Circular Element Finite Element Method (CE-FEM): A Purely Geometric, Area-Preserving Continuum Mechanics Framework

Abstract:This manuscript introduces the Circular Element Finite Element Method (CE-FEM), a novel computational mechanics framework that replaces classical polynomial shape functions and Gauss quadrature with an exact geometric circle-to-ellipse transformation. Founded on the axiomatic invariant that a material circle (in 2D) deforms into a rotated ellipse under load with exact area conservation (πab≡πR2πab≡πR2, maintained to 10−1610−16 double-precision machine tolerance), the method extracts continuum strains directly from cardinal nodal finite differences. By reinterpreting zero-energy hourglass modes as physical bending displacement fields, an analytical gradient stiffness matrix [Ke]grad=π(D11+D33)t16(v1v1T+v2v2T)[Ke]grad=16π(D11+D33)t(v1v1T+v2v2T) is derived, elevating convergence from O(h)O(h) to O(h2)O(h2) without numerical integration. This upload includes the theoretical derivation alongside fully functional MATLAB implementations for solid elasticity, incompressible Venturi pipe dynamics, aerodynamic wing flows (NACA 4412), hydraulic force multiplication (Pascal's Law), and thermal smoke plumes.