feigen

Stiffness eigenmodes of a single finite element: feigen (German for figs) solves K·φ = λφ, names every mode and plots it. Rigid-body and hourglass modes show up as zero eigenvalues.

fpm run -- -e qua -d 1          # 4-node quad, one cubature point
# cubature points: 1, zero modes: 5
#  mode    eigenvalue  match  name
#     1   0.00000E+00  100.%  translation x  u=(1,0)
#     ...
#     4   0.00000E+00  100.%  bending x  u=(xy,0)      <- hourglass
#     ...
# wrote qua_p1_lag_d1.png
fpm run -- -e hex -p 2 -f serendipity -o hex20.svg
fpm run -- --help

The -- separates fpm's options from feigen's; after fpm install, run feigen -e hex directly. The same steps are a library, module feigen:

q = cubature(CUB_QUA, 1)                    ! one point: reduced integration
s = shapefunc(q, 1)                         ! 4-node quadrilateral
K = stiffness(s, q, s%coords, material(2, 1._dp, 0.3_dp))
call modes(K, lambda, phi)
print *, count(lambda == 0)                 ! 5: 3 rigid-body + 2 hourglass
call name_modes(s%coords, lambda, phi, names, match)
call plot_modes(s, s%coords, lambda, phi, names, 0.2_dp, 'modes.svg')

Coding agents: the complete API reference is llms.txt, and contributor instructions are in AGENTS.md.

Features

• Lines, triangles, quadrilaterals, tetrahedra, hexahedra and wedges of any order, and serendipity quadrilaterals and hexahedra (fortran_shapefuncs).
• Any cubature degree, default 2*order, to show the hourglass modes of reduced integration (fortran_cubatures).
• Reference or distorted geometry: --coords reads one node per line.
• Isotropic linear elastic material; plane strain in 2D.
• Mode names (translation, rotation, expansion, tension/compression, shear, bending, torsion, warping, hourglass) with a match percentage; complete for linear line, quad and hex elements.
• One PNG, SVG or PDF with a subplot per mode, named after the element by default (qua_p1_lag_d2.png), drawn with fplotlib.

Conventions: degrees of freedom node by node, Voigt order xx, yy, zz, xy, xz, yz, and eigenvalues below 1e-10*max|λ| are exactly zero.

Installation

With fpm 0.13 or later:

git clone https://github.com/willklausler/feigen && cd feigen
fpm install --profile release       # puts feigen in ~/.local/bin

To use the module in your own fpm project, add to your fpm.toml:

[dependencies]
feigen = { git = "https://github.com/willklausler/feigen", tag = "v0.1.0" }

There is no make build: the dependencies stdlib, fplotlib and FLAP are fpm packages.

Documentation

• User manual and API documentation, generated with FORD (tools/doc.sh)
• llms.txt: the API reference and the program's options as one plain-text file, also the manual's source
• man ./man/feigen.3 (man/feigen.3)
• Example programs in example/ (fpm run --example --all) and the program in app/ (fpm run)

Compilers

CI tests every push on:

Compiler	Linux	macOS	Windows
gfortran 14	yes	yes	yes
gfortran 13	yes
ifx 2026	yes

nvfortran, LLVM flang 21 and ifx on Windows cannot build the dependencies; see llms.txt.

Why feigen?

Checking the eigenvalues of an element stiffness matrix is a standard test of a new element: the number of zero eigenvalues must equal the number of rigid-body modes, and any extra zero is a spurious (hourglass) mode [1, 2]. The test is usually a throwaway script. General finite element codes compute eigenmodes of an assembled model, usually vibration modes with a mass matrix, and do not name them. feigen runs the test for every element of fortran_shapefuncs from one command line, names the modes and draws them.

References

1. R. D. Cook, D. S. Malkus, M. E. Plesha, R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, 2002.
2. K.-J. Bathe, Finite Element Procedures, Prentice Hall, 1996.
3. D. P. Flanagan, T. Belytschko, A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, International Journal for Numerical Methods in Engineering 17(5), 679–706, 1981. doi:10.1002/nme.1620170504
4. O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 6th ed., Butterworth-Heinemann, 2005.

The stiffness matrix and the plane-strain material are standard [2, 4]. The hourglass fields follow Flanagan and Belytschko [3]. The code has not been independently reviewed.

Status and support

Version 0.1.0. The API may still change before 1.0; changes are recorded in CHANGELOG.md. Bug reports, questions and pull requests are welcome on GitHub; see CONTRIBUTING.md. To cite this package, use CITATION.cff.

License

MIT. The logo is cropped from a photo by Tijana Drndarski on Unsplash, under the Unsplash License.