Modelowanie hierarchiczne i metoda elementów skończonych do adaptacyjnej analizy struktur złożonych

• Autorzy: Zboiński G.
• Języki publikacji: PL

Abstrakty

PL

Niniejsza praca omawia tematykę modelowania hierarchicznego oraz metodę elementów skończonych do adaptacyjnej analizy struktur złożonych. Celem ogólnym pracy był rozwój usystematyzowanej teorii i technik numerycznych do automatycznej selekcji modeli obliczeniowych dla wybranej klasy problemów sprężystych mechaniki ciała stałego i mechaniki konstrukcji, obejmujących występujące w budowie maszyn, a także w budownictwie lądowym, złożone układy mechaniczne. Rozpatrywane były struktury składające się zarówno z części bryłowej (rozbudowanej przestrzennie), części grubościennej i/lub cienkościennej (powłokowej lub płytowej) oraz stref przejściowych. Wprowadzone modele obliczeniowe, zastosowane do każdej z wymienionych części różniły się między sobą wykorzystaniem wybranej teorii płyt, powłok lub teorii trójwymiarowej, a także parametrami numerycznymi metody elementów skończonych takimi jak: lokalne stopnie aproksymacji p i q, odpowiednio w kierunkach wzdłużnych i poprzecznym oraz lokalnym parametrem gęstości sieci h. Prezentowana praca obejmuje zarówno oparta na podejściu trójwymiarowym hierarchię modeli, odpowiednie aproksymacje metodą elementów skończonych, jak i szacowanie błędów sterujących procesem adaptacji oraz same procedury adaptacyjne. Pracę podzielono na osiem zasadniczych części, które w sposób logiczny odzwierciedlają przyjętą metodykę badawczą. We wprowadzającej części I przedstawiono przesłanki podjęcia tematu, zaprezentowano problemy badawcze związane z tematyką oraz omówiono dotychczasowy stan badań, a w oparciu o niego dokonano sformułowania celów pracy. W części II opisano ideę trójwymiarowego modelowania hierarchicznego, w tym hierarchię modeli opartych na podejściu trójwymiarowym oraz podstawy adaptacyjnych metod elementów skończonych, ze szczególnym uwzględnieniem adaptacyjności typu p i q, h oraz adaptacyjności modelu. Część III zawiera omówienie hierarchii elementów skończonych do adaptacyjnego modelowania i analizy struktur złożonych. Przedstawia ona algorytmy elementów skończonych opartych na podejściu trójwymiarowym: bryłowych, powłokowych hierarchicznych, powłokowych pierwszego rzędu i przejściowych, przeznaczonych odpowiednio do modelowania części masywnych, cienko- i grubościennych oraz stref przejściowych struktur złożonych. Część IV dotyczy szacowania błędów aproksymacji i modelowania oraz błędu całkowitego. Omówiono w niej definicje lokalne i globalne błędów oraz ich estymatory, a także algorytmy: obliczania wyrównoważonych, liniowych funkcji rozkładu naprężeń międzyelementowych oraz rozwiązywania problemów lokalnych, wykorzystywane w estymacji błędów. Część V przedstawia istotę trzech zjawisk: niewłaściwej granicy rozwiązania, lockingu oraz warstwy brzegowej oraz narzędzia numeryczne do wykrywania oraz oceny intensywności i zasięgu tych zjawisk. W części VI zamieszczono opis izotropowych procedur adaptacyjnych ciał trójwymiarowych oraz opracowano anizotropowe schematy adaptacyjne dla części bryłowych, powłokowych i przejściowych struktur złożonych, w zakresie zbieżności asymptotycznej. Następnie wprowadzono modyfikacje tych procedur w przypadku wystąpienia trzech zjawisk, omówionych w części V. Kolejna, VII część monografii przedstawia uogólniony algorytm adaptacyjnej analizy struktur złożonych, programy komputerowe służące tej analizie oraz wybrane przykłady numeryczne, potwierdzające efektywność zaproponowanych procedur. Pracę zamyka część VIII, w której omówiono wnioski z badań oraz dokonano podsumowania rozważań.

EN

The presented work concerns the problems of hierarchical modelling and finite element methods for adaptive analysis of complex structures. The main objective of the work was the development of the systematic theory and numerical techniqiues for automatic selection of computational models for a certain class of elastic problems of solid and structural mechanics, including complex mechanical systems occuring both in mechanical and civil engineering. The work considers structures consisting of solid (extended in three dimensions) parts, thin- or thick-walled parts (plates and shells), as well as of solid-to-shell, or solid-to-plate transition zones. The proposed computational models, applied to each such a part, differ with the utilization of the selected plate, shell or three-dimensional theory, the local longitudinal and transverse approximation orders p and q, and in the local mesh dimension h. The presented work includes a 3D-based hierarchy of models, proper hierarchical finite element approximations, error estimation for control of the adaptive process, and finally adaptive procedures within each part of complex structures. The work was divided into eight parts, which reflect the anticipated research methodology in a logical manner. In the first, introductory part, the reasoning for undertaking the subject is presented. Furthermore, the research problems dealing with the subject are established, and the state of the art is described, basing on which the research goals are formulated. The second part contains the description of the idea of 3D-based hierarchical modelling of complex structures, including the hierarchy of 3D-based solid, shell and transition models, and the basis of the adaptive finite element methods with a special emphasis on p, q and h adaptivity, and model adaptivity as well. The third part of the work presents a hierarchy of finite elements for adaptive modelling and analysis of complex structures. This part describes the algorithms of the 3D-based solid, hierarchical shell, first order shell, and solid-to-shell transition finite elements assigned for adaptive modelling of solid parts, thin- and thick-walled parts as well as of transition zones of complex structures. The fourth part of the work concerns the approximation, modelling and total error estimation. It introduces definitions of the local and global errors as well as of their estimators and indicators. This part also describes the algorithms for calculation of the equilibrated, linear interelement stress dis tribution and for the solution of local problems, which are both utilized in error estimation. In the fifth part the genesis of the three phenomena of the improper solution limit of the 3D model, the locking, and the boundary layer is elucidated. Moreover, the numerical tools for detection and assessment of intensity and range of these phenomena are elaborated. The sixth part of the work contains a description of the isotropic in its nature, adaptive procedure for 3D bodies. This part also includes the presentation of the own anisotropic adaptive procedures for solid, shell and transition parts of complex strucures, which procedures correspond to asymptotic range of convergence. Then, some modifications due to the three phenomena from the fifth part are proposed. The seventh part of the monograph proposes a generalized algorithm for adaptive analysis of complex structures and presents the computer code for such an analysis. In addition, numerical tests proving the effectivity of the elaborated algorthms are included. The work is completed by eighth part which formulates final conclusions and describes the significance of the obtained results.

Słowa kluczowe

PL

techniki numeryczne; mechanika ciała stałego; modelowanie hierarchiczne; elementy skończone; struktury złożone; modelowanie matematyczne

• Wydawca: Wydawnictwo Instytutu Maszyn Przepływowych PAN
• Czasopismo: Zeszyty Naukowe Instytutu Maszyn Przepływowych Polskiej Akademii Nauk w Gdańsku
• Rocznik: 2001
• Tom: nr 520/1479
• Strony: 3--304
• Opis fizyczny: Bibliogr. 359 poz., rys., tab.

Twórcy

autor

Zboiński G.

• Instytut Maszyn Przepływowych PAN w Gdańsku

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