Structural shape sensitivity analysis : relationship between material derivative and control volume approaches

The material derivative and control volume approaches for structural shape design sensitivity analysis are presented, analyzed, and compared. Starting with a continuum formulation and a general response functional needing sensitivity analysis, the two approaches are derived. It is shown that the final design sensitivity expression for one approach can be obtained from the final expression for the other. Thus, the two approaches are theoretically equivalent. Discretizations of the continuum expressions are presented, and it is shown that the two approaches can lead to different implementations for numerical calculations. A unified interpretation is developed in which the explicit design variations (partial derivatives with respect to the design variables) of the internal and external forces are the major calculations needed to implement the design sensitivity analysis. The discretized forms of the continuum sensitivity expressions are also compared with the ones obtained by starting with the discretized model ab initio. This comparison shows that these two approaches lead to similar expressions for numerical calculations. Therefore, the exact same procedures can be used for computer implementations for both of the approaches. The presented analyses give insights for implementation of the design sensitivity analysis theory with the finite element analysis programs.