In this article, we focus on a generative design algorithm for area-to-point (AP) conduction problems in a Lagrangian framework. A physically meaningful continuous area to point path solution is generated through an adaptive growth procedure, which starts from the source point and extends spreading the whole conduction domain. This is achieved by using a set of special moving morphable components (MMCs) whose contour and skeleton are described explicitly by parameterized level-set surfaces. Unlike in the conventional methods where topology optimization was carried out in an Eulerian framework, the proposed optimizer is Lagrangian in nature, which is consistent with classical shape optimization approaches, giving great potential to reduce the total number of design variables significantly and also yielding more flexible modeling capability to control the structural feature sizes. By doing this, the growth elements are separated from the underlying finite element method (FEM) grids so that they can grow toward an arbitrary direction to form an optimized area-to-point path solution. The method is tested on an electromagnetic bandgap (EBG) power plane design example; both simulation and experiment verified the effectiveness of the proposed method.