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Reševanje močno povezanih inženirskih problemov z uporabo avtomatskega odvajanja : doktorska disertacija
Hudobivnik, Blaž (Author), Korelc, Jože (Mentor) More about this mentor... This link opens in a new window, Mikoš, Matjaž (Thesis defence commission member), Brank, Boštjan (Thesis defence commission member), Turk, Goran (Thesis defence commission member), Kegl, Marko (Thesis defence commission member)

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Abstract
V doktorski disertaciji je predstavljen pristop k formulaciji in reševanju močno povezanih inženirskih problemov po metodi končnih elementov z uporabo tehnologije avtomatskega odvajanja, kar nam omogočata orodji AceGen in AceFEM. Prikazano je, da je možno poljubno šibko obliko diferencialnih enačb povezanega problema preoblikovati v skalarno funkcijo, t. i. psevdo-potencial. Z uporabo avtomatskega odvajanja in ustreznih izjem pri odvajanju se lahko iz psevdo-potenciala avtomatsko izpeljejo enačbe problema in konsistentna tangentna matrika končnega elementa, ki zagotovijo kvadratično konvergenco Newton-Raphsonove iteracijske metode. Hkrati taka formulacija problema vodi do izjemno hitrih in računsko natančnih kod končnih elementov. Z velikim številom fizikalnih polj se soočimo s problemom naraščanja programske kode končnega elementa z vsakim dodanim poljem. Problem smo rešili z aditivno razdelitvijo celotnega povezanega problema na posamezne podprobleme, katerih koda končnega elementa bo zapisana v ločenem končnem elementu na način, da se ohrani kvadratična konvergenca Newton-Raphsonove iteracije. Na računskih primerih termo-hidro-mehanskih problemov je pokazano, da je ločena formulacija primerno učinkovita in računsko enakovredna skupni formulaciji. Ločena formulacija končnih elementov je lastnost sekvenčnega pristopa, zato smo na različnih primerih pokazali, da je enovito oz. celovito reševanje polnega sistema učinkovitejše od ločenega oz. sekvenčnega reševanja. V doktorski disertaciji je predstavljen tudi nov pristop k izračunu matričnih funkcij. Te so nujne za formulacijo nelinearnih mehanskih problemov, kot so nekateri hiperelastični modeli (npr. Henckyjev in Ogdenov model) in natančen opis evolucije plastičnega tečenja v primeru velikih deformacij. Predstavljena je nova metoda avtomatske izpeljave poljubne matrične funkcije in njenega prvega in drugega odvoda za matrike dimenzije 3 _ 3 z realnimi lastnimi vrednostmi. Opisana metoda nudi alternativo formulacijam, ki temeljijo na lastnih vrednostih, saj je rodovna funkcija za razliko od lastnih vrednosti stabilna in gladka. Rodovna funkcija je izražena z lastnimi vrednostmi matrike, zato je v okolici večkratnih lastnih vrednosti razvita v potenčno vrsto. Kreirana je bila tudi knjižnica podprogramov za izračun standardnih matričnih funkcij v zaključeni obliki. S tem lahko izbrane matrične funkcije pri formulaciji problemov obravnavamo kot elementarne funkcije. Na posameznih matrikah in različnih kombinacijah hiperelastičnih in elasto-plastičnih modelov smo pokazali, da so izpeljane matrične funkcije in njeni odvodi točni in natančni, formulacija pa je učinkovita.

Language:Slovenian
Keywords:grajeno okolje, gradbeništvo, disertacije, avtomatsko odvajanje, povezani problemi, inženirski problemi, velike deformacije, termo-hidro-mehanski problem, kvadratična konvergenca, matrične funkcije, hiperelastični modeli, Henckyjev model, Ogdenov model, model Cam-Clay, Metoda končnih elementov MKE
Work type:Doctoral dissertation (mb31)
Tipology:2.08 - Doctoral Dissertation
Organization:FGG - Faculty of Civil and Geodetic Engineering
Year:2016
Publisher:[B. Hudobivnik]
Number of pages:XXVIII,159 str.
Place:Ljubljana
COBISS.SI-ID:7393121 Link is opened in a new window
Views:1284
Downloads:346
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Secondary language

Language:English
Title:Automatic differentiation based solution of strongly coupled problems in engineering : doctoral thesis
Abstract:
The doctoral thesis presents an approach for the formulation and solution of strongly coupled engineering problems with the finite element method using the automatic differentiation technique that the software tools AceGen and AceFEM enables. It has been shown that it is possible to transform arbitrarily weak form of differential equation of coupled problems into scalar function pseudo-potential. By using the automatic differentiation and appropriate exceptions in the differentiation procedure, the equations of the problem and the consistent tangent matrix of finite element can be automatically derived from the pseudo-potential, which ensures quadratic convergence of the Newton-Raphson iterative procedure. At the same time such formulation of the problem leads to an extremely fast and accurate finite element codes. With a large number of physical fields we are faced with the problem of increasing size of element software code with each added field. The problem was solved by additive split of the total problem to individual subproblems, for which the code is derived inside separate final element in a manner that preserves the quadratic convergence of the Newton-Raphson iteration. It has been shown on the numerical examples of the thermo-hydro-mechanical problems that separate formulation is suitably efficient compared to the unified formulation. The separate formulation of finite elements is a property of sequential approach. Therefore, we have shown on several examples that unified solution of the full system is more efficient than sequential solution procedure. Additionally, a new approach to the evaluation of matrix functions is presented. These are necessary for the formulation of non-linear mechanical problems, such as certain hyperelastic models (e.g. Hencky and Ogden models) and the exact evolution of plastic flow in the case of large deformations. A new method of automatic derivation of an arbitrary matrix function and its first and second derivatives of the matrix of dimensions 3 _ 3 with real eigenvalues is presented. The described method provides an alternative to the formulations based on the eigenvalues, because the generating function is stable and smooth compared to the eigenvalues. The generating function is a function of the eigenvalues of matrix. Therefore it is expanded into power series in the vicinity of multiple eigenvalues. A library of subroutines which calculate the standard matrix functions in closed form was created. Thus, matrix functions can be considered as elementary functions when formulating problems. We have shown on individual matrices and various combinations of hyperelastic and elasto-plastic models that the derived matrix function and its derivatives are accurate and precise, and the formulation is efficient.

Keywords:building environment, civil engineering, thesis, automatic differentiation, coupled problems, engineering problems, large strains, thermo-hydro-mechanical problem, finite strains, quadratic convergence, matrix functions, Hyperelastic models, Ogden model, Cam-Clay model, finite element model FEM

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