Recoverable Identifier

arXiv:2605.04884 · detector doi_compliance · incontrovertible · 2026-05-19 14:04:33.019903+00:00

Paper page Integrity report arXiv Try DOI

Evidence text

Zhu, Y., Wang, Y., Zhang, X., and Kang, Z., “A New Form of Forbidden Frequency Band Constraint for Dynamic Topology Optimization,”Structural and Multidisciplinary Optimization, Vol. 65, No. 4, 2022. doi:10.1007/s00158- 022-03220-1. A. Real Formulation of the Eigenvalue Problem The complex eigenvalue problem Eqs. (6) and (4) can be decomposed into real and imaginary components. The resulting system of equations is written in terms of real numbers only as ˆr(v;w 0,x) =   ˆrr ˆri ˆrm ˆrp   =   Jqr−λrqr +λiqi Jqi−λrqi−λiqr q⊺ r qr +q ⊺ i qi−1 e⊺ kqi   ,v=   qr qi λr λi   .(34) The subscriptsrandidenote the real and imaginary parts of the eigenvalue equation, respectively; the subscriptsmandpdenote the magnitude and the phase residual, respectively. More details on the eigenvalue problem setup are provided by He et al. [29]. B. Top-Level Adjoint Details The coefficient matrix for the top-level adjoint equation Eq. (16) is ∂ˆr ∂v ⊺ =   J−λrIλ iI−q r qi −λiI J−λrI−qi −qr 2q⊺ r 2q⊺ i 0 0 0e ⊺ k 0 0   ⊺ .(35) Whenf=λ r, we have∂f/∂v= [0,0,1,0]. When a function of the eigenvector is used, such as Eq. (13), the partial derivative is∂f/∂v= [˜q⊺ r, ˜q⊺ i,0,0]. Forf=λr, the analytic adjoint solution is given by Eq. (18) in the main text. If˜uis any (unnormalized) left eigenvector, then enforcingu∗q=−1is achieved by the scaling u= ˜u −˜u∗q.(36) Similarly, whenf=λi (relevant when certain frequencies need to be avoided [79]), the adjoint solutio

Evidence payload

{
  "printed_excerpt": "Zhu, Y., Wang, Y., Zhang, X., and Kang, Z., \u201cA New Form of Forbidden Frequency Band Constraint for Dynamic Topology Optimization,\u201dStructural and Multidisciplinary Optimization, Vol. 65, No. 4, 2022. doi:10.1007/s00158- 022-03220-1. A. Real ",
  "reconstructed_doi": "10.1007/s00158-022-03220-1.A.Real",
  "ref_index": 79,
  "resolved_title": null,
  "verdict_class": "incontrovertible"
}