| The sensitivity of the objective functional, required in gradient-based optimization algorithms, is often hard to find explicitly. One method to determine the sensitivities called "Fixed Basis Function" Finite Element analysis assumes that during sensitivity analysis the finite element nodes will not change or move with the varying domain. Instead, they are accommodated by the boundary conditions on the adjacent nodes; therefore it evaluates the stationary derivatives of the objective function. In this work computer implementation of Fixed Basis Function plus Material Derivative approach, Complex Step and Finite Difference method is presented in one and two dimensions for a comparison study. The Fixed Basis Method produces very accurate results compared to the exact stationary derivative for the beam and converges for the plate problem. It is also very cost effective. Except for the fixed basis method, all other methods include the effect of domain change in the sensitivity and therefore produce convective derivatives. |
