Application of computer algebra in symbolic computations and boundary-value problems of the theory of shells

• Autorzy: Walentyński R.

Warianty tytułu

PL

Zastosowanie algebry komputerowej w obliczeniach symbolicznych i zagadnieniach brzegowych teorii powłok

• Języki publikacji: EN

Abstrakty

EN

The contribution consists of two parts. The first part deals with the application of the computer algebra system MATHEMATICS and the package for tensor analysis MathTensor in symbolic computations in the theory of shells. The second part is devoted to finding an approximate solution of shell boundary value problems. The contribution is neither a theory of shells nor does it aspire to be a theory of approximation. The problem related to shells, being an object of interest - the description of bodies in a curved space, is the background of considerations: 1) the possibility of showing various tools of computer assistance in symbolic .computations, 2) a proposal of the application of some approach to the Least Squares Method in search for approximate solutions applying computer algebra. The consideration in the first part is preceded by a presentation of basic relations of the considered theory of shells. One approach to the theory of shells has been explored, but ways of the application of computer algebra tools presented in the contribution may be further developed in other theories. The aim of this part of the contribution is to present an effective application of computer algebra system capabilities of formulating equations and their adaptation for further numerical computations. It has been shown that computer algebra is not "a wizard box" for an automatic derivation of expressions, but an assistant device, which helps a conscious research worker to obtain desired and reliable results. Ways of the application of advanced tools of symbolic computations have been presented, which permit to receive final expressions in the desired and possibly simplest form. Moreover, it was presented that there is no need and that it is not advisable to neglect "ad hoc" any terms in expressions. Thanks to computer algebra applications it has been possible to determine the constitutive relations of shells, which satisfy the last equation of equilibrium (2.51). The reliability of results is an important aspect of this part of the contribution. Thus, special emphasis is put on the scrutiny of the obtained formulas. The computer algebra system is a program and only a program. Although it is an advanced technological product, the entire responsibility for the results is up to the user. The attention of the contribution is focused reasonably on linear problems and shells of simple shapes, as its aim is to present a rational application of the tool, the computer algebra system. However, it has been pointed out how it is possible to extend the range of consideration to nonlinear tasks. Such approaches to the tasks for computer algebra have been shown, that the computations might be completed successfully and as quickly as possible. The results of symbolic computations presented in the contribution are differential equations in terms of displacements and other relations for a cylindrical shell. The approximation of equations derived in the first part is considered in the next part of the contribution. An application of some approach to the Least Squares Method to search an approximate solution of the differential equation system is presented. In this approach to the method the functional, which is an object of minimization, is appended with terms taking into account boundary conditions. This makes it possible to simplify considerably the implementation of the method algorithm within the computer algebra environment and permits to approximate multidimensional tasks with a discontinuous boundary. Approximations of such tasks have been shown in several other contributions of the Author. Within the elements of implementation selected solutions of the application of advanced tools of the system have been presented, which permits to speed up the computational process, in particular the integration of polynomial expressions and the construction of a system matrix of algebraic equations and its decomposition. In boundary-value problems of shells, especially the considered long cylindrical shells the phenomenon of a boundary layer occurs, so that direct a approach to the problem by means of methods of numerical approximation slowly converges with an actual result. The application of the two step approach based on membrane approximation fails, due to the bending flexibility of cylindrical shells in the parallel direction. These difficulties have been overcome applying the proposed approach of the Least Squares Method. By means of computational experiment a boundary-condition phenomenon has been discovered - that is connected not only with the method, but also with the character of the differential equations - which permits to approximate described the problem with a tenth-order operator taking into account only four boundary conditions applied in the membrane approach. The obtained approximation has been called base solution and is feasible in most of the domain, excluding the boundary layer. This discovery becomes the basis of the two-step approach proposal, presented by examples of two tasks related to cylindrical shells. This approximation is as stable as that obtained by means of the method based on membrane approximation and simultaneously free from the already mentioned disadvantages of this approach. In the first step the task is approximated taking into account selected boundary conditions. Neglected boundary conditions are satisfied locally in the second step. Additional methods of global and local error evaluation are shown, They allow, among others, to find a phenomenon of false convergence. An effective parameter of error evaluation is the value of the minimized functional. The Least Squares Method is a global approach. Its results are functions and in this context it is an analytical approach. Therefore, there is no problem with differentiation and integration, the problem of interpolation and extrapolation does not occur, either. The estimation of a global and local error of approximation is straightforward. Thanks to that the approach is free from disadvantages of discrete methods and in times of computer algebra development may become a tool of scrutiny and moreover an alternative to numerical methods. The most important feature is the discovered possibility of a two-step solution of tasks with a boundary layer. Thus, it may be applied in other tasks of mechanics and mathematical physics, where similar phenomena occur. Opposite to asymptotic approaches the method requires neither the setting up of assumptions of small parameter nor to lower the order of the differential operator. Thus, it may be applied to the problems of a wider class. The contribution shows a wide spectrum of computer algebra applications starting with formulating equations of the problem, through numerical computations, to the publication of results . The computer algebra system has been employed in the production of most contribution graphics and formulas. The processing of results is a crucial technical element connected with scientific work. The Author hopes that the presented considerations might become at least an inspiration to apply modern tools of computer assistance in processes of formulating and solving complex problems of mechanics and physics.

