  The aim of this article is to introduce a new methodology for constructing
morphings between shapes that have identical topology. This morphing is
obtained by deforming a reference shape, through the resolution of a sequence
of linear elasticity equations, onto the target shape. In particular, our
approach does not assume any knowledge of a boundary parametrization.
Furthermore, we demonstrate how constraints can be imposed on specific points,
lines and surfaces in the reference domain to ensure alignment with their
counterparts in the target domain after morphing. Additionally, we show how the
proposed methodology can be integrated in an offline and online paradigm, which
is useful in reduced-order modeling scenarii involving variable shapes. This
framework facilitates the efficient computation of the morphings in various
geometric configurations, thus improving the versatility and applicability of
the approach. The methodology is illustrated on the regression problem of the
drag and lift coefficients of airfoils of non-parameterized variable shapes.
