Discrete Adjoint Formulation for Continuum Sensitivity Analysis

Abstract

Continuum Sensitivity Analysis (CSA) is an approach for calculating analytic derivatives. A direct CSA formulation is advantageous for computing derivatives of many state variables or performance functions. An adjoint formulation of CSA is beneficial for computing derivatives with respect to many design variables, although adjoint CSA boundary conditions are often problematic. For the proposed continuum-discrete hybrid adjoint approach, the adjoint variable is introduced after discretization which simplifies boundary conditions. The sensitivity boundary conditions for the hybrid CSA are posed in terms of the continuum state variables. Thus, the hybrid adjoint formulation of CSA results in design derivatives that are as accurate as those obtained from direct CSA, in addition to making the analysis efficient for the case of large number of design variables. Two test cases, first of an axial bar and second of a cantilever beam modeled with solid elements, illustrate how the hybrid adjoint formulation inherits the benefits of the direct and adjoint CSA formulations. This is also the first application of CSA to obtain design derivatives nonintrusively using three dimensional (3-D) Spatial Gradient Reconstruction (SGR) method for 3-D solid elements.