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http://yadda.icm.edu.pl:80/baztech/element/bwmeta1.element.baztech-article-BPB4-0033-0002

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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. 
Treść / Zawartość http://prace.ippt.gov.pl/
Warianty tytułu
Języki publikacji EN
Abstrakty
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.
Twórcy
autor Kowalczyk, P.
Bibliografia
[1] ABAQUS v. 6.Ą, Theory Manuał ABAQUS Inc., Pawtucket, RI, USA, 2003.
[2] L. Anand. Constitutive eąuations for hot-working of metals. Int. J. Plasticity, 1:213-231, 1985.
[3] H.J. Anttinez. Bulk-Metal Forming Processes — From Computational Modelling via Sensitivity analysis to Tool Shape Optimization. Prace IPPT PAN, Warszawa, 2001, habilitation thesis.
[4] H.J. Antunez and M. Kleiber. Sensitivity of forming processes to shape param-eters. Comput Methods Appl Mech. Engng., 137:189-206, 1996.
[5] R.J. Asaro. Micromechanics of crystals and poły ery st ais. Adv. Appl. Mech., 23:1-115, 1983.
[6] S. Badrinarayanan and N. Zabaras. A sensitivity analysis for the optimal design of metal-forming processes. Comput. Methods Appl. Mech. Engng., 129:319-348, 1996.
[7] D. Balagangadhar and D.A. Tortorelli. Design of large deformation steady elasto-plastic manufacturing processes. Part II: Sensitivity analysis and optimization. Int. J. Numer. Methods Engng., 49:933-950, 2000.
[8] K.J. Bathe. Finite Element Procedures in Engineering Analysis. Prentice Hall, 1995.
[9] M.P. Bends0e and J. Sokołowski. Sensitivity analysis and optimization of elastic-plastic structures. Engng. Optim., 11:31-38, 1987.
[10] M.P. Bends0e and J. Sokołowski. Design sensitivity analysis of elastic-plastic problems. Struci. Mech., 16:81-102, 1988.
[11] C. Bischof, H.M. Biicker, B. Lang, A. Rasch, and J.W. Risch. Extending the func-tionality of the general-purpose finite element package SEPRAN by automatic differentiation. Int. J. Numer. Methods Engng., 58:2225-2238, 2003.
[12] C. Bischof, A. Carle, P. Khademi, and A. Mