Accelerated nonlinear finite element method for analysis of isotropic hyperelastic materials nonlinear deformations

In this work, an approximate Jacobian matrix is proposed based on the total Lagrangian formulation of Finite Element Method for isotropic hyperelastic materials. The approximate Jacobian matrix can take the place of the exact Jacobian matrix in the Newton–Raphson method to avoid frequent construction and factorization of the Jacobian matrix. A new Quasi–Newton method employing the approximate Jacobian matrix is developed to significantly improve the convergence rate. The proposed method was tested on three examples with the combinations of three hyperelastic material models and three element types. The results show that the proposed method is more efficient than ABAQUS and CALCULIX (an open-source FEM software package) on all of the tests without loss of accuracy. It is up to 100 times faster than the traditional Quasi–Newton method, and at least 2.5 times faster than ABAQUS.