AN ADJOINT VARIABLE METHOD FOR SENSITIVITY ANALYSIS OF NON-LINEAR ELASTIC SYSTEMS

SUMMARY An adjoint variable method for design sensitivity analysis of non-linear elastic systems is presented. The method uses domain parameterization and a mutual form of the Hu-Washizu energy principle, and extends results reported in a recent work for linear elastic systems to non-linear elasticity. Non-linearities due to finite deformations and non-linear, hyperelastic constitutive models are considered. In contrast to other methods for non-linear sensitivity analysis, the present formulation can be applied with force, displacement or mixed approximate solution methods. The mutual energy expression used in the adjoint sensitivity derivation is developed from a non-linear extension of the Hu-Washizu energy functional and yields a linear governing equation for the adjoint system. This has important ramifications for the computational cost of a sensitivity analyses of non-linear systems: excluding the cost of determining the response of the system, the cost of a sensitivity analysis for a non-linear system is essentially the same as that for a linear system. Finite element implementation of the resulting sensitivity expressions is discussed, and two numerical examples are presented. The first example involves large deformations of a Mooney-Rivlin body, while the second involves design sensitivity analysis for mixed solution methods.