Static analysis

Linear elastostatic problem \(\mathbf{K}\, \mathbf{u} = \mathbf{f}\), driven by Model.solve.

Algorithm

1. Assemble the global stiffness \(\mathbf{K}\) and force \(\mathbf{f}\) from the per-element ke and apply_force calls (see Global assembly).

2. Eliminate Dirichlet DOFs. The free / constrained partition gives \(\mathbf{K}_{ff}\, \mathbf{u}_f = \mathbf{f}_f - \mathbf{K}_{fc}\, \mathbf{u}_c\) — the reduced system the backend factorises (see Boundary-condition elimination).

3. Factor and solve \(\mathbf{K}_{ff}\) through the registered linear backend (see Linear-solver backends). Backend dispatch order: Pardiso → CHOLMOD → MUMPS → UMFPACK → SuperLU, first-installed wins.

4. Reaction recovery — \(\mathbf{r}_c = \mathbf{K}_{cf}\, \mathbf{u}_f + \mathbf{K}_{cc}\, \mathbf{u}_c - \mathbf{f}_c\). Populates StaticResult.reaction.

5. Stress / strain recovery is lazy — the static result returns a DOF-indexed displacement plus a thin accessor protocol that calls Model.eel(u) / compute_nodal_stress on demand.

Public API

• femorph_solver.Model.solve() — the main entry point.

• femorph_solver.solvers.static.StaticResult — the return type, with displacement / reaction / free_mask arrays.

• femorph_solver.io.static_result_to_grid() — scatter the DOF vector onto (n, 3) UX/UY/UZ point data on a copy of the input grid.

Verification cross-references

The verification gallery exercises every static-solve path:

• Single-hex uniaxial tension — Hooke’s law + Poisson check — single-hex Hooke’s law.

• Cantilever tip deflection — Euler-Bernoulli closed form — Euler-Bernoulli cantilever.

• Cantilever under uniformly distributed load — Euler–Bernoulli closed form — uniform-distributed load.

• Cantilever beam under a tip moment — tip moment, parabolic deflection.

• Propped cantilever under uniformly-distributed load — statically-indeterminate beam.

• Simply-supported beam under a central point load — simply-supported beam.

• Clamped-clamped beam under a central point load — clamped-clamped beam.

• Lamé thick-walled cylinder under internal pressure — Lamé thick-cylinder.

• Clamped square plate under uniform pressure (NAFEMS LE5) — NAFEMS LE5.

• Simply-supported plate under uniform pressure (Navier series) — Navier plate.

• NAFEMS LE1 — elliptic membrane (plane stress) — NAFEMS LE1 plane-stress.

Implementation: femorph_solver.solvers.static.

References

• Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002) Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, §2.10 (Dirichlet partitioning).

• Bathe, K.-J. (2014) Finite Element Procedures, 2nd ed., §3.4 (boundary-condition imposition), §8 (linear elastic static analysis).

• Saad, Y. (2003) Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, §3 (sparse direct solvers).