Sensitivity analysis of linear elastic systems using domain parameterization and a mixed mutual energy principle

A unified approach for explicit structural design sensitivity analysis of linear elastic systems is presented. First-order sensitivity expression involving the complete set of design variable, including shape design parameters, are derived for general response functional. Both direct differentiation and adjoint variable methods are developed in a general, variational context. A domain parameterization method, related to the mixed Eulerian-Lagrangian kinematic description, is introduced for the treatment of shape variation problems. This approach is an alternative to the material derivative methods presented elsewhere in the literature. The adjoint variable derivation is based on a mixed energy principle, the mutual Hu-Washizu energy principle, which is formally stated and proved. The design sensitivity derivation based on this principle unifies and extends several of the adjoint variable methods found in the literature. The derivations presented here are extended to nonlinear systems in a subsequent paper. The sensitivity expressions are valid for use with force, displacement or mixed approximate solutions of the elasticity equations. Other formulations are generally only consistent with displacement solution methods. Numerical implementation of the design sensitivity methods is discussed and illustrative examples are included.