An adjoint variable method for sensitivity analysis of non‐linear elastic systems

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.