A Karhunen-Loève Least-Squares Technique for Optimization of Geometry of a Blunt Body in Supersonic Flow

Accuracy and efficiency of the Karhunen-Loève (KL) method is compared to a pseudospectral method employing global Lagrange interpolating polynomials for a two-dimensional linear heat conduction problem and for the supersonic flow of an inviscid, calorically perfect ideal gas about an axisymmetric blunt body. For the heat conduction problem, efficiency and accuracy of a KL Galerkin model, pseudospectral approximation, and a second order finite difference approximation are compared for two sets of boundary conditions: one with an infinite number of modes and a second with a finite number of modes. For both the infinite and finite mode boundary conditions, the KL Galerkin model achieves an accuracy consistent with the underlying pseudospectral solver in as few as five modes. Not including the cost of sampling the design space and building the KL model, the KL Galerkin model is an order of magnitude faster than the pseudospectral method for the finite mode boundary condition, two orders of magnitude faster than the pseudospectral method for the infinite mode boundary condition, and five orders of magnitude faster than the finite difference for both types of boundary conditions at an accuracy level of 10^{-4} as measured against a separation of variables solution. In the supersonic blunt body problem, the bow shock is fitted, and the Euler equations are solved in generalized coordinates using both a KL least-squares model and a pseudospectral method. Not including the cost of sampling the design space and building the KL model, the KL least-squares method requires almost half the CPU time as the pseudospectral method to achieve the same level of accuracy. Single variable design problems are solved for both the heat conduction problem and the supersonic blunt body problem using the pseudospectral solver and the KL model; in both problems, the KL model optimal design predictions are within the expected level of accuracy of the KL models. For both problems, a response surface which is a quadratic polynomial fit of the design problem objective function versus the geometric design variable is built from a design space sampling of three pseudospectral solutions. For the heat conduction problem, the KL model is significantly more accurate than the response surface over most of the design space, while for the blunt body design problem, the KL model is slightly less accurate than the response surface. Since the KL model requires more CPU time, the single variable blunt body design problem posed here offers no advantage over a response surface in terms of CPU time. Nevertheless, the accuracy of the response surface is strongly dependent on the problem, whereas the KL method achieves a consistently low level of error for both the heat conduction and supersonic blunt body problems considered.