Novel first and second order numerical differentiation techniques and their application to nonlinear analysis of Kirchhoff–Love shells

Numerical simulation based on FEM/IGA methods is the standard approach for the approximated solution of applied physical problems. In this context, the differentiation of the numerical counterpart of mechanical fields is required. Moreover, the differentiated function can have a complicated shape, depend on many variables and change within the process. Many state-of-the-art numerical differentiation methods are not suitable for this kind of applications and the common way is to exploit analytical differentiation. Thus, an on-the-fly differentiation method is desirable particularly when the process is complicated and when new mechanical models are under development. In this paper, a new method is proposed for a precise computation of the gradient and Hessian. This method has been applied to nonlinear analysis of Kirchhoff–Love shells, which can be considered as an appropriate test bench to prove the reliability in relevant physical context. Numerical experiments show the advantages of the proposed techniques with respect to standard approaches.