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fix(output): the geometric curvature and section moment of a stretched line - #99
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… a stretched line The element energy is EI kappa^2/2 per unstretched length with kappa the curvature of the stretched line, so the recovered m, the conjugate of the tangent r', has the normal length EI kappa/(1 + e)^2. The node outputs read that length as the moment and divided it by EI, which put Curv 2e and BendMom e below the values they are defined by. The energy does not change under a rotation of the whole line, so the moment that balances the forces is r' x m, of length M = EI kappa/(1 + e). BendMom is now M, Curv is (1 + e) M/EI, the reciprocal of the bend radius, and StfMom the sleeve's share of M, with the stretch |r'| of the node's tangent. Tension, positions and forces do not change. Under OutputRecovery element the exact traces and EI x curvature stand on unsleeved elements. driver_curvature_stretch holds both against the elastica of this energy, strain swept to 1.2 %: the curvature error falls from 2.30 % to 0.006 % and the moment error from 1.15 % to under 0.001 %. driver_bend_stiffener takes the same elastica: the cable moment moves from 0.18 % to 0.01 % of the root value and the node positions from 0.6 to 0.03 mm. Signed-off-by: Jae Hoon Seo <jaehoon.seo@inha.ac.kr>
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Summary
The node outputs of a finite-EI line read the conjugate of the tangent as the moment. The element energy is
EI κ²/2per unstretched length, so that conjugate has normal lengthEI κ/(1+ε)², which putCurv2ε andBendMomε below their definitions.Curv(andCurvX,CurvY) is now the geometric curvature of the stretched line, the reciprocal of the bend radius.BendMomis the moment that balances the forces,EI κ/(1+ε), andStfMomthe sleeve's share of it.BendMom (1+ε) = EI × Curv, so the two no longer agree exactly; the difference is the strain.Result changes
Tension, positions and forces do not change. On the 12 example decks with finite-EI lines only the curvature and moment columns move, at most 0.024 % and 0.012 %, and at most 0.0026 % at peak curvature. On a soft line the change is about 2ε: 2.3 % at 1.17 % strain.
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New test
driver_curvature_stretchagainst the elastica of the same energy, at EA 2e6 (strain up to 1.17 %):CurvBendMomBendMomequals the spring moment at 1.00000 (0.98944 before).driver_bend_stiffener, its reference now on the same energy: cable moment 0.18 → 0.01 % of root, node positions 0.6 → 0.03 mm.OPENBLAS_CORETYPE=Prescott; no new compiler warnings.The CHANGELOG entry lands with the release batch.