Mesh Movement for a Discrete-Adjoint Newton-Krylov Algorithm for Aerodynamic Optimization

A grid movement algorithm based on the linear elasticity method with multiple increments is presented. The method is relatively computationally expensive but is exceptionally robust, producing high-quality elements even for large shape changes. It is integrated with an aerodynamic shape optimization algorithm that uses an augmented adjoint approach for gradient calculation. The discrete-adjoint equations are augmented to explicitly include the sensitivities of the mesh movement, resulting in an increase in efficiency and numerical accuracy. This gradient computation method requires less computational time than a function evaluation and leads to significant computational savings as dimensionality is increased. The results of the application of these techniques to several large deformation and optimization cases are presented. Nomenclature A = coordinates of the airfoil surface E = modulus of elasticity f = external forces G = coordinates of the interior grid nodes J, F = objective functions i = increment number K = stiffness matrix L = Lagrangian l = length of a side of a triangle n = number of increments P = potential energy Q = flow variables R = radius of a circumscribed circle R = flow residual r = residual of the grid movement equations s = semiperimeter of a triangle u = element displacements V = element volume X = design variables = boundary , = adjoint vector = radius of an inscribed circle = stress tensor = element shape quality = spatial domain Subscripts e = belonging to an element t = belonging to the entire system jQ = Q is held constant in the differentiation = subtriangular element inside a quadrilateral Superscripts ^ = known variable on the boundary T = transpose