We investigate the growth and self-organization of hetero-epitaxial quantum dots (QDs) on different patterned substrates and show that it may lead to uniform arrays of dots with different localizations. We consider the surface instability enforced by elasticity and capillarity with the wavelength lambda(ATG) and study here the influence of the pattern shape. Starting with the results for a generic sinusoidal pattern, we investigate more realistic Gaussian-like pattern shapes. Even if the evolution is intrinsically non-linear, and thence not described by the superposition principle, we show that for a pit-like pattern, the QD morphologies may be rationalized by the results of the generic sinusoidal profile with appropriate length scales. The growth of islands both on the pattern peaks or valleys is shown to result from the racing between the wetting and buried elastic dipole interactions. (C) 2015 Elsevier B.V. All rights reserved.