Uogólnienia metody elementów skończonych w inżynierskich symulacjach numerycznych ośrodka nieciągłego i dyskretnego

Jakubowski, J.

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Uogólnienia metody elementów skończonych w inżynierskich symulacjach numerycznych ośrodka nieciągłego i dyskretnego

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Jakubowski J.

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Warianty tytułu

Finite element method generalizations applied to numerical simulations of discontinuous and discrete solid models

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Abstrakty

Klasyczna metoda elementów skończonych jest metodą wydajną i uniwersalną, z zastosowaniem wielu dostępnych pakietów również łatwą w użyciu. W zastosowaniu do symulacji wprost nieciągłości i nieciągłego masywu skalnego, ma jednak pewne istotne ograniczenia. W ciągu ostatnich kilkunastu lat pojawiły się uogólnienia tej metody stosujące generalnie rzecz ujmując nowe metody aproksymacji, z których wiele opartych jest na tzw. podziale jedności. W rezultacie, na bazie metody elementów skończonych i metody różnic skończonych powstały metody bezsiatkowe, metody wzbogaconej aproksymacji metody elementów skończonych i metoda rozmaitości numerycznych. Wszystkie te metody mają zdolność naturalnego odwzorowania nieciągłości bez kłopotliwych operacji przebudowy siatki. Każda z nich jest zdolna do symulacji ośrodka ciągłego, ośrodka nieciągłego oraz rozpadu ośrodka w jednym, spójnym schemacie numerycznym (każda z nich w innym). Po uzupełnieniu o algorytmy rozpoznawania kontaktów metody te nabierają cech metod elementów dyskretnych. Są to na razie rozwiązania laboratoryjne, nad którymi pracują matematycy, numerycy i programiści, które nie trafiły jeszcze w ręce inżynierów. W przyszłości mogą mieć szerokie zastosowanie w symulacjach budowlanych i geomechanicznych i ze względu na swoje cechy mogą stanowić alternatywę dla metody elementów skończonych, metody elementów odrębnych i innych popularnych inżynierskich metod symulacyjnych. W artykule omówiono wyżej wymienione metody i perspektywy ich zastosowania do symulacji ośrodka nieciągłego i dyskretnego w szczególności nieciągłego masywu skalnego.

Traditional finite element method is efficient and universal numerical simulation method, and implemented with many available software packages, also easy to use. Applied to simulations of discontinuities and discontinuous rock mass, it has got serials limitations. For the last several years some generalizations of this method have been developed with the use of new approximation techniques, particularly partition of unity. As a result of these developments mesh-free methods (MFree), enriched approximation methods (GFEM, XFEM) and numerical manifold method has been developed, basing on finite element method and finite difference method approaches. All the three groups of methods listed above have ability to model discontinuities without challenging and expensive remeshing. All of them can simulate continuous medium, discontinuous medium and model disintegration within a single numerical schema (each of them within different one). Completed with contact detection algorithms, they meet criteria of discrete element method. The above mentioned methods are still in their very early stages of development and many theoretical and practical problems need to be solved before they will be used in Civil Engineering and Rock Mechanics for practical applications. In the future, due to their advantages, they can offer an alternative for finite element method, distinct element method and other popular engineering simulation methods. The article presents the above mentioned methods and their possible applications for discontinuous and discrete medium simulation, particularly for the simulation of discontinuous rock mass.

Słowa kluczowe

symulacje numeryczne mechanika skał symulacja nieciągłego masywu skalnego metoda elementów skończonych uogólniona metoda elementów skończonych rozszerzona metoda elementów skończonych metody bezsiatkowe metoda rozmaitości

numerical simulations rock mechanics discontinuous rock mass simulation finite elements method generalized finite element method extended finite element method meshfree methods element-free methods manifold method

Wydawca

Wydawnictwa AGH

Czasopismo

Górnictwo i Geoinżynieria

Rocznik

2010

Tom

R. 34, z. 2

Strony

325--340

Opis fizyczny

Bibliogr. 67 poz., rys.

Twórcy

autor

Jakubowski J.

Katedra Geomechaniki, Budownictwa i Geotechniki, Wydział Górnictwa i Geoinżynierii, Akademia Górniczo-Hutnicza, Kraków

Katedra Geomechaniki, Budownictwa i Geotechniki, Wydział Górnictwa i Geoinżynierii, Akademia Górniczo-Hutnicza, Kraków

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