The aim of this article is to introduce a new methodology for constructing morphings between shapes that have identical topology. The morphings are obtained by deforming a reference shape, through the resolution of a sequence of linear elasticity equations, onto every 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 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.

翻译：本文旨在提出一种构建具有相同拓扑结构形状间变形映射的新方法。该变形映射通过求解一系列线性弹性方程，将参考形状变形至每个目标形状而获得。特别地，本方法无需预设任何边界参数化信息。此外，我们展示了如何在参考域中对特定点、线和面施加约束，以确保变形后它们与目标域中的对应部分对齐。另外，我们论证了所提方法如何融入离线与在线计算范式，这对涉及可变形状的降阶建模具有重要意义。该框架能够高效计算不同几何构型下的变形映射，从而提升了方法的通用性与适用性。本文以非参数化可变形状翼型的升力与阻力系数回归问题为例，对所提方法进行了演示。