A novel computational technique for modelling material and moving weak discontinuities (referred to as interfaces) is presented. In the new approach, based on the finite element method, the non-conforming mesh is used, in which spatial discretization of the computational domain is not compatible with the internal geometry of the modelled body. As a result, an appropriate treatment must be applied to elements cut by the interface to account for the presence of two material phases with typically different parameters. In the proposed method [1], these elements are replaced by so-called laminated elements which are entirely composed of laminated microstructures with the volume fraction of both phases and the lamination orientation determined by the interface position within the finite element. This method, named laminated element technique (LET), has been validated with numerous numerical examples covering two- and three-dimensional problems, involving linear elasticity as well as elasto-plastic cases within finite deformations. The generalization of LET to problems involving moving interfaces became feasible through its coupling with the phase-field method (PFM). The resulting computational technique, named LET-PF [2], exploits the advantages of PFM while simultaneously reducing some of its limitations. In contrast to PFM, in LET-PF phases of both materials are mixed within a thin layer of only one-element thickness. This allows for obtaining results of higher accuracy compared to the conventional PFM or results of comparable accuracy but using coarser meshes, thus reducing the computational cost. The performance of LET-PF is demonstrated through problems typical for micromechanics of heterogeneous materials. These examples are limited to two-dimensional cases within linear elasticity. Results obtained using LET-PF have been subjected to an extensive comparative analysis, where they have been confronted with results from PFM and, if possible, with analytical solutions. This comparison leads to the conclusion that, in certain cases, LET-PF eliminates limitations of the conventional phase-field method, thus providing a significant advantage. [1] J. Dobrzański, K. Wojtacki, S. Stupkiewicz. Lamination-based efficient treatment of weak discontinuities for non-conforming finite element meshes. Comput. Struct. (2024) 291:107209. [2] J. Dobrzański, S. Stupkiewicz. Towards a sharper phase-field method: A hybrid diffuse–semisharp approach for microstructure evolution