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Inverse shape optimization problems and application to airfoils. (English) Zbl 1167.49321
Summary: We consider a set of parameterized planar arcs $$(x(t), y(t)) (0 \leq t \leq 1)$$, satisfying certain smoothness, regularity and monotonicity conditions (in particuler $$x(t)$$ is monotone increasing, and $$y(t)$$ positive and unimodal), and a functional $$J (y)$$ involving an adjustable weighting function $$\omega (t)$$ and a positive constant $$\alpha > 1$$. We first prove the strict convexity of the functional for $$\alpha \geq 2$$. Under the less stringent condition $$\alpha > 1$$, we derive the stationarity condition and the formal expression for the Hessian, and prove that if a point exists at which the functional is stationary w.r.t. variations in $$y = y(t)$$, for fixed $$x = x(t)$$, then it is unique and realizes a global minimum; the functional is then unimodal. We also observe that the stationarity condition (Euler-Lagrange equation) is an integral differential equation depending only on the arc shape and not on the parameterization per se, which gives the variational problem a certain intrinsic character. Then, we solve the inverse problem: given an admissible parameterized arc, we construct a smooth weighting function $$\omega (t)$$ for which the stationarity condition is satisfied, thus making the functional unimodal, and derive certain asymptotics. A numerical example pertaining to optimum-shape design in aerodynamics is computed for illustration.

MSC:
 49N45 Inverse problems in optimal control 49Q10 Optimization of shapes other than minimal surfaces 76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing 35B37 PDE in connection with control problems (MSC2000) 49M37 Numerical methods based on nonlinear programming