PL

Praca składa się z dwóch części. W części pierwszej przedstawiono zastosowanie systemu algebry komputerowej MATHEMATICA oraz pakietu analizy tensorowej MathTensor do obliczeń symbolicznych w teorii powłok. Część druga dotyczy znajdowania rozwiązania przybliżonego zagadnienia brzegowego powłok. Praca nie jest teorią powłok, ani też nie aspiruje do miana teorii aproksymacji. Zagadnienia związane z powłokami, z uwagi na przedmiot zainteresowali - opis ciał w zakrzywionej przestrzeni, stanowią kanwę rozważań z uwagi na: 1) możliwość pokazania różnorodnych narzędzi wspomagania komputerowego w zakresie obliczeń symbolicznych, 2) propozycję zastosowania pewnego ujęcia Metody Najmniejszych Kwadratów do znajdowania rozwiązań przybliżonych z wykorzystaniem algebry komputerowej. Rozważania części pierwszej poprzedzono zestawieniem podstawowych zależności omawianej teorii powłok. Omówiono jedno z podejść do teorii powłok, jednak przedstawione w pracy sposoby zastosowania narzędzi algebry komputerowej mogą być wykorzystane w innych teoriach. Celem tej części pracy jest przedstawienie możliwości efektywnego wykorzystania systemu algebry komputerowej w formułowaniu równań oraz ich przygotowaniu do dalszych obliczeń numerycznych. Pokazano, że algebra komputerowa nie jest "czarodziejską skrzynką" do automatycznego uzyskiwania wyrażeń ale asystentem pomagającym świadomemu badaczowi na uzyskanie żądanych i wiarygodnych wyników. Przedstawiono tu sposoby wykorzystania szeregu zaawansowanych narzędzi obliczeń symbolicznych, które pozwalają na uzyskanie końcowych wyrażeń w żądanej i możliwie najprostszej postaci. Ponadto pokazano, że nie trzeba, ba nawet nie należy pomijać "ad hoc" jakichkolwiek wyrazów w wyrażeniach. Dzięki wykorzystaniu algebry komputerowej udało się wyznaczyć związki konstytutywne powłok spełniające ostatnie równanie równowagi (2.51). Ważnym aspektem tej części pracy jest problem wiarygodności wyników. Dlatego też, szczególną uwagę poświęca się weryfikacji otrzymanych wzorów. System algebry komputerowej jest programem i tylko programem i mimo swojego zaawansowania technicznego całą odpowiedzialność za wyniki ponosi jego użytkownik. W pracy uwaga została skupiona celowo na zagadnieniach liniowych i powłokach o prostych kształtach, gdyż jej celem jest przedstawienie racjonalnego wykorzystania narzędzia, jakim jest system algebry komputerowej. Wskazano jednakże, w jaki sposób można rozszerzyć zakres rozważań na przypadki zadań nieliniowych. Pokazano sposoby takiego formułowania zadań dla systemu algebry komputerowej, aby obliczenia zakończyły się powodzeniem i przebiegały możliwie jak najszybciej. Wynikiem obliczeń symbolicznych przedstawionych w pracy są przemieszczeniowe równania różniczkowe i inne związki dla powłoki walcowej. Aproksymacja równań otrzymanych w części pierwszej jest przedmiotem rozważań następnej części opracowania. Przedstawiono tu zastosowanie pewnego ujęcia Metody Najmniejszych Kwadratów do znajdowania rozwiązania przybliżonego układu równań różniczkowych. W tym ujęciu metody funkcjonał, będący przedmiotem minimalizowania, uzupełniono o wyrazy uwzględniające warunki brzegowe. Pozwala to na znaczne uproszczenie wdrożenia algorytmu metody w obrębie środowiska algebry komputerowej oraz umożliwia aproksymację zadań wielowymiarowych z nieciągłym brzegiem. Aproksymacje tego typu zadań pokazano w kilku innych pracach Autora. W ramach przedstawionych elementów wdrożenia metody przedstawiono wybrane rozwiązania wykorzystania zaawansowanych narzędzi systemu algebry komputerowej pozwalające na przyśpieszenie procesu obliczeniowego, w szczególności całkowania wyrażeń wielomianowych i budowy macierzy układu równań algebraicznych i jej dekompozycji. W zagadnieniu brzegowym powłok, a w szczególności rozważanych długich powłok walcowych, występuje zjawisko warstwy brzegowej, które sprawia, że podejście bezpośrednie do problemu z użyciem metod aproksymacji