Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to c hanges in problem domain or shape. A finite element displacement formulatio n is adopted and title point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, t he method is referred to as a "fixed basis function" finite clement shape s ensitivity analysis. This approach avoids the requirement of explicit or ap proximate differentiation of finite clement matrices and vectors and the di fficulty or errors resulting from such calculations. Effectively, the sensi tivity to boundary shape change is determined exactly; thus, the accuracy o f the solution sensitivity is dictated only by the finite element mesh used . The evaluation of sensitivity matrices and force vectors requires only mo dest calculations beyond those of the reference problem finite element anal ysis; that is, certain boundary integrals and reaction forces on the refere nce location of the moving boundary are required. In addition, the formulat ion provides the unique family of element domain changes which completely e liminates the inclusion of grid sensitivity from the shape sensitivity calc ulation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C-0 problem; the extension to three-dime nsional problems is straight-forward.