A new Controllable Structure Generation Scheme (CSGS) based on discrete Gaussian quadrature space and velocity is presented and used to generate multiple-phase random isotropic homogenous and shape-constrained anisotropic heterogeneous structures. The primary advantage of the new CSGS over the existing random structure generation growth method is the ability to model a wide variety of structures by controlling the shape through relatively simple constraint indexes. The growth speed probability function is introduced to control the mesoscopic porosities and mixture/separation of material phases. The model is applied to generate four packed structure types (shapeless random, separated solid shapes, separated random-filled shapes, and random-mixture-filled shapes). Three-dimensional steady and transient thermal diffusion are simulated by Non-Dimensional Lattice Boltzmann Method (NDLBM). The steady state results are compared to measured data available in the published literature. The transient results reveal how the mesoscopic shape of a structure impacts thermal diffusion. With equivalent macroscopic volume fractions, structures with higher mesoscopic volume fractions of high conductivity phases possess higher effective thermal conductivity/diffusivity because there is greater connectivity of the higher conductive material at mesoscopic scale. (C) 2016 Elsevier Ltd. All rights reserved.