We report computational results on the formation of complex nanostructures on surfaces of coherently strained thin films that are deposited epitaxially on semiconductor substrates with periodic pit patterns, where individual pits have the shape of an inverted truncated pyramid. The simulated nanostructures consist of quantum dot molecules (QDMs) containing quantum dots (QDs) arranged in one-dimensional (1D) and two-dimensional (2D) regular arrays, as well as elongated QDs. We use a continuum-scale three-dimensional (3D) kinetic model that has been validated experimentally and perform self-consistent simulations of film surface morphological evolution according to the model. The results of our computational analysis are supported by a nonlinear stability theory that provides a comprehensive explanation for the observed nanopattern formation on the film surface as the outcome of a recently introduced "tip-splitting" instability that accompanies the well-known Stranski-Krastanow instability during epitaxial growth. Emphasis is placed on the effects of systematically varying geometrical design parameters of individual pyramidal pits in periodic pit patterns, including film thickness, period of the pit pattern, pit depth, length and width of the pit opening, and inclination of the pit wall on the resulting film surface nanopattern formation. We show that complex nanostructures are formed on the film surface, including ridges and quantum dot molecules inside film surface pits, as well as elongated QDs at the rims of the pits, as a result of varying the pit opening dimensions and the pit wall slope. Through our simulation predictions, we demonstrate that the geometrical parameters of the substrate pit pattern can be tuned properly in order to achieve a precise control of the ordered nanostructure patterns that form on the film surface. Our findings have important implications for designing optimal semiconductor surface patterns toward enabling future nanofabrication technologies.