Adjoint Sensitivity Theory for the Finite Element Method

Abstract

Sensitivity analysis is an important aspect of many model studies. The objective is to determine how sensitive model results are to perturbations in the model input or system parameters. The input of concern generally includes boundary conditions, forcing functions, model parameters and for transient analyses, initial conditions. The sensitivity of the model state variables can be investigated or alternatively the sensitivity of a selected performance measure or response function may be of interest. The performance measure which may be written as

$$P = P(\left\{ {{\kern 1pt} \phi } \right\},{\kern 1pt} {\kern 1pt} \left\{ {{\kern 1pt} \alpha } \right\})$$

(1)

is a function of the system state { φ} and the system parameters { α}. For some problems, P may be independent of some or all of the parameters { α}. In this paper { φ} and { α} represent column vectors of finite element nodal state variables and system parameters respectively.