Complex variable sensitivity methods for finite element analysis

Complex variable differentiation methods offer many advantages over traditional finite differencing. These advantages include higher accuracy and greater stability. Numerical differentiation can be used to perform sensitivity analyses for engineering design problems. Shape sensitivity analysis of finite element models has become very common, since it can be used to perform design optimization. Finite differencing methods can be very difficult to implement in conjunction with finite element analysis due to the meshing problems it can create. For this reason, the complex variable differentiation methods, complex Taylor series expansion and Fourier differentiation, may be better suited than finite differencing for shape sensitivity analysis. One-dimensional and two-dimensional finite element codes have been written in Matlab, and shape sensitivity analysis using both complex variable methods and traditional finite differencing have been conducted. It was observed that for almost all cases, the accuracy of the numerical solution limits the accuracy of the numerical derivatives. This means that the increased accuracy of the complex variable methods is limited by the error in the solution. Complex Taylor series expansion still has several advantages over finite differencing, including a reduced number of sample points, and no re-meshing requirements.