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Prace Instytutu Podstawowych Problemów Techniki PAN

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Sensitivity analysis in finite element coputations of elasto-plasticity

Autorzy Kowalczyk, P. 
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EN Until early nineties of the 20th century, computational engineering problems were commonly understood as tasks in which input parameters (like geometry, material and sizing properties, initial state variables, etc.) were defined and as-signed certain numerical values, while the response - space-time distributions of state variables and various functionals thereof - were sought for as solutions of certain initial-boundary value problems in their either analytical forms or space-time discretized at various levels. Highly advanced methods were (and continue to be) developed in which the wide use of computers has turned out to be indispensable for any sophisticated assessment of realistic structural be-haviour. Particular difficulties have been encountered and gradually overcome in the case of problems with complex geometries, non-classical (unilateral) bound-ary conditions and complicated, strongly nonlinear and path-dependent materiał behaviour. Among many methodologies of approximate solution of nonlinear problems of mechanics, the finite element method (FEM) is widely recognized as most ef-fective and universal tool. Its success has been proved in industrial practice - it seems hardly imaginable now that design process of any structure would not include extensive numerical simulations of its mechanical behaviour performed with the use of a nonlinear finite element codę. Fundamentals and application power of the method, developed sińce early seventies, have been presented in anumber of monographs and textbooks, like e.g. [8, 25, 46, 49, 61, 94, 105, 141], as well as in manuals to specific commercial implementations, e.g. [1]. It is stressed that abilities of the methodology include both statical and dynamical analysis, various types of geometrie and materiał nonlinearities, complex bound-ary condition types, solid-fluid mechanical interactions, as well as couplings with electrical, thermal and magnetic phenomena. Plastic phenomena oceurring in many structural and geotechnical materials have always belonged to most challenging tasks in computational engineering. Strongly nonlinearl, path- and frequently also rete- and temperature-dependent character of plastic deformation combined with geometric nonlinearities and discontinuous changes in material behaviour at limit points between elastic and inelastic regimes require advanced mathematical models and appropriate computational solution algorithms. Starting from works of Tresca [123], Huber [47] and von Mises [86] who had laid fundamentals for modern theory of plasticity, large progress has been made in both understanding the micromechanical nature of plastic flow and in its mathematical modelling in the macro scale. Formulation of mathematical theory of slip-induced plasticity in metal crystals [5, 43, 45] was accompanied by development of models including large deformations [75,76,93], anisotropy [42, 50, 89], porosity and void nucleation [24, 34, 126]l, nonlinear hardening and cyclic effects [19, 26 79], and viscoplastic phenomena [2, 39, 102, 127]. For granular materials (rocks. soils) and metallic foams, where plasticity is mainly friction-induced, a variety of non-associated and associated flow models were developed [16, 28, 30, 109]. The citations above refer to only milestone publications and are by no means intended to be exhaustive - extensive reviews of constitutive models of plasticity can be found e.g. in monograph [44, 57, 81, 139]. Elasto-plastic and elasto-viscoplastic constitutive relations are flow equations and are thus formulated in terms of rates of state variables. In numerical applications they require appropriate integration in time. The standard approach employs the "elastic predictor-plastic corrector" shceme, in which trial elastic finite increment of tress is, whenever violating the yield condition, corrected by a plastic counterpart determined from approximate time integration of the flow equation. Such return-mapping algorithms were first proposed in the form of radial return schemes for cylindrical flow surfaces in the space of pricipal stress components (e.g. Huber-Mises yield condition), cf. [131], and [74] (account for strom hardening). Return mapping strategies for more general plasticity models were developed in e.g. [53, 116] (plane stress state), [110, 112, 114] (general smooth and non-smooth yield surfaces), [87, 104] (for yield surfaces with corners), [14, 15] (for the Cam-clay plasticity model), [103, 130] (for viscoplastic models), and [52] (for granular materials). A critical issue regarding efficiency of iterative solution schemes in such algorithms is determination fo the consistent tangent stiffness operator [111, 115, 116]. This frequently appears to be a complex task, e.g., in a number of large-deformation formulations, evaluation of such an operator requires unique differentiation of tensor spectral decomposition routines, including cases of multiple eigenvalues. The monograph [113] contains an exhaustive review of return-mapping algorithms and discussion of related computational issues, including consistent differentation. In some practical steady plastic flow applications (forging, extrusion), the constitutive relations do not need to be time-integrated - instead, they are spatially discretized in the ratę form and only state variable rates are of inter-est as the analysis response, cf. [140] for fundamentals and [117] for practical applications of the 'flow approach' in computational plasticity. In the recent two decades, we have been witnessing emergence of a very fruit-ful branch in computational mechanics - the parameter sensitwity analysis (SA). Mathematical fundamentals of the method were presented in numerous ar-ticles and books; let us only mention here milestone monographs [22, 32, 37, 41, 65] and collections [66, 108]. The aim of SA is to find the relationship between parameters defining the system at hand and the system response. The main