numerycznej wolno zbiega się do poprawnego wyniku. Zastosowanie podejścia dwuetapowego opartego na przybliżeniu błonowym okazuje się zawodne z uwagi na podatność powłok walcowych na zginanie w kierunku równoleżnikowym. Trudności te udało się pokonać z zastosowaniem proponowanego ujęcia Metody Najmniejszych Kwadratów. W wyniku eksperymentu obliczeniowego odkryto zjawisko warunku brzegowego - jak się okazuje związane nie tylko z metodą, ale i charakterem równań różniczkowych - pozwalające na aproksymację problemu opisanego operatorem różniczkowym dziesiątego rzędu z uwzględnieniem jedynie czterech warunków brzegowych - stosowanych w podejściu błonowym. Uzyskana aproksymacja nazwana rozwiązaniem bazowym jest poprawna na większości dziedziny problemu z wyjątkiem warstwy brzegowej. To odkrycie stało się podstawą zaproponowania podejścia dwuetapowego, przedstawionego na przykładach dwóch zadań dotyczących powłok walcowych. Przybliżenie to jest stabilne i szybko zbieżne tak jak to otrzymywane w metodzie opartej na przybliżeniu błonowym i jednocześnie wolne od już wspomnianych wad tego podejścia. W pierwszym etapie zadanie jest przybliżane z wybranymi warunkami brzegowymi. Pominięte warunki brzegowe są spełniane lokalnie w drugim kroku. W pracy przedstawiono interpretację fizyczną zaobserwowanego zjawiska obliczeniowego oraz dokonano próby jego wyjaśnienia od strony matematycznej. Przy okazji wskazano na potencjalnie szerokie pole zastosowań metody. Pokazano dodatkowe metody oceny błędu lokalnego i globalnego aproksymacji pozwalające między innymi wykryć zjawisko pozornej zbieżności. Efektywnym parametrem oceny błędu jest wartość minimalizowanego funkcjonału. Metoda Najmniejszych Kwadratów jest globalnym podejściem do problemu. Wynikiem rozwiązania są funkcje i w tym sensie jest metodą analityczną. Dlatego też nie ma kłopotu z różniczkowaniem lub całkowaniem, nie występuje też problem interpolacji i ekstrapolacji. Łatwo też ocenić lokalny i globalny błąd aproksymacji. Dzięki temu podejście jest wolne od wad metod dyskretnych i w dobie rozwoju systemów algebry komputerowej może stać się narzędziem weryfikacyjnym, a nawet alternatywą dla metod numerycznych. Najistotniejszą cechą jest jednak dostrzeżona możliwość rozwiązywania dwuetapowego zadań z warstwą brzegową. Może zatem znaleźć zastosowanie w innych zagadnieniach mechaniki i fizyki matematycznej, w których występują podobne zjawiska. W przeciwieństwie do podejść asymptotycznych metoda nie wymaga stawiania założeń o małym parametrze oraz obniżania rzędu operatora różniczkowego. W związku z tym może być wykorzystana do aproksymacji szerszej klasy zadań. Praca pokazuje szerokie spektrum wykorzystania systemu algebry komputerowej od formułowania równań problemu, poprzez obliczenia numeryczne, do publikacji wyników. System algebry komputerowej został wykorzystany do przygotowania większości grafiki występującej w pracy oraz wzorów. Opracowanie wyników do publikacji stanowi istotny, techniczny element związany z pracą naukową. Autor żywi nadzieję, że przedstawione w pracy rozważania staną się przynajmniej inspirujące do wykorzystania nowoczesnych narzędzi wspomagania komputerowego w procesie formułowania oraz rozwiązywania złożonych problemów mechaniki i fizyki.

Słowa kluczowe

EN

theory of shells; symbolic computation; boundary value problem; Least Squares Method; tensor analysis

PL

teoria powłok; algebra komputerowa; obliczenia symboliczne; zagadnienie brzegowe; metoda najmniejszych kwadratów; analiza tensorowa

• Wydawca: Wydawnictwo Politechniki Śląskiej
• Czasopismo: Zeszyty Naukowe. Budownictwo / Politechnika Śląska
• Rocznik: 2003
• Tom: z. 100
• Strony: 13--198
• Opis fizyczny: Bibliogr. 236 poz.

Twórcy

autor

Walentyński R.

• Katedra Teorii Konstrukcji Budowlanych Politechniki Śląskiej, 44-100 Gliwice, ul. Akademicka 5, tel. [phone]

Bibliografia

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