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.

AMD

Approximate Minimum Degree ordering — Amestoy, Davis & Duff 1996, SIAM J. Matrix Anal. Appl. 17 (4), 886–905. Heuristic node reordering that minimises fill-in for symmetric direct factorisation.

ARPACK

ARnoldi PACKage — Sorensen, Lehoucq, Yang, et al. 1996, SIAM J. Sci. Comput. 17 (4). Implicitly-restarted Arnoldi / Lanczos method for sparse eigenproblems. Default sparse eigen backend in scipy and femorph-solver.

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.

axisymmetric

2D revolution about an axis: every quantity depends on \((r, z)\) only, no \(\theta\) dependence. Halves the DOF count of the equivalent 3D model on rotor / vessel geometries. Cook 2002 §6.7.

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.

bandwidth

For a sparse matrix \(A\), the maximum \(|i - j|\) for which \(A_{ij} \neq 0\). Bandwidth-reducing orderings (Cuthill-McKee) were the predecessor to AMD / METIS for direct solvers. Cook 2002 §13.3.

BEAM2

2-node Hermite-cubic Euler-Bernoulli (or Timoshenko) beam. Cook 2002 Table 16.3-1. 6 DOF per node (3 translations + 3 rotations).

Cholesky

Triangular factorisation \(A = L L^\top\) for symmetric positive-definite \(A\). Half the operations of LU and no pivoting needed; the SPD direct solvers (CHOLMOD, Pardiso) ship a sparse Cholesky path.

CHOLMOD

Sparse supernodal Cholesky factorisation — Davis & Hager 2009, ACM Trans. Math. Softw. 35 (3). Part of SuiteSparse. Best-in-class single-process SPD Cholesky for medium-large problems; a reliable fallback when Pardiso is unavailable.

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.

condition number

For a symmetric positive-definite \(A\), the ratio \(\kappa_2(A) = \lambda_\text{max} / \lambda_\text{min}\) of extreme eigenvalues. Bounds the relative error amplification of a linear solve and the convergence rate of CG-family iterative solvers. See ill-conditioning.

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).

constitutive matrix

\(D\) (sometimes \(C\)) — the linear elastic stress-strain matrix \(\sigma = D \varepsilon\). 6 × 6 in 3D Voigt form; depends only on \(E\) and \(\nu\) for an isotropic material.

COO

Coordinate (triplet) sparse storage — (row_index, col_index, values). Easy to construct and accumulate during element-by-element FE assembly; converted to CSR before being handed to a solver.

CSC

Compressed Sparse Column — transpose layout of CSR. Preferred input format for many sparse direct solvers (CHOLMOD, UMFPACK).

CSR

Compressed Sparse Row storage — three arrays (values, col_index, row_pointer) representing a sparse matrix. Fast row-by-row matrix-vector multiplication. scipy.sparse.csr_array is the canonical femorph-solver assembly target.

cyclic faces

The two θ-bounding surfaces of a sector, paired one-to-one under the per-step rotation \(R(\alpha)\). Cyclic- symmetry constraints lock corresponding pairs to the same displacement scaled by \(e^{i 2\pi k / N}\).

cyclic-symmetry sector

The base wedge of an \(N\)-fold periodic rotor — one of \(N\) identical sectors that tile the full \(360°\) geometry under rotation. Modelling one sector with cyclic-phase BCs at the cyclic faces is the canonical reduction; femorph-solver wraps the workflow in CyclicModel.

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/....

DOF

Degree of freedom — a single scalar nodal displacement / rotation that the solver actually solves for. A 3D solid has 3 DOFs per node (UX, UY, UZ); a beam or shell has 6 (UX, UY, UZ, ROTX, ROTY, ROTZ).

drilling DOF

Out-of-plane rotation about the surface normal of a shell element. Standard QUAD4 shells (DKT) have 5 DOFs per node and no drilling rotation; coupling to a solid that carries a 6th rotational DOF requires an explicit constraint or a shell formulation that includes drilling.