interest of the analyst is here not just to ąuantify the response for a given set of input parameters, but also to evaluate influence of possible variations of the paraineters on the corresponding change in the response. In practice this means determination of response gradients with respect to the system parameters, al-though higher order derivatives may also be a subject of interest [27, 38, 40], e.g. in the critical state analysis [90, 91]. Depending on the naturę of input parameters and their variations considered in the sensitivity analysis, we may distinguish between different types of SA in mechanical problems. Probably the most frequently encountered one in the literaturę is the design sensitivity analysis (DSA) in which the system input parameters are understood as the design parameters, i.e. the quantities whose particular numerical values as well as their changes depend on arbitrary decisions of the designer. Another branch of SA is the imperfection sensitvity analysis (ISA) in which variations of the system input parameters are independent of the designer's will and have usually random character. Mathematical essence in both cases is the same - the sensitivity analysis consists in determination of first- or higher-order gradients of response functionals with respect to a certain set of input scal ar ąuantities. Thus, the names are freąuently exchanged and the term 'design variation' happens to be used to describe imperfection, while 'design parameters' may as well denote e.g. probability distribution parameters (means, standard deviations, etc.) of random input ąuantities. In this thesis, 'design parameters' will be referred to as scalar input parameters of any type defining the system under consideration. Conseąuently, the term DSA will be considered equivalent to SA in generałl The area of applications of sensitivity analysis is wide and still increasing. Computational power of contemporary computer hardware and maturity of mathematical formulations allow to obtain response sensitivity information for large-scale nonlinear mechanical problems as numerically cheap add-on to the primary analysis results. It is thus widely belived that any robust computer simulation of a mechanical system will be considered incomplete as long as it is not complemented with studies on response sensitivity with respect to input parameter variations. Natural areas of application for DSA are the parametric 'what-if' studies at the stage of structural design, and the gradient-based design optimization in which the 'best' set of design parameter values is sought for according to certain optimality criteria [37]. System identification problems in which the unknown values of design parameters are to be found, given the response values, are another area of applications of DSA, comceptually very similar to the design optimization. ISA may again find its applications in system identification,l but it can also be employed in error analysis, reliability-based optimization, as well as in stkochastic analysis of the system response. This study is devoted to methods of sensitivity analysis in computational problems of elasto-plaslticity. Evaluation of sensitivity gradients for solutions embedding a so wide spectrum of nonlinear phenomena is a tempting challenge for researchers and, on the other hand, it responds to increasing needs of industry, where elasto-plastic behaviour of materials often plays the crucial role from the point of view of the safety, economy and ergonomy of the final product. Main attention is focused on the analytical methods of sensitivity, i.e. on determination of analytical response gradients with respect to design parameters (contrary to their approximate determination with finite-difference formulae at small design perturbations). Publications on sensitivity analysis in these classes of problems had been appearing from late 80's. DSA for elasto-plasticity described with the independent deformation theory was discussed in [9, 10]. For path-dependent problems, which iclude most of the elasto-plastic constitutive theories, Ryu et al. [107] were the first to point out the most important conseąuence of path-dependence for sensitivity analysis - that sensitivity gradients are path-dependent, too, i.e. at each time instant they depend on the response itself as well as on its sensitivity at previous time instants. Computational aspects of DSA for small-strain elasto-plasticity were then discussed in [13, 73, 77, 85, 92, 95, 106, 128, 132, 135] (rate-independent models), and in [17, 35, 62-65, 67, 129] (rate-dependent models. Reduced dimension (plane stress) formulation was presented in [68]. Formulation applying to a general rate-type constitutive model of soil was discussed in [31]. Important issues specific to the analysed class of problems, discussed in the publications, include the crucial role of the algorithmic constitutive tangent stiffness operator (consistent with the time integration scheme) [31, 62, 64, 67, 68, 128, 129] and the problem of discontinuity of sensitivity gradients [67. 68. 77]. For large deformation elasto-plastic models sensitivity analysis was discussed in [118] (rigid-plastic approach only), [51, 96, 124, 125] for one-dimensional case (trusses), [21, 59, 65, 72, 78, 134] for 3-d models based on additive elastic-plastic strain decomposition, and finally in [6, 7, 23, 58, 60, 71, 80, 119, 133] for models based on multiplicative elastic-plastic decomposition of the deformation gradient. Formulations of sensitivity analysis in the velocity description, including the flow approach concept, can be found in [3, 4, 29, 54, 65, 82, 136, 137]. The references above include both statical, dynamical and structural stability formulations, applicable in the formalism of the finite as well as the boundary element method. At the level of constitutive formulations there are no significant differences between these areas of applications. Besides, they include formulations of both sizing and shape parameter sensitivity analysis. Despite different naturę and way of description of both the types of input parameters, the shape and non-shape sensitivity can be apparently treated in a uniform manner, upon introduction of the reference volume concept [62].
