Glossary#

Concise definitions of the integration / formulation / verification terms that recur across the gallery and the reference chapters. Every term cites the textbook chapter or seminal paper where the definition comes from, so a term cross-reference in the docs lands somewhere the reader can audit.

asymptotic#: A closed-form result valid in a limit (slender, thin, infinite, …); the FE captures the next-order correction beyond the limit (Timoshenko shear, Mindlin transverse shear, finite-width finite-thickness, …) and so converges to a slightly different number than the closed form on a thick / wide / finite-domain mesh.
B-bar#: Selective-reduced integration on the dilatational (volumetric) component of the strain field — Hughes 1980, International Journal for Numerical Methods in Engineering 15 (9), 1413–1418. Replaces the volumetric \(\partial u_i / \partial x_i\) averaged across each element. Cures volumetric locking but does not cure shear locking on bending-dominated problems.
BEAM2#: 2-node Hermite-cubic Euler-Bernoulli (or Timoshenko) beam. Cook 2002 Table 16.3-1. 6 DOF per node (3 translations + 3 rotations).
closed form#: An exact analytical solution to an idealised version of the problem (Euler-Bernoulli, Kirchhoff, Lamé, Saint-Venant, …) used as the verification reference. Always carries an asymptotic regime (slender beam, thin plate, infinite domain, small strain) that the FE truthfully captures the corrections to — so a small \(O(h)\) discrepancy at the FE side is not a bug.
consistent mass#: Mass matrix obtained by integrating \(\rho\, N_i\, N_j\) over the element with the same shape functions used for the stiffness — produces a non-diagonal \(M\). Required for accurate higher-mode frequency recovery (Cook 2002 §11.2).
devpi#: The internal Python package mirror at http://devpi.femorph-devpi.svc.cluster.local/root/femorph/+simple/. Used as the sole index by every CI workflow on the cluster — proxies PyPI under /root/pypi/+f/... and serves the company’s own private wheels under /root/femorph/+f/....
EAS#: Enhanced Assumed Strain — Simo and Rifai 1990, Int. J. Numer. Methods Eng. 29 (8), 1595–1638. Augments the strain field with element-internal “enhanced” modes that are condensed out at element level, so the global DOF count is unchanged. Cures both shear and volumetric locking on regular hex meshes; partially cures shear locking on moderately distorted meshes (compare the Cook’s-membrane gallery example).
enhanced strain#: Synonym for EAS.
Galerkin error#: For a smooth solution \(u\) and a \(p\)-degree polynomial element, \(\|u - u^h\|_{L^2} = O(h^{p+1})\) and \(\|u - u^h\|_{H^1} = O(h^p)\). Strang & Fix 2008 An Analysis of the FEM §3.7. HEX8 (\(p = 1\)) gives \(O(h^2)\) displacement-norm convergence; HEX20 (\(p = 2\)) gives \(O(h^3)\).
HEX20#: 20-node serendipity quadratic hexahedron (8 corners + 12 mid- edge). ANSYS catalogue name SOLID186. Higher-order accuracy at the cost of 2.5× DOF — needed to close through-thickness bending error on plate-like geometries without relying on integration tricks.
HEX8#: 8-node trilinear isoparametric hexahedron. ANSYS catalogue name SOLID185. Default workhorse element for structural elasticity in femorph-solver. See HEX8 — 8-node trilinear hexahedron for the formulation.
lumped mass#: Diagonal approximation of the consistent mass that puts each element’s mass on its own nodes only. Cheaper to invert (explicit dynamics, eigenvalue power-iteration). Trades accuracy for diagonal structure — Cook 2002 §11.4.
octant symmetry#: Three-plane reflective symmetry exploited on a 1/8 sphere / 1/8 cube model. Pin \(u_x = 0\) on \(x = 0\), \(u_y = 0\) on \(y = 0\), \(u_z = 0\) on \(z = 0\). Reduces a 3D pressure-vessel problem to a single octant FE model.
patch test#: A smallest-mesh-that-could-still-fail test of element correctness — apply prescribed displacements that should produce uniform strain and check that the solver recovers exactly that (Irons 1972 — Cook 2002 §6.4). See Irons constant-strain patch test.
plain Gauss#: Full-tensor 2 × 2 × 2 (or 3 × 3 × 3 for HEX20) Gauss-Legendre quadrature applied to the standard isoparametric stiffness integral. Exact for the polynomial part of the integrand on an undistorted hex. Suffers shear locking on bending problems (Hughes 1987 The Finite Element Method §2.7; Cook 2002 Concepts and Applications of FEA §6.13) and volumetric locking as \(\nu \to 0.5\).
QUAD4#: 4-node bilinear isoparametric quadrilateral, 2D plane-stress / plane-strain element. ANSYS catalogue name PLANE182.
quarter symmetry#: Two-plane reflective symmetry on a 1/4 model. Standard for plate / shell / cylinder benchmarks.
Saint-Venant#: Adjective for a quasi-uniform stress / strain regime far from load-application boundaries (Saint-Venant 1855). In FE practice: away from concentrated-load nodes, the field smoothes back to the closed-form distribution. See Cantilever Saint-Venant torsion — rectangular cross-section.
selective-reduced integration#: Umbrella term for any quadrature scheme that uses a lower-order rule on a specific strain component (deviatoric vs volumetric, transverse-shear vs in-plane) than on the rest. B-bar is the canonical example for the volumetric component. See Cook 2002 §6.13 + §6.14.
shear locking#: Spurious stiffening of low-order isoparametric elements under bending. The root cause: the parametric basis cannot represent the linear transverse-shear distribution that pure bending demands without simultaneously introducing a non-zero shear strain at the Gauss points (Hughes 1987 §2.7; Cook 2002 §6.6). Symptoms vanish as the mesh refines, but error is \(O(1)\) on coarse meshes — see Shear-locking demonstration — HEX8 integration variants.
volumetric locking#: Spurious stiffening as Poisson’s ratio \(\nu \to 0.5\) when the kinematics constrain the dilatational mode. The constant-pressure component of the strain field has no FE basis to live in (Cook 2002 §6.6; Hughes 1980). Cured by B-bar or EAS.