A model reduction technique for the direct FE2 method using equivalent quadratic-RVEs with a radial basis function (RBF) surrogate model

In this paper, equivalent quadratic-representative volume elements (EQ-RVEs) in the direct FE 2 framework are proposed to efficiently reduce the computational cost for the nonlinear multiscale analysis of hyperelastic materials. In the offline stage, training loads are applied to discretized RVEs to obtain boundary displacement and force snapshots of EQ-RVEs for the construction of a radial basis function (RBF) surrogate model. The RBF surrogate model provides explicit forms of the internal force vector and the tangent stiffness matrix of EQ-RVEs, ensuring stable and robust convergence in Newton-Raphson iterations for solving nonlinear equations. EQ-RVEs are implemented as a user element subroutine (UEL) to link the reduced micro-scale elements to macro-scale elements in the commercial software Abaqus through multi-point constraints (MPCs). Benchmark tests indicate that the proposed EQ-RVE direct FE 2 method preserves the accuracy of full FE 2 simulations while reducing computation time up to 99% compared to single-scale models, offering a practical and scalable tool for the nonlinear multiscale analysis of composite structures.