Coupled problem of the inverse design and constraint optimization Árpád Veress a,⇑ , Attila Felföldi a , Tamás Gausz a , László Palkovics b a Department of Aircraft and Ships, Budapest University of Technology and Economics, Sztoczek u. 6, J ép. 4. em. 426, H-1111 Budapest, Hungary b Knorr-Bremse R&D Center Budapest, Major u. 69, H-1119 Budapest, Hungary article info Keywords: CFD Inverse design method Optimization Wing profile abstract Two dimensional inviscid compressible flow solver has been implemented, validated and extended for completing inverse design based optimization problems. Euler equations are used for modelling basic physics. The standard cell centred finite volume method has been applied with Roe’s approximated Riemann solver, MUSCL (Monotone Upstream Schemes for Conservation Laws) approach and Mulder limiter. Stratford’s separation pre- diction method with constrained SQP (Sequential Quadratic Programming) optimization is implemented to evaluate the optimum pressure distribution at strong adverse pressure gradient flow conditions. A particular inverse design method is developed to complete wall surface modification till the optimum target pressure distribution is reached by means of inverse, wall modification and direct algorithms. The direct solver is validated and the cou- pled system of the inverse design method with constraint optimization is tested over the NACA 65-410 airfoil. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Today, the benefit of Computational Fluid Dynamics (CFD) has been used intensively in design process, by which the number of experimental cycle can be substituted or reduced significantly and so the cost, capacity and time consumption of developments can be reduced. Furthermore, the virtual topology allows higher flexibility in model generation by means of parameterization, which gives an especial opportunity to use CFD tools in automatic design and optimization problems. Beside the developments of the central core of the fluid dynamics solvers [14,18], the different optimization techniques, coupled with CFD, are also under intensive research [19]. In case of direct optimization techniques, an attempt has been made to find the optimal solution. They typically utilize some sort of search technique (gradient-based optimizer), stochastic based algorithms (e.g. evolutionary strategies, genetic algorithms), artificial neural networks or some other optimization methods. These procedures can be computationally expensive because several flow solutions must be calculated to specify the direction of deepest descent, fitness of individuals in the population, etc. in order to determine the shape changes [21]. Furthermore, the required number of flow solutions increases dramatically with the number of design variables. In case of a specific set of the inverse design-type methods, the geometry modification is based on the prescribed set of the pre-defined variables at the wall by simple, fast and robust algorithms, which makes them especially attractive amongst other optimization techniques. The wall modification can be completed within much less flow solutions for inverse design techniques than for direct optimization methods. Hence, the inverse design methods typically being much more computa- tionally efficient and they are very innovative to be used in practice. The main drawback of inverse design methods is that the designer should create target (optimum in a specific sense) pressure or velocity distributions that should correspond to the design goals and meet required aerodynamic characteristics. However, it can be difficult to specify the required pressure 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.110 ⇑ Corresponding author. E-mail addresses: averess@rht.bme.hu (Á. Veress), laszlo.palkovics@knorr-bremse.com (L. Palkovics). Applied Mathematics and Computation 219 (2013) 7115–7126 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc