A Priori Node Placement Using Chebyshev Polynomial Zeros for Non-Uniform Finite Element Meshing

This study proposes a non-uniform meshing strategy for finite element (FE) discretization aimed at improving mechanical accuracy under controlled nodal budgets in structural analyses with localized field gradients. Unlike quasi-uniform meshing, the proposed approach uses a priori node placement based on the zeros of Chebyshev polynomials. Their natural clustering near the endpoints of [−1, 1] is mapped onto the physical domain to generate a non-uniform distribution by construction, without adaptive refinement or iterative remeshing. The main contribution therefore lies not in the Chebyshev points themselves, but in their use as a deterministic, solution-independent rule for placing the vertices of low-order FE meshes. Three meshing strategies are compared: quasi-uniform, graded, and Chebyshev-based meshes. The graded mesh deliberately concentrates nodes in predefined critical regions, whereas the Chebyshev-based mesh follows a predefined analytical placement rule. Performance is assessed against converged fine-mesh FE reference solutions using relative errors in displacement and von Mises stress, local and peak-stress errors, mesh-quality indicators, stiffness-matrix conditioning, and computational cost. For the two-hole plate at Ntot=2200 the relative displacement error decreases from 1.28×10−2 for the quasi-uniform mesh to 8.48×10−3 for the Chebyshev-based mesh, while the graded mesh reaches 7.75×10−3. Overall, the Chebyshev-based strategy improves displacement accuracy relative to quasi-uniform meshing in the investigated configurations, although the geometrically graded mesh generally achieves lower errors and improvements in stress accuracy are not systematic.