The paper describes the optimization procedure supporting the designing process of geometry of gradient coatings basing on numerical simulation of internal stress and strain distributions in the coating and substrate. In mathematical model the gradient coating is represented by the so-called transition functions describing the change of physico-chemical parameters such as Young's modulus, Poisson's ratio, thermal expansion coefficient and the density as a function of the spatial variables. The object of optimization is system composed of a CrN/CrCN gradient coating and Cr interlayer between the CrN /CrCN coating and the steel substrate deposited on nitrided 4140 steel substrate. Decision variables are: the parameters of the of curvature of transition function , thickness of gradient coating and the thickness of the Cr interlayer. Optimization was carried out under pre-defined fixed continuous external loads and created decision criteria were the functions of the state of stress and strain in the coating and the substrate. Using the optimization procedure the sets of optimal parameters (Pareto sets) of the PVD gradient coating/nitrided substrate systems, due to the adopted decision criteria were determined. The analysis of the obtained optimal solutions (Pareto-optimal sets) was carried out using the "utopian solution method". It was also examined the technological stability of the Pareto-optimal solutions (nondominated) by analyzing the number of direct neighbors of these solutions in the decision variables space.
In the paper was proposed optimization procedure supporting the prototyping of the geometry of multi-module CrN /CrCN coatings, deposited on substrates from 42CrMo4 steel, in respect of mechanical properties. Adopted decision criteria were the functions of the state of internal stress and strain in the coating and substrate, caused by external mechanical loads. Using developed optimization procedure the set of optimal solutions (Pareto-optimal solutions) of coatings geometry parameters, due to the adopted decision criteria was obtained. For the purposes of analysis of obtained Pareto-optimal solutions, their mutual distance in the space of criteria and decision variables were calculated, which allowed to group solutions in the classes. Also analyzed the number of direct neighbors of Pareto-optimal solutions for the purposes of assessing the stability of solutions.Keywords: multi-module coatings, optimization, Pareto sets, internal stress W pracy została zaproponowana procedura optymalizacyjna wspomagająca dobór geometrii wielomodułowych powłok CrN/CrCN, osadzonych na podłożu ze stali 42CrMo4, pod kątem właściwości mechanicznych. Przyjęte kryteria decyzyjne były funkcjami stanu naprężeń oraz odkształceń wewnętrznych w powłoce i podłożu, powstałych na skutek mechanicznych obciążeń zewnętrznych. Wykorzystując opracowaną procedurę uzyskano zbiór optymalnych wartości parametrów geometrii powłok (rozwiązania Pareto-optymalne), ze względu na przyjęte kryteria decyzyjne. Do celów analizy otrzymanych rozwiązań określono ich wzajemne odległości w przestrzeni kryteriów i zmiennych decyzyjnych, co umożliwiło pogrupowanie rozwiązań w klasy. Analizowano również liczbę bezpośrednich sąsiadów rozwiązań Pareto-optymalnych w celach oceny stabilności rozwiązań.
The multi-objective optimization procedure of geometry of TiAlN/TiN/Cr multilayer coatings was created. The procedure was applied to the multilayer coatings subjected to constant tangential and normal loads (Hertzian contact). In physical model Cr, TiN and TiAlN layers were treated as a continuous medium, thus in mathematical description of the stress and strain states in the coatings a classical theory of stiffness was used. Decisional variables used in procedure were thicknesses of Cr, TiN and TiAlN layers and decisional criteria were functions of the stress and strain fields in the coating and substrate. Using created optimization procedure, Pareto set of optimal values of layers' thicknesses were determined. Additionally, two methods of analysis of Pareto-optimal set were introduced and discussed.
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