Local boundary value problem on element based-<i>posteriori</i> error estimator for finite element mesh refinement of axial deformation of friction beams

Purpose The variable cross-section and boundary conditions of beams make deformation more complex, and conventional analytical methods fail; the solutions of numerical methods based on meshes, such as finite element method, have solution errors, especially in domains with varying cross-sections or complex boundaries, which eliminates solution smoothness and leads to significant errors. Therefore, the quality of mesh generation needs to be optimized. Design/methodology/approach In this study, an h -version adaptive finite element algorithm of second-order ordinary differential equation for axial deformation of friction beams with variable cross-section and boundary is proposed, which uses the posteriori error estimation based on the strategy for division of sub-problems and solution of boundary value problems. The detailed procedure and pseudo code of adaptive finite element algorithm are further provided, to form the scheme of the local boundary value problem on element based- posteriori error estimator for finite element mesh refinement of axial deformation of friction beams. Findings Some typical numerical examples are designed to verify the effectiveness and reliability of the proposed h -version adaptive finite element algorithm. The cases of friction beams with uniform, linear cross-sections and constant, linear, sinusoidal, parametric, arctangent displacement boundaries are considered to validate the robustness of the proposed algorithm. This algorithm uses dense mesh in domains with larger solution errors to suit the displacement changes, and the final results may satisfy the pre-specified error tolerance. Once the displacement violently changes caused by the more complex cross-section function and boundaries, more computation steps are required to reduce the solution errors. Originality/value The proposed algorithm performs good generality and applicability, which can be extended to two- and three-dimensional problems, using the local boundary value problem on element based- posteriori error estimator to estimate the solution error within the element and perform adaptive element subdivision.