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).

effective mass

Per-mode contribution to the total mass mobilised by an excitation: \(M_\text{eff,i}^{(d)} = (\Gamma_i^{(d)})^2 \cdot \varphi_i^\top M \varphi_i\). Sum across modes equals the total mass; cumulative-effective-mass curve drives the design-code “n-modes-needed” selection.

enhanced strain

Synonym for EAS.

fill-in

Non-zeros introduced into the Cholesky / LU factor that are not present in the original sparse matrix. Drives the memory footprint of a direct solve; minimised by node reordering (AMD / METIS) before factorisation.

free DOF

A DOF whose value is unknown — the solver computes it. Complement of prescribed DOF.

full integration

Quadrature rule that integrates the polynomial part of the element’s stiffness exactly on an undistorted reference element. HEX8 → 2 × 2 × 2 Gauss; HEX20 → 3 × 3 × 3 Gauss. Susceptible to shear locking and volumetric locking. Synonym for plain Gauss.

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 \(O(h^3)\).

generalised eigenvalue problem

\(A x = \lambda B x\) for two square matrices \(A, B\). The femorph-solver modal solver attacks the symmetric-definite case \(K \varphi = \omega^2 M \varphi\) with \(K, M\) SPD.

generalised plane strain

Plane-strain plus a uniform out-of-plane stretch — captures the average \(\varepsilon_z\) of a finite-length body loaded in its plane. ANSYS PLANE183 KEYOPT(3)=5.

harmonic index

For a cyclic-symmetric rotor with \(N\) sectors, the integer \(k \in \{0, 1, …, \lfloor N/2 \rfloor\}\) that labels the per-sector phase advance \(e^{i 2\pi k / N}\) of a cyclic eigenmode. Thomas 1979, Int. J. Numer. Methods Eng. 14 (1). Each \(k\) is a separate (smaller) eigenproblem on the base sector.

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.

hourglass

Spurious zero-energy mode of an under-integrated element — a deformation pattern that produces zero strain at every Gauss point because the rule has too few points. Named for the diagonal-stripe pattern on a 2 × 2 grid of reduced-integration QUAD4s. Belytschko, Liu, Moran 2014, Nonlinear Finite Elements, §8.4.

HRZ lumping

Hinton-Rock-Zienkiewicz mass-lumping scheme — Hinton, Rock, Zienkiewicz 1976, Earthquake Eng. Struct. Dyn. 4 (3), 245–249. Scales the diagonal of the consistent mass to match the element’s total mass. More accurate than row-summed lumping for higher-order elements; used by femorph-solver’s lumped mass path.

ill-conditioning

\(\kappa(A) \gg 1\). Common FE causes: large element- size ratios, near-incompressible materials without B-bar / EAS, near-singular constraint sets, mixed beam-and-solid coupling. A direct solve still produces an answer; an iterative solve may stall.

isoparametric

Element formulation that uses the same shape functions to map both the geometry (parametric → physical coordinates) and the displacement field (parametric → displacement). Bathe 1996, Finite Element Procedures, §5.3. The canonical FE element class.

Jacobian

Matrix \(J_{ij} = \partial x_i / \partial \xi_j\) mapping parametric coordinates to physical coordinates. Its determinant gives the volumetric scale factor for the Gauss-quadrature integration. Distorted elements have small or near-singular Jacobians and produce poor conditioning.

Lanczos

Tridiagonalisation algorithm at the heart of ARPACK’s symmetric eigenpath. Cornelius Lanczos, 1950, J. Res. NBS 45, 255–282. Builds a Krylov subspace \(\{v, Av, A^2 v, …\}\) and reduces \(A\) to a tridiagonal matrix whose extreme eigenvalues approximate the extreme eigenvalues of \(A\).

LOBPCG

Locally Optimal Block Preconditioned Conjugate Gradient — Knyazev 2001, SIAM J. Sci. Comput. 23 (2), 517–541. Block iterative eigensolver that finds the lowest \(k\) eigenpairs of a symmetric definite pencil without an explicit matrix factorisation; needs only a matrix-vector product and an SPD preconditioner.

LU

Triangular factorisation \(A = L U\) with \(L\) unit-lower-triangular and \(U\) upper-triangular. Solver of last resort for asymmetric matrices; SuperLU / UMFPACK are sparse implementations.

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.

MAC

Modal Assurance Criterion — Allemang & Brown 1982, Proc. IMAC. Scalar in \([0, 1]\) measuring the cosine-squared of two mode-shape vectors: \(\text{MAC}(\varphi_a, \varphi_b) = |\varphi_a^\top \varphi_b|^2 / (\varphi_a^\top \varphi_a)(\varphi_b^\top \varphi_b)\). Used to pair up modes between two solves (different mesh, different solver) when raw indices won’t line up.

