Abstract
In the paper the shape optimization problem for the static, compressible Navier-Stokes equations is analyzed. The drag minimizing of an obstacle immersed in the gas stream is considered. The continuous gradient of the drag is obtained by application of the sensitivity formulas derived in the works of one of the co-authors. The numerical approximation scheme uses mixed Finite Volume - Finite Element formulation. The novelty of our numerical method is a particular choice of the regularizing term for the non-homogeneous Stokes boundary value problem, which may be tuned to obtain the best accuracy. The convergence of the discrete solutions for the model under considerations is proved. The non-linearity of the model is treated by means of the fixed point procedure. The numerical example of an optimal shape is given.
Highlights
One of the main applications of the theory of compressible viscous flows [27] is the optimal shape design in aerodynamics
The theory leading to the formulas for the shape derivatives is based on a series of papers by P.I
Find a solution, to the following boundary value problem posed in the variable domain Ωε = B \ Sε, for the shape parameter ε ∈ (−δ, δ) with δ > 0: Re
Summary
One of the main applications of the theory of compressible viscous flows [27] is the optimal shape design in aerodynamics. For the approximation of solutions the method of Finite Volumes is used, what required supplementing the results of [10],[8] with some additional analysis. This analysis constitutes the main novelty of the paper. Compressible Navier-Stokes, drag minimization, Finite Volume method, Finite Element method. Let B ⊂ R2 be a doubly connected, bounded hold all domain with a smooth boundary Σ = ∂B.
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