Transient thermomechanical sensitivity analysis using a complex-variable finite element method

A complex-variable transient thermomechanical element was developed and used to compute highly accurate sensitivities of the thermal and stress time-dependent responses of a thick-walled cylinder with temperature-dependent material properties with respect to fifteen material, shape, and loading parameters using the complex Taylor series expansion method (CTSE). The analysis was conducted by employing a complex-variable user element within the Abaqus commercial finite element software, herein called ZFEM, using real and imaginary nodal degrees of freedom (DOF). The real coordinates denote the spatial coordinates, as typical, and the imaginary nodal DOF denote the perturbation of the geometry for shape sensitivities. Upon completion, the imaginary DOF hold the partial derivatives of the temperature and the displacements with respect to the parameter of interest. Sensitivities of stresses were obtained using the displacement sensitivities. The sensitivities were verified by a comparison with finite difference estimates using the built-in Abaqus elements. The numerical results demonstrate the superior accuracy, ease of use, and reduced run-time of ZFEM compared to the finite difference method. The conclusions show that ZFEM can be utilized to obtain time-dependent sensitivities with high accuracy and precision for transient thermomechanical analyses that are computationally more efficient in comparison to finite difference methods.