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Recovery of the rod cross section shape

Vladislav V. Kravchenko, Sergii M. Torba, Alexander O. Vatulyan

TL;DR

This work presents a direct NSBF-based approach to the inverse problem of recovering a rod’s cross-section $F(x)$ from vibration data. By recasting the problem as a Sturm–Liouville inverse problem and exploiting NSBF representations, the method identifies the first coefficient $g_0(x)$, from which $F(x)$ is obtained via $F(x)=F(0)ig(g_0(x)+1ig)^2$. The algorithm combines end-point coefficient recovery, spectral data completion, and a second linear system to recover $g_0(x)$ across the domain, yielding a practical, numerically stable route to shape reconstruction. The approach is validated on multiple cross-section profiles and demonstrates robustness to data density and moderate noise, with performance enhanced by a stabilizing functional $R_N$ and careful truncation. This framework connects inverse elasticity problems to NSBF-based reconstruction and offers a direct, computation-lean path to determining rod geometry from frequency-domain measurements.

Abstract

A direct method for solving the inverse problem of determining the shape of the cross section of a rod is proposed. The method is based on Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. The first coefficient of the representation is sufficient for the recovery of the unknown function. A system of linear algebraic equations for finding this coefficient is obtained. The proposed method leads to an efficient numerical algorithm.

Recovery of the rod cross section shape

TL;DR

This work presents a direct NSBF-based approach to the inverse problem of recovering a rod’s cross-section from vibration data. By recasting the problem as a Sturm–Liouville inverse problem and exploiting NSBF representations, the method identifies the first coefficient , from which is obtained via . The algorithm combines end-point coefficient recovery, spectral data completion, and a second linear system to recover across the domain, yielding a practical, numerically stable route to shape reconstruction. The approach is validated on multiple cross-section profiles and demonstrates robustness to data density and moderate noise, with performance enhanced by a stabilizing functional and careful truncation. This framework connects inverse elasticity problems to NSBF-based reconstruction and offers a direct, computation-lean path to determining rod geometry from frequency-domain measurements.

Abstract

A direct method for solving the inverse problem of determining the shape of the cross section of a rod is proposed. The method is based on Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. The first coefficient of the representation is sufficient for the recovery of the unknown function. A system of linear algebraic equations for finding this coefficient is obtained. The proposed method leads to an efficient numerical algorithm.

Paper Structure

This paper contains 8 sections, 1 theorem, 54 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem of determining the shape of the cross section of the rod
  3. Reformulation of Problem A
  4. Solution of Problem B
  5. Representations of solutions
  6. Linear algebraic systems for coefficients at the end points
  7. Algorithm for solving Problem A
  8. Numerical examples

Key Result

Theorem 4.1

The solutions $\varphi(\rho,x)$ and $S(\rho,x)$ and their derivatives with respect to $x$ admit the following series representations where $j_{k}(z)$ stands for the spherical Bessel function of order $k$ (see, e.g., AbramowitzStegunSpF). For every $\rho\in\mathbb{C}$ all the series converge pointwise. For every $x\in\left[ 0,\pi\right]$ the series converge uniformly on any compact set of the com

Figures (8)

  • Figure 1: The original rod cross section area $F$ from Example \ref{['ExParabolic']} together with the recovered ones using different values of $N_1=N_2=N$ in \ref{['equations 1']} and clean or noisy data. On this figure we illustrate the importance of restricting the number of unknowns taken on Step \ref{['AlgStep3']}. The potential was better recovered from the noisy data by taking smaller values of $N_1=N_2=1$ than from the clean data with larger values of $N_1$ and $N_2$.
  • Figure 2: Illustration to the work of Step \ref{['AlgStep3']} of the proposed algorithm for the inverse problem in Example \ref{['ExParabolic']}. Values of the functions $Q_N$ and $R_N$ given by \ref{['FunctionalOrig']} and \ref{['Functional opt']} are presented for both clean and noisy data
  • Figure 3: Relative errors of the recovered rod cross section area $F$ from Example \ref{['ExParabolic']} using different values of $N_1=N_2=N$ in \ref{['equations 1']} and clean or noisy data.
  • Figure 4: Relative errors of the recovered rod cross section area $F$ from Example \ref{['ExExp']} using "clean" values $f(\omega)$ given in different sets of points $\Omega_j$, $j\in\{1,2,3,4\}$. The parameter $N$ for each plot indicates the value of $N_1=N_2=N$ taken in \ref{['equations 1']} and obtained by minimizing the function $R_N$\ref{['Functional opt']}.
  • Figure 5: Same as Figure \ref{['Ex2RelErrClean']}, but "noisy" data was used.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 4.1: KNT
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Example 6.1
  • Example 6.2
  • Example 6.3
  • Example 6.4