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Supplemental material outlines finite-difference methods for analyzing pipeline responses during tunnel interactions, aiming for reproducibility.
This supplemental material provides detailed finite-difference discretization formulas, joint-node derivations, virtual-node treatments, and global matrix definitions supporting the time-dependent pipeline response analysis presented in the associated manuscript entitled “Longitudinal Time-dependent Response of Existing Pipelines Triggered by Tunnel Down-crossing Considering Pipeline-soil Interface Gap Evolution.” The material is intended to supplement the theoretical formulations in the main manuscript and to improve the reproducibility of the proposed analytical framework.
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Zhang et al. (2026) studied this question.
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