Finite Element Sensitivity for Large Deformation Problems Using the Complex Variable Finite Element Method

The complex variable finite element method (ZFEM) has been enhanced to compute sensitivity derivatives of output variables of interest with respect to shape, material properties, and load parameters in large deformation elasticity. ZFEM performs finite element operations employing complex variables and yields complex variable outputs. The real component of the complex variable outputs provides typical response variables, while the imaginary component contains their sensitivities with respect to an input parameter of interest. In this study, two developments have been added to ZFEM, particularly: (a) a complex-variable element for solving large deformation elastic problems based on the updated Lagrangian formulation and (b) a complex-variable hyperelastic material routine implementing the Mooney-Rivlin model. These developments were incorporated within ABAQUS incremental-iterative procedure for the solution of nonlinear elastic systems. ZFEM’s capability to compute sensitivities within nonlinear geometric analyses was demonstrated by comparing ZFEM sensitivities to those obtained through direct differentiation of analytical solutions for well-known solid mechanics problems under large deformation. First, sensitivities with respect to material, loading, and shape parameters were obtained for a cantilever beam subjected to a concentrated load at the free end. Then, ZFEM was employed to determine the influence of the Mooney-Rivlin material constants on the stress levels experienced by a hyperelastic cube undergoing axial loading. In both numerical examples, ZFEM results were in excellent agreement with the analytical solution. These new capabilities of ZFEM can facilitate the sensitivity analysis of metals under manufacturing processes (e.g. rolling), crack propagation in hyperelastic incompressible materials, development of constitutive models for rubber-like materials (e.g. soft tissues and skin), and many other.