PL Niniejsza rozprawa jest poświęcona metodom analizy wrażliwości parametrycznej w obliczeniach numerycznych zagadnień sprężysto-plastycznych. Określenie gradientów wrażliwości dla rozwiązań obejmujących tak szeroki wachlarz zjawisk nieliniowych, zarówno geometrycznych jak materiałowych, stanowi w}'-zwanie dla badaczy, a jednocześnie odpowiedź na potrzeby przemysłu, gdzie zjawiska sprężysto-plastyczne w materiałach odgrywają istotna rolę z punktu widzenia bezpieczeństwa, kosztów i funkcjonalności produktu. W rozprawie przedstawione są przede wszystkim analityczne metody wrażliwości, w których gradienty wyznaczane są ściśle, w przeciwieństwie do metod opierających się na przybliżeniach metody różnic skończonych dla małych perturbacji parametrów projektowych. Rozważania przedstawione w pracy dotyczą analizy wrażliwości pierwszego rzędu w geometrycznie liniowych oraz nieliniowych zagadnieniach obliczeniowych izotermicznej sprężysto-plastyczności metali, między innymi również z uwzględnieniem zjawisk lepkoplastycznych. Treść rozprawy jest w dużej mierze kompilacją wyników oryginalnych badań naukowych autora, prowadzonych w latach 1994-2006 w Instytucie Podstawowych Problemów Techniki PAN, przy współpracy z kolegami z Zakładu Metod Komputerowych, i opublikowanych w pracach [64, 65, 67, 68, 71-73, 132, 133]. Trzon rozprawy stanowią rozważania na temat układów statycznych, choć dyskusja obejmuje również rozszerzenia przedstawionych sformułowań w kierunku obliczeń dynamicznych. Problemy wrażliwości w zagadnieniach np. stateczności lub analizy drgań, w których niezbędne jest wyznaczanie wartości i wektorów własnych układu dyskretnego, nie są w pracy podejmowane. Wszystkie zagadnienia brzegowo-początkowe rozważane w rozprawie są dyskutowane w kontekście przybliżonych sformułowań dyskretnych opart}'ch na metodzie elementów skończonych (MES). Ograniczenie to nie umniejsza ogólnego charakteru rozważań i wniosków, przynajmniej dla przypadku wrażliwości na parametry materiałowe i wymiarowe. W zagadnieniach wrażliwości kształtu, kluczowe pojęcie niezależnej od parametrów projektowych geometrycznej konfiguracji bazowej jest w niniejszej pracy utożsamione z konfiguracja bazową izoparametrycznego elementu skończonego - stąd dyskusja tej grupy zagadnień ogranicza się do sformułowania dyskretnych w ujęciu MES. Rozdział 1 zawiera wprowadzenie do tematy, z omówieniem historii i obecnego stanu badań w dziedzinie wrażliwości parametrycznej zagadnień spężysto-plastycznych. Rozdział 2 prezentuje ogólną ideę analizy wrażliwości nieliniowych układów mechanicznych. Zostało tu przedstawione ogólne sformułowanie problemu podstawowego, zarówno w ujęciu ciągłym, jak też w dyskretnym, oraz wprowadzone zostały pojęcia parametrów projektowych i funkcjonałów stanu. Rozróżniono następnie dwa fundamentalne podejścia do analizy wrażliwości: ciągłe i dyskretne, z dyskusja na temat ich zalet i wad, i wskazaniem pewnych metod pośrednich. W dalszej części rozprawy rozważano dokładniej jedynie podejście dyskretne, z dwiema alternatywnymi strategiami rozwiązania: metodą bezpośredniego różniczkowania (DDM) i metodą układu sprzężonego (ASM). Przedyskutowano w sposób krytyczny podejścia: analityczne, półanalityczne oraz różnic skończonych, do wyznaczania gradientów wrażliwościowych. Na koniec podniesiona została kwestia różniczkowalności rozwiązań sprężysto-plastycznych względem parametrów projektowych i możliwych w konsekwencji tego faktu nieciągłości rozwiązań wrażliwościowych. Poprawne sformułowanie zagadnienia wrażliwości wymaga dogłębnego zrozumienia sformułowania zagadnienia