Nonintrusive continuum sensitivity analysis for fluid applications

Abstract

Continuum Sensitivity Analysis (CSA) provides an analytic method of computing derivatives for structures, fluids and fluid-structure-interaction problems with respect to shape or value parameters. CSA does not require mesh sensitivity to calculate local shape sensitivity, thereby contributing to its computational efficiency. CSA involves solving linear sensitivity equations with the corresponding sensitivity boundary conditions. Spatial gradients of flow variables are required to construct the sensitivity boundary conditions, the accuracy of which significantly affects the flow shape derivatives. A method called Spatial Gradient Reconstruction (SGR) is used to accurately compute the flow spatial gradients that are further used to set the sensitivity boundary conditions. Here we present a nonintrusive CSA formulation for calculating the material derivatives of flow variables with respect to shape design parameters. Nonintrusive CSA means CSA without any modification to the black-box program used for the CFD flow analysis. Moreover, nonintrusive CSA is flow solver agnostic, in that, knowledge of the solver's discretization is not needed. An example of two-dimensional flow over a NACA0012 airfoil demonstrates the application of nonintrusive CSA to steady flow problems involving Euler (compressible inviscid) equations over unstructured grids with finite volume spatial discretization. Accuracy was studied by investigating convergence for a family of six high-quality grids, ranging from four thousand to four million cells. Factors such as the accuracy of the flow spatial gradients, flow sensitivity boundary conditions, weak implementation of the boundary conditions in the finite volume framework, and the use of an approximate flux Jacobian matrix, all of which affect the accuracy of the material derivatives, are discussed. A novelty of this work is that the sensitivity analysis is done nonintrusively using FLUENT and SU2 software to establish the use of black-box codes for obtaining flow derivatives using the CSA approach.