  Modeling the spontaneous evolution of morphology in natural systems and its
preservation by proportionate growth remains a major scientific challenge. Yet,
it is conceivable that if the basic mechanisms of growth and the coupled
kinetic laws that orchestrate their function are accounted for, a minimal
theoretical model may exhibit similar growth behaviors. The ubiquity of surface
growth, a mechanism by which material is added or removed on the boundaries of
the body, has motivated the development of theoretical models, which can
capture the diffusion-coupled kinetics that govern it. However, due to their
complexity, application of these models has been limited to simplified
geometries. In this paper, we tackle these complexities by developing a finite
element framework to study the diffusion-coupled growth and morphogenesis of
finite bodies formed on uniform and flat substrates. We find that in this
simplified growth setting, the evolving body exhibits a sequence of distinct
growth stages that are reminiscent of natural systems, and appear spontaneously
without any externally imposed regulation or coordination. The computational
framework developed in this work can serve as the basis for future models that
are able to account for growth in arbitrary geometrical settings, and can shed
light on the basic physical laws that orchestrate growth and morphogenesis in
the natural world.
