Multicomplex variable differentiation in probabilistic analysis and finite element models of structural dynamic systems
The multicomplex-step differentiation method has many advantages when compared to finite differencing and the complex-step differentiation method. By using multicomplex variables instead of real variables, arbitrary order derivatives are obtained from the coefficients of the now multicomplex function, provided that the functions are holomorphic in the variable(s) of interest. Since this is a relatively new method, it has not yet been explored in probabilistic analysis and structural dynamic systems of finite elements. For this reason, a new multicomplex Score Function and multicomplex Infinitesimal Perturbation Analysis for higher order probabilistic sensitivities calculations is presented. In addition, a multicomplex Newmark-beta method for computing any order derivative from a dynamic system of finite elements is given. The multicomplex-step derivatives from the multicomplex finite element analysis are then used to perform a probabilistic finite element analysis. Several numerical examples are used to illustrate each of the novel methodologies.