Reliability-Oriented Modeling of Bellows Compensators: A Comparative PDE-Based Study Using Finite Difference and Finite Element Methods

Yerzhan Y. Sarybayev, Doszhan Y. Balgayev (), Denis Y. Tkachenko, Nikita V. Martyushev (), Boris V. Malozyomov, Baurzhan S. Beisenov and Svetlana N. Sorokova
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Yerzhan Y. Sarybayev: Department of Technological Machines and Equipment, Institute of Energy and Mechanical Engineering, Satbayev University, Almaty 050013, Kazakhstan
Doszhan Y. Balgayev: Department of Technological Machines and Equipment, Institute of Energy and Mechanical Engineering, Satbayev University, Almaty 050013, Kazakhstan
Denis Y. Tkachenko: Department of Mechanical Engineering, Institute of Energy and Mechanical Engineering, Satbayev University, Almaty 050013, Kazakhstan
Nikita V. Martyushev: Department of Information Technologies, Tomsk Polytechnic University, 634050 Tomsk, Russia
Boris V. Malozyomov: Department of Electrotechnical Complexes, Novosibirsk State Technical University, 630073 Novosibirsk, Russia
Baurzhan S. Beisenov: Department of Technological Machines and Equipment, Institute of Energy and Mechanical Engineering, Satbayev University, Almaty 050013, Kazakhstan
Svetlana N. Sorokova: Department of Mechanical Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia

Mathematics, 2025, vol. 13, issue 21, 1-30

Abstract: Bellows compensators are critical components in pipeline systems, designed to absorb thermal expansions, vibrations, and pressure reflections. Ensuring their operational reliability requires accurate prediction of the stress–strain state (SSS) and stability under internal pressure. This study presents a comprehensive mathematical model for analyzing corrugated bellows compensators, formulated as a boundary value problem for a system of partial differential equations (PDEs) within the Kirchhoff–Love shell theory framework. Two numerical approaches are developed and compared: a finite difference method (FDM) applied to a reduced axisymmetric formulation to ordinary differential equations (ODEs) and a finite element method (FEM) for the full variational formulation. The FDM scheme utilizes a second-order implicit symmetric approximation, ensuring stability and efficiency for axisymmetric geometries. The FEM model, implemented in Ansys 2020 R2, provides high fidelity for complex geometries and boundary conditions. Convergence analysis confirms second-order spatial accuracy for both methods. Numerical experiments determine critical pressures based on the von Mises yield criterion and linearized buckling analysis, revealing the influence of geometric parameters (wall thickness, number of convolutions) on failure mechanisms. The results demonstrate that local buckling can occur at lower pressures than that of global buckling for thin-walled bellows with multiple convolutions, which is critical for structural reliability assessment. The proposed combined approach (FDM for rapid preliminary design and FEM for final verification) offers a robust and efficient methodology for bellows design, enhancing reliability and reducing development time. The work highlights the importance of integrating rigorous PDE-based modeling with modern numerical techniques for solving complex engineering problems with a focus on structural integrity and long-term performance.

Keywords: bellows compensator; stress-strain state; partial differential equations; finite difference method; von Mises criterion; buckling analysis; axisymmetric shell (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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