podstawowego. Rozdział 3 przedstawia zatem szczegółowo zagadnienie statycznej deformacji ciała sprężysto-plastycznego, a w szczególności sformułowania konstytutywne dla małych i dużych deformacji w ujęciu prędkościowym i przyrostowym, oraz sformułowanie dyskretne globalnych równań równowagi w ujęciu MES. Wprawdzie materiał ten niemal nie zawiera oryginalnych wyników autora, jednak jego umieszczenie w pracy jest niezbędne, gdyż następujące w kolejnym rozdziale wyprowadzenie równań wrażliwości zawiera liczne odwołania do notacji i szczegółów równań zagadnienia podstawowego. W Rozdziale 4 przedstawiono wyprowadzenie sformułowania wrażliwości dla tak zdefiniowanego problemu podstawowego. Jest to zasadnicza, oryginalna część rozprawy. Podniesione zostały w szczegółności aspekty obliczeniowe analizy wrażliwości, między innymi kluczowa rola algorytmicznej, konsystentnej macierzy stycznej. Omówiono szczegółowo kwestię wrażliwości na parametry kształtu. Pokazano, że nie ma istotnych sprzeczności między sformułowaniami dla parametrów wpływających i niewpływających na początkową geometrię układu (podejście zunifikowane). Następnie zróżniczkowano równania pojawiające się w procedurach konstytutywnych sprężysto-plastyczności względem zmiennych wejściowych (zależnych od parametrów projektowych) w celu otrzymania całkowitych i cząstkowych pochodnych projektowych zmiennych stanu, niezbędnych w globalnym sformułowaniu analizy wrażliwości. Podkreślono wniosek o liniowym (przynajmniej na kroku obliczeniowym) charakterze rozwiązania wrażliwościowego, nawet w przypadku silnych nieliniowości w równaniach konsystencji modelu konstytutywnego. Wyprowadzone sformułowania przedstawiono w postaci zamkniętych algorytmów obliczeniowych, gotowych do implementacji od ręki w programie komputerowym. Przykłady obliczeniowe, zamieszczone na końcu rozdziału, ilustrują zaprezentowane algorytmy i inspirują d}'skusję na temat szczegółowych kwestii omówionych w rozdziale. Rozdział 5 obejmuje rozszerzenie rozważań z poprzednich rozdziałów na przypadek analizy dynamicznej układów sprężysto-plastycznych. Przedstawiono równania analizy podstawowej oraz wyprowadzone z nich równania analizy wrażliwości w ujęciu metody elementów skończonych. Przedyskutowano dwie strategie obliczeń dynamicznych: niejawną i jawną, szczególną uwagę poświęcając tej drugiej, jako bardziej popularnej w praktycznych obliczeniach inżynierskich, a jednocześnie wymagającej nieco innego podejścia do analizy wrażliwości, niż dotychczas omówione. Wyprowadzenie równań matematycznych i podanie algorytmów obliczeniowych zostało zilustrowane kolejnymi przykładami obliczeniowymi. Rozdział 6 zawiera uwagi na temat praktycznej implementacji omówionych algorytmów analizy wrażliwości w programie metody elementów skończonych. Wskazano zarówno na trudności (duża ilość niezbędnych dodatkowych instrukcji i procedur), jak też na zalety (np. zautomatyzowany charakter dużej części pracy do wykonania przez programistę). Przedyskutowano kwestię możliwości automatycznej generacji kodu na różnych poziomach sformułowań. W Rozdziale 7 sformułowano wnioski i przewidywane kierunki dalszych badań. Niniejsza praca jest rozprawą habilitacyjną autora.
Słowa kluczowe
PL inżynieria materiałowa   programowanie matematyczne  
EN materials engineering   differential equation  
Wydawca Instytut Podstawowych Problemów Techniki PAN
Czasopismo Prace Instytutu Podstawowych Problemów Techniki PAN
Rocznik 2006
Tom nr 7
Strony 1--212
Opis fizyczny Bibliogr. 141 poz.
autor Kowalczyk, P.
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