Discrete aerodynamic sensitivity analysis on decomposed computational domains

Abstract

A new scheme is presented for solving the large sparse unsymmetric systems of linear equations which arise from the application of the aerodynamic sensitivity equation to large two-dimensional as well as three-dimensional design optimization problems. Performance comparisons between a generalized minimum residual (GMRES) method and a direct solution method are performed on a single computational domain. Results show that, due to the use of preconditioning, the GMRES method requires almost the same memory allocation as that required by the direct method. Moreover, it is found that for many right-hand sides of the above systems, the GMRES method is significantly less efficient than the direct method. Since large systems cannot be solved by direct methods on a single domain, due to the computer memory limitations, the computational domain in the new scheme is divided into small manageable ones and each one is solved separately. In addition to the savings in memory requirements, this scheme can be applied easily to complex-geometry problems which cannot be represented by single computational grids. The computational performance of this scheme is assessed by considering the flow over a transonic airfoil. The sensitivity equation is solved for three cases where the computational domains are represented by different numbers of subdomains. Identical results are obtained in all cases without any visible effect due to the subdivisions of the computational grids. This indicates the scheme's high accuracy in solving the above system on decomposed computational domains.