METIS

Multi-level graph-partitioning library — Karypis & Kumar 1998, SIAM J. Sci. Comput. 20 (1). Used by Pardiso / MUMPS to compute nested-dissection orderings on large sparse matrices; beats AMD on bandwidth-bound problems.

modal assurance criterion

Synonym for MAC.

modal damping

Per-mode damping ratio \(\zeta_i\) applied directly in the modal-superposition formulation. More flexible than Rayleigh (can match measured per-mode ratios) but requires the eigendecomposition.

modal superposition

Approximate solution of the full FE system by projecting onto the lowest \(n_\text{modes}\) eigenmodes — see Harmonic (frequency-response) analysis. Cook 2002 §11.6. Cheaper than direct frequency response by one factorisation per frequency; loses accuracy when the truncated tail contributes significantly to the response.

mode participation factor

For mode \(i\) and excitation direction \(d\), \(\Gamma_i^{(d)} = \varphi_i^\top M r^{(d)} / (\varphi_i^\top M \varphi_i)\) where \(r^{(d)}\) is the unit influence vector along \(d\). Quantifies how strongly each mode is excited by a base motion in direction \(d\). Cook 2002 §11.5.

multi-frontal

Sparse-direct factorisation strategy that decomposes the elimination tree into “fronts” (small dense sub-matrices) and processes them in parallel. Used by Pardiso, MUMPS, SuperLU. Davis 2006, Direct Methods for Sparse Linear Systems, §6.

MUMPS

MUltifrontal Massively Parallel sparse direct Solver — Amestoy et al. 2001, SIAM J. Matrix Anal. Appl. 23 (1). Distributed-memory multifrontal direct solver; alternative to Pardiso on machines without MKL.

NAFEMS

National Agency for Finite Element Methods and Standards. Publishes a series of independently-sourced FE verification benchmarks (LE-1 … LE-10, FV-1 … FV-32, etc.) used as the cross-vendor reference suite. See Verification.

nested dissection

Recursive divide-and-conquer node ordering — George 1973, SIAM J. Numer. Anal. 10 (2), 345–363. Splits the matrix graph along a vertex separator and orders the separator last. Asymptotically optimal fill-in for 2D / 3D meshes.

nodal diameter

Engineering name for the harmonic index \(k\) — physically the number of stationary diameter lines the mode shape produces on the full rotor under the \(k = 0, 1, …, N/2\) family.

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.

parametric space

The element’s reference domain in canonical coordinates (e.g. the bi-unit cube \(\xi, \eta, \zeta \in [-1, 1]\) for HEX8). All shape functions are written in parametric space and pulled back to physical space through the Jacobian.

Pardiso

PARallel DIrect SOlver — multifrontal sparse direct linear solver, Schenk & Gärtner 2002, Future Generation Computer Systems 20 (3), 475–487. Shipped with Intel MKL. Fastest SPD direct backend on most x86 hardware; the linear() registry’s first choice when linear_solver="auto".

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.

pivoting

Permutation step in LU / Cholesky factorisation to maintain numerical stability. Required when the matrix has small or zero diagonal entries; SPD direct solvers (CHOLMOD, Pardiso) skip it for symmetric positive-definite inputs.

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 \(\nu \to 0.5\).

plane strain

2D simplification with \(\varepsilon_z = \varepsilon_{xz} = \varepsilon_{yz} = 0\) (no out-of-plane strain). Models a slice through a long prismatic body (tunnel, dam, pipe). Cook 2002 §3.5.

plane stress

2D simplification with \(\sigma_z = \sigma_{xz} = \sigma_{yz} = 0\) (no out-of-plane stress). Models thin plates loaded in their plane. Cook 2002 §3.5.

prescribed DOF

A DOF whose value is fixed by a Dirichlet boundary condition (clamped, pinned, applied displacement). Model.fix and the APDL D card register prescribed DOFs. Excluded from the free-DOF eigen / linear solve and recovered from the reaction-force back-substitution.

PRIMME

PReconditioned Iterative MultiMethod Eigensolver — Stathopoulos & McCombs 2010, ACM Trans. Math. Softw. 37 (2). Block Davidson family with adaptive method-switching. Tends to beat ARPACK on tightly-clustered spectra typical of cyclic-symmetric rotors.

profile

Sum of the row-bandwidth contributions in a sparse matrix. Storage cost of a banded direct factor scales with profile, not just bandwidth.

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.

Rayleigh damping

Proportional damping form \(C = \alpha M + \beta K\), with two scalars \(\alpha, \beta\). Rayleigh 1894, Theory of Sound, vol. 1. Convenient because it preserves modal orthogonality (per-mode damping ratio \(\zeta_i = \alpha / (2\omega_i) + \beta \omega_i / 2\)) and only needs one extra K + M assembly.

reduced integration

Quadrature with one fewer point per direction than full integration. Cures shear / volumetric locking but introduces hourglass modes that need stabilisation.

residual flexibility

Static-correction approximation for the modal-superposition truncation error. Add \((K^{-1} - \sum_i \varphi_i \varphi_i^\top / \omega_i^2) F\) to the modal sum to recover the static contribution of every mode above \(n_\text{modes}\) in one extra solve. Cook 2002 §11.7.

residual norm

For a candidate solution \(\tilde{u}\), the L₂ norm of \(r = F - K \tilde{u}\). Iterative-solver convergence criterion (CG, GMRES, MINRES); also the diagnostic of last resort for “did the direct solver actually solve it?”.

rigid-body mode

A non-trivial displacement field that produces zero strain — translation along an axis, rotation about an axis. An unconstrained 3D solid has 6 rigid-body modes (3 translations + 3 rotations) which appear in the modal spectrum at \(\omega = 0\) and need a small negative shift-invert shift to keep the factor non-singular.

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.

shift-invert

Spectral transformation that turns the generalised eigenproblem \(K \varphi = \omega^2 M \varphi\) into \((K - \sigma M)^{-1} M \varphi = (1/(\omega^2 - \sigma)) \varphi\), so an iterative solver (ARPACK, LOBPCG) converges toward eigenvalues near \(\sigma\) rather than the smallest in absolute value. Indispensable for grounded FE problems where the lowest physical mode sits well above zero (Saad 2011, Numerical Methods for Large Eigenvalue Problems, §4.3).

strain-displacement matrix

\(B = \partial N / \partial x\) — the matrix that maps nodal displacement to strain at a Gauss point. The K-matrix integrand is \(B^\top D B \, |J| \, w\). Bathe 1996 §5.3.4.

Sturm sequence

Counting algorithm that gives the number of eigenvalues of \(K - \sigma M\) below a shift \(\sigma\) from the signature of its LDL factor. Used as a sanity check that a modal solve returned every mode in the band; ARPACK can miss closely-clustered eigenvalues and Sturm-counting catches it.

SuperLU

Sequential / shared-memory unsymmetric direct solver — Demmel et al. 1999, SIAM J. Matrix Anal. Appl. 20 (3). Bundled with scipy and so the universal fallback when no faster backend is installed.

UMFPACK

Unsymmetric MultiFrontal sparse direct solver — Davis 2004, ACM Trans. Math. Softw. 30 (2). Part of SuiteSparse. Used for asymmetric / non-SPD problems and as a fallback when CHOLMOD’s symmetric path doesn’t apply.

Voigt notation

Compressed engineering convention for symmetric 3 × 3 stress / strain tensors as 6-vectors: \((\sigma_{xx}, \sigma_{yy}, \sigma_{zz}, \sigma_{xy}, \sigma_{yz}, \sigma_{xz})\). Engineering shear \(\gamma_{ij} = 2 \varepsilon_{ij}\) is used for strain so \(\sigma : \varepsilon = \sigma^\top \varepsilon\) (with the engineering-shear convention).

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.

Wilson-Q6

Wilson, Taylor, Doherty & Ghaboussi 1973 incompatible-modes formulation for QUAD4 — predecessor to EAS, adds two parametric quadratic bubbles to the displacement field and condenses them at element level. ANSYS PLANE182 KEYOPT(1)=2.

zero-energy mode

General term for a non-rigid-body deformation that produces zero strain energy. hourglass modes are zero-energy modes of reduced-integration elements; rigid- body modes are zero-energy modes of every element by construction.