This paper is devoted to the conception and general structure of the integrated computational environment for constructing multi-dimensional large grids (with 1010 nodes and more) for high-performance solutions of interdisciplinary direct and inverse mathematical modelling problems in computational domains with complicated geometrical boundaries and contrast material properties. This includes direct and inverse statements which are described by the system of differential and/or integral equations. The constructed computational grid domain consists of subdomains featuring a grid, which may be of different types (structured or non-structured); discretization at the internal boundaries can be consistent or non-consistent. The methodology of such quasi-structured meshes makes it possible to use various algorithms and codes in the subdomains, as well as different data structure formats and their conversion. The proposed technologies include grid quality control, the generation of dynamic grids adapted to singularities of input geometric data of structures and multigrid approaches with local refinements, taking into account information about the solution to be obtained. The balanced grid domain decomposition, based on hybrid programming at the heterogeneous clusters with distributed and hierarchical shared memory, supports scalable parallelization. In addition, the paper outlines the technological requirements to provide a successful long-life cycle for the proposed computational environment. In a sense, the considered development presents a stable software ecosystem (integrated grid generator DELAUNAY) for supercomputing modelling in the epoch of big data and artificial intellect.