Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to changes in problem domain or shape. A finite element displacement formulation is adopted and title point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, the method is referred to as a "fixed basis function" finite clement shape sensitivity analysis. This approach avoids the requirement of explicit or approximate differentiation of finite clement matrices and vectors and the difficulty or errors resulting from such calculations. Effectively, the sensitivity to boundary shape change is determined exactly; thus, the accuracy of the solution sensitivity is dictated only by the finite element mesh used. The evaluation of sensitivity matrices and force vectors requires only modest calculations beyond those of the reference problem finite element analysis; that is, certain boundary integrals and reaction forces on the reference location of the moving boundary are required. In addition, the formulation provides the unique family of element domain changes which completely eliminates the inclusion of grid sensitivity from the shape sensitivity calculation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C-0 problem; the extension to three-dimensional problems is straight-forward.