High-order Chebyshev finite element for efficient stability assessment of tunnel structures

This paper presents an advanced locking-free high-order Chebyshev Finite Element Method (CFE) for accurately assessing the stability of rectangular tunnels in incompressibility constraints. Building on the strengths of the Spectral Element Method (SEM) and Legendre Spectral Finite Elements (LSFEs), this approach leverages Chebyshev polynomials to construct high-order shape functions that provide kinematic flexibility to enforce the incompressibility constraint. These shape functions enable rapid convergence through p-refinement, improving accuracy without increasing the number of elements, while maintaining computational efficiency for complex geotechnical problems. The stability assessment employs an upper bound limit analysis, where CFE is used to approximate the displacement field, formulated as a second-order cone program (SOCP) and solved through numerical optimization techniques. Tailored for 2D stability evaluation, this approach can efficiently handle varying boundary conditions and complex geometries through element mapping and high-order Chebyshev polynomials, ensuring robust and precise results. Validation against established benchmarks, including analytical solutions and standard finite element methods, highlights the reliability and accuracy of this method, making it a valuable tool for practical engineering applications. The study targets two geotechnical scenarios: uniform and depth-dependent undrained shear strength conditions. This high-performance framework also provides a robust foundation for future research in geotechnical stability analysis.