Sensitivity analysis of any hyperelastic evaluation functions coupled with adjoint method and automatic differentiation

This study introduces a new sensitivity analysis method for the topology optimization of a static hyperelastic material, which combines the adjoint variable method with automatic differentiation (AD). The adjoint variable method, frequently used in sensitivity analysis, requires mathematical formulations. Therefore, any changes in the design problem require reformulating the sensitivity analysis and updating the calculation program. The proposed method allows for the calculation of design sensitivities without being tied to specific evaluation functions, constitutive laws, or interpolation methods. This method effectively addresses the considerable memory requirements often associated with AD. To showcase the versatility of the proposed approach, we assessed both the compliance and the maximum von Mises stress of the second Piola–Kirchhoff stress tensor. We examined two hyperelastic materials: St. Venant-Kirchhoff, Neo-Hookean, and Mooney–Rivlin. For broader applicability, we used the discrete material optimization (DMO) method to address multimaterial problems, evaluating the adaptability in the interpolation of material properties based on the design variables. Through numerical examples, we validated the sensitivity analysis, analyzed the computational time and memory usage, and confirmed the efficacy of the proposed method. Examples involving two-dimensional problems highlight the practical application of this method in topology optimization.