Complex variable methods for shape sensitivity of finite element models

Shape sensitivity analysis of finite element models is useful for structural optimization and design modifications. Complex variable methods for shape sensitivity analysis have some potential advantages over other methods. In particular, for first order sensitivities using the complex Taylor series expansion method (CTSE), the implementation is straightforward, only requiring a perturbation of the finite element mesh along the imaginary axis. That is, the real valued coordinates of the mesh are unaltered and no other modifications to the software are required. Fourier differentiation (FD) provides higher order sensitivities by conducting an FFT analysis of multiple complex variable analyses around a sampling radius in the complex plane. Implementation of complex variable sensitivity methods requires complex variable finite element software such that complex nodal coordinates can be used to implement a perturbation in the shape of interest in the complex domain. All resulting finite element outputs such as displacements, strains and stresses become complex and accurate derivatives of all finite element outputs with respect to the shape parameter of interest are available. The methodologies are demonstrated using two-dimensional finite element models of linear elasticity problems with known analytical solutions. It is found that the error in the sensitivities is primarily defined by the error in the finite element solution not the error in the sensitivity method. Hence, more accurate sensitivities can be obtained through mesh refinement.