io.BytesIO()
s.save(buf)
data = buf.getvalue()

# offering the file for download
from js import Blob, Object, URL, document
from pyodide.ffi import to_js
blob = Blob.new(to_js([data]), to_js({'type': 'application/zip'},
                                     dict_converter=Object.fromEntries))
link = document.createElement('a')
link.href = URL.createObjectURL(blob)
link.download = 'plate.shell.zip'
link.click()

# loading the bytes back, e.g. from a file uploaded by the user
s2 = load(io.BytesIO(data))

In Pyodide the field outputs run serially, without OpenMP.

Saving and loading

Shell, StiffPanelBay and MultiDomain are saved with save() to a zip file with the inputs in JSON and the arrays, such as the results, in NumPy's .npy format, see panels.json_io. The files are independent of the Python version, readable outside Python and safe to load from untrusted sources, unlike pickle:

from panels.shell import load
s.name = 'plate'
s.save()              # writes plate.shell.zip
s2 = load('plate')

The pickle files saved by older versions are no longer loaded, since a pickle file can execute arbitrary code; one from a trusted source can be converted by loading it with pickle.load() and saving the object again.

History

See CHANGELOG.md for the details of each version.

• version 0.14.0 (2026-10-26)
  • BUG: the mass matrices used shell.rho in the closed-form kernels and the opposite sign of the laminate offset; they now use the moments of the density of the laminate, exact for per-ply densities
  • BUG: TSDT with a laminate offset, the transverse shear vanishes at the faces of the laminate, and the connections tie the rotations at the common reference surface
  • BUG: BladeStiff1D torsion with GJ = 4 bf D66 and coupling S1 with the condensed stiffness about the mid-plane of the flange; StiffPanelBay.calc_kA no longer skew-symmetrizes; one load per panel in the cylinder assembly functions; MultiDomain.force_out_plane and MultiDomain.plot fixed
  • Documentation of the theory in the docstrings, new page on the TSDT with an offset, validation notebook of the snap-through of a shallow circular arch (Fung and Kaplan, 1952)
  • Requires composites>=0.10.0 and structsolve>=0.9.3, whose residual is R = F_int - F_ext, as in the documentation of panels, and whose arc-length steps no longer double back along the path (since 0.9.2)
• version 0.13.11 (2026-10-16)
  • BUG: the analytical constitutive stiffness matrix of the CLPT models ignored force_orthotropic_laminate, inconsistently with the internal forces and the numerical matrices
  • Shell.ABD is calculated from the laminate and read-only; new attribute Shell.ABDnxny for a stiffness that is not that of the laminate, constant or one matrix per integration point, used by all the kernels, also within a MultiDomain, and by Shell.stress(), and checked to have symmetric blocks as any laminate; new Shell.integration_points()
  • Documentation: the derivation of the tangent stiffness matrix of the CLPT models with Donnell kinematics, formerly a LaTeX file, is part of the theory page of the tangent stiffness
  • Shell.calc_kTv() and MultiDomain.calc_kTv(): product of the tangent stiffness matrix by a vector without assembling it, for all the models, at the cost of one evaluation of the internal forces; used for the displacement control with prescribed displacements imposed exactly
  • The numerically integrated kernels evaluate the approximation functions once per integration point instead of in their innermost loops, with identical results: the non-linear matrices, the connections, the cohesive zone and the stiffeners are assembled several times faster
  • Cohesive zone 'SB_TSL': rules of the discretisation of the cohesive domain, tsl_resolution(), tsl_margin() and tsl_discretization(), and a warning when the cohesive domain is under-resolved; the decay rate of the opening is (k_o (1/D11_1 + 1/D11_2)/4)**0.25, (k_o/(2 D11))**0.25 for the symmetric DCB (the tag 0.13.9 used the rate of one panel on a rigid foundation, (k_o/(4 D11))**0.25), and the margin ahead of the crack front is 3 l_cz + 8/lambda
  • No upper limit for the number of Gauss-Legendre points of MultiDomain.calc_results() and MultiDomain.force()
  • calc_kw_tsl() requires k_i, tau_o and G1c
  • Maintenance: .gitignore without the runs of thesis_timing.py
  • Requires structsolve>=0.8.0
• version 0.12.0 (2026-10-14)
  • Prescribed displacements imposed exactly with the null-space method, MultiDomain(..., pd_method='null-space'), as inhomogeneous linear constraints c = Z c_r + c_p0 + lbd*c_p1: linear static, linear buckling, Newton-Raphson and arc-length analyses under displacement control, reactions MultiDomain.reaction_pd() and MultiDomain.reaction_resultant(), also for a single Shell; prescribed rotations; the penalty method remains the default
  • BUG: fkCld_xcte and fkCld_ycte placed the penalty stiffnesses of ku and kv at the rows and columns of w
  • Requires structsolve>=0.7.0
• version 0.11.1 (2026-10-08)
  • Saving and loading in a zip file with JSON and NumPy arrays, replacing pickle: Shell.save(), StiffPanelBay.save(), MultiDomain.save(), also to io.BytesIO, and panels.json_io
  • Pyodide support: await micropip.install("panels") in the browser, and the whole test suite runs in Pyodide in GitHub Actions
  • Requires composites>=0.9.21 and structsolve>=0.6.1
• version 0.11.0 (2026-10-07)
  • Follower (hydrostatic) pressure loads, Shell.add_pressure_load(..., follower=True), with the unsymmetric load stiffness kCfollower in Shell and MultiDomain
  • Theory documentation, verification tests and validation notebooks of the follower pressure
  • Interactive 3D plots of multi-domain assemblies with plotly, MultiDomain.plot3d()
  • Improved Cython and compiler flags
  • TSDT uses Laminate.Dts and Laminate.Fts, requires composites>=0.9.12 and structsolve>=0.5.0
• version 0.9.0 (2026-09-24)
  • Null-space connection, exact, new default, as alternative to penalty-based connection
• version 0.8.0 (2026-09-23)
  • FSDT and TSDT for plate and cylindrical shells
  • Sanders-Koiter for cylindrical shells
  • Shell and Multidomain connections supporting FSDT, TSDT and Sanders-Koiter kinematics
  • Many improvements in the documentation
  • New tests based on literature
  • BUG fixes on aerodynamic matrix of cylindrical shells and mass matrix of stiffener 1D
• version 0.7.1 (2026-09-21)
  • Fixed the cohesive zone of the MultiDomain connection 'SB_TSL', which could not capture the onset of failure of the DCB: tangential separation with the rotation of each panel, tractions with the uncorrected separation and separation correction as in the thesis of D'Souza (2024)
  • Consistent tangent of the cohesive zone, MultiDomain.calc_kT_TSL(), and faster assembly of the 'SB_TSL' secant stiffness
  • The kernels fkCSB11_dmg, fkCSB12_dmg and fkCSB22_dmg take the distances dt and db from each mid-surface to the interface
  • Validation of the cohesive zone against mode I DCB results in the literature, in notebooks/, with a convergence study
  • Fixed StiffPanelBay.save() with composites>=0.9.0
• version 0.6.21 (2026-09-18)
  • Fixed the mass of the base of BladeStiff2D, which was missing, and the section forces of MultiDomain.force(), which lacked the Jacobian
  • StiffPanelBay.calc_kA() sums the contribution of every panel, and calc_kA, calc_cA, calc_fext, save, get_size and uvw_stiffener of StiffPanelBay and Shell.calc_cA no longer raise
  • Shell.strain() and Shell.stress() work for a plate with r=None
  • Line coverage of the tests raised from 67 % to 92 %
  • Fixed StiffPanelBay.add_panel and the partial-domain integration of Shell.calc_kC, calc_kG and calc_kM
  • The limits Shell.x1, x2, y1, y2 of the integration domain default to None instead of the sentinels -1 and +1, which silently integrated the full domain for a limit at 1.0 and ignored a limit given on one side only; the old -1 now raises
  • Fixed the MultiDomain skin-base connections: the 'SB' penalty was a hard-coded mm-unit value, the top and bottom panels depended on their order in the assembly, and the connection dictionaries were modified
  • Penalty constants of each MultiDomain connection can be given with 'kt' and 'kr'
  • Reproduction of Stamatelos and Labeas with plate domains only in tests/multidomain/test_stamatelos_labeas_2023_multidomain.py, using every MultiDomain connection
  • Fixed the argument handling of the stiffness API: an invalid c or c_cte no longer reaches the nogil kernels, and a model that needs a radius raises instead of computing with r = 0
  • Reproduction of Stamatelos and Labeas (Computation 2023, 11, 110) in notebooks/stamatelos_labeas_2023.ipynb
  • Tangent stiffness matrix that is the exact derivative of the internal force vector, giving quadratic convergence to the Newton-Raphson method
  • Shell and MultiDomain can be passed directly to structsolve.Analysis, including the arc-length methods
  • Gauss-Legendre points and weights from scipy.special.roots_legendre
  • Modules of the models renamed without the _bardell suffix, and the legacy _bardell model names removed from panels.modelDB.db
  • Sphinx documentation with usage examples
• versions 0.5.x
  • Damaged multidomain connections
• up to version 0.3.x
  • Multidomain methods for pristine structures

License

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

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