Second-order design sensitivity analysis using diagonal hyper-dual numbers

Although sensitivity analysis provides valuable information for structural optimization, it is often difficult to use the Hessian in large models since many methods still suffer from inaccuracy, inefficiency, or limitation issues. In this context, we report the theoretical description of a general sensitivity procedure that calculates the diagonal terms of the Hessian matrix by using a new variant of hyper-dual numbers as derivative tool. We develop a diagonal variant of hyper-dual numbers and their arithmetic to obtain the exact derivatives of tensor-valued functions of a vector argument, which comprise the main contributions of this work. As this differentiation scheme represents a general black-box tool, we supply the computer implementation of the hyper-dual formulation in Fortran. By focusing on the diagonal terms, the proposed sensitivity scheme is significantly lighter in terms of computational costs, facilitating the application in engineering problems. As an additional strategy to improve efficiency, we highlight that we perform the derivative calculation at the element-level. This work can contribute to many studies since the sensitivity scheme can adapt itself to numerous finite element formulations or problem settings. The proposed method promotes the usage of second-order optimization algorithms, which may allow better convergence rates to solve intricate problems in engineering applications.