Annular and Disk Finite Elements for Two-Dimensional Elastostatic Problems

This paper presents a new annular finite element and circular disk super element for solving two-dimensional (2-D) elastostatic problems. Addressing the geometric approximation errors inherent in standard polynomial-based elements, the proposed annular element employs an exact geometric mapping based on the polar coordinate system, ensuring a theoretically consistent representation of circular boundaries. The displacement field is interpolated using standard Q8 shape functions to maintain compatibility. Furthermore, to enhance computational efficiency for large-scale particulate systems, a disk super element is developed by applying static condensation to eliminate internal degrees of freedom. This technique significantly reduces the global system size without compromising the high-fidelity internal stress analysis. The accuracy of the annular element is rigorously verified against analytical solutions for thick-walled cylinders and disks with stress concentrations. Finally, the efficiency and physical consistency of the disk super element are demonstrated through the simulation of large-scale disk packing problems, highlighting its potential for modeling granular composites and microstructures.