Structural Design Sensitivity Analysis: Continuum and Discrete Approaches

Abstract

A unified approach for structural design sensitivity analysis involving both shape and sizing variables is presented. Starting with a continuum formulation and a general response functional needing sensitivity analysis, the direct variation and adjoint approaches are derived. Discretization of the continuum expressions for the two approaches is presented and the numerical implementation aspects are discussed. The discretized forms of the continuum sensitivity expressions are compared with the ones obtained by starting with the discretized model ab initio. This comparison shows that the two approaches give similar discretized expressions for numerical calculations. Therefore, exactly same procedures can be used for computer implementation of both the approaches. The continuum approach, however, gives certain insights that would not be possible with only the discrete approach. The presented analyses and insights lead to a unified view point for numerical implementation of design sensitivity analysis which is quite straightforward with existing or new finite element analysis codes. The explicit design variations (partial derivatives with respect to the design variables) of the internal and external nodal forces are the major calculations needed to implement the design sensitivity analysis. An implementation scheme is suggested that is quite general and simple needing minimal programming.