A continuous adjoint formulation with emphasis to aerodynamic-turbomachinery optimization

This paper summarizes progress, recently made in the Lab. of Thermal Turbomachines of NTUA, on the formulation and use of the continuous adjoint methods in aerodynamic shape optimization problems. The basic features of state of the art adjoint methods and tools which are capable of handling arbitrary objective functions, cast in the form of either boundary or field integrals, are presented. Starting point of the presentation is the formulation of the continuous adjoint method for arbitrary integral objective functionals in problems governed by arbitrary, linear or nonlinear, first or second order state pde's; the scope of this section is to demonstrate that the proposed formulation is general without being restricted to aerodynamics. It is noticeable that, regardless of the type of functional (field of boundary integral) the expressions of its gradient with respect to the design variables include boundary integrals only. Thus, the derived adjoints can be used with either structured or unstructured grids and there is no need for repetitive remeshing or computation of field integrals which increase the CPU cost and deteriorate the computational accuracy. Then, the presentation focuses on aerodynamic shape optimization problems governed by the compressible fluid flow equations, numerically solved through a time-marching formulation and an upwind discretization scheme for the convection terms. Two design problems, namely the inverse design of a 2D cascade at inviscid flow conditions (used as a test bed for the assessment of three descent algorithms based on the same gradient information) and the design optimization of a 3D peripheral compressor cascade for minimum viscous losses are presented. For the latter, the flow is turbulent and the field integral of entropy generation, recently proposed by the same authors, is used as objective function