An ill-posed inverse problem of a wing with locally given velocity data and its analysis

Abstract

Using locally given vertical velocity data around a wing, an inverse formulation is presented to solve a lifting problem. The inverse problem is expressed by a Fredholm integral equation of the first kind. In this paper, the kernel of the integral equation gives a Hilbert–Schmidt integral operator, and therefore the occurrence of ill-posedness in the sense of stability cannot be avoided in a normal topology. This difficulty is solved by using the regularization method for ill-posed problems. A composition mapping is introduced so that local velocity data can be available for this inverse problem. In this paper, the ill-posed inverse problem of a wing is studied using the Landweber–Friedman's regularization method within the framework of linear potential theory. A numerical example demonstrates that only with locally given velocity data is the regularization method accurate and suitable for the present physical problem of an inverse mathematical formulation. Therefore, the lifting problem can be solved by using a locally given fluid velocity instead of a wing geometry.