Higher-order finite element analysis of FG-TPMS sandwich thin-walled box beams using Al-Salam-Chihara Ritz polynomials

The growing demand for lightweight structures with tailorable mechanical properties has driven the development of advanced cellular materials. Motivated by this demand, a higher-order finite element formulation is developed within the Ritz method for the static and free vibration analyses of thin-walled box beams made of functionally graded triply periodic minimal surface (FG-TPMS) materials, in which the material properties vary continuously through the wall thickness according to the TPMS density distribution. Three TPMS sheet-based architectures, namely Primitive (P), Gyroid (G), and I-graph–wrapped package graph (IWP), are considered, along with laminated composite face sheets. A higher-order thin-walled beam theory describes the kinematics of the beam, and the governing equations are derived using the variational principle in conjunction with the Ritz approximation. Al-Salam-Chihara polynomials are employed as admissible functions, and their performance is systematically evaluated in terms of numerical stability, convergence rate, and computational efficiency. The parameter q = 0.5 is identified as optimal, yielding lower condition numbers and faster computation than the classical Chebyshev polynomials. A fixed thin-walled beam formulation is also introduced to ensure accuracy in thick beam configurations; neglecting this correction results in errors of up to 13.5% in clamped–clamped beams. A unified hp -refinement framework incorporating h-, p-, and hp- strategies is presented, with p -refinement demonstrating significantly faster convergence than h -refinement for smooth domains. Numerical results reveal that the P-type TPMS architecture provides the highest torsional resistance, while shear deformation contributions reach up to 78.5% for the IWP architecture under clamped–clamped boundary conditions. The effects of core thickness ratio and TPMS pattern type on natural frequencies and displacements are also discussed in detail. These findings provide practical guidance for selecting suitable TPMS architectures and material gradation patterns to improve shear resistance and structural efficiency in lightweight sandwich thin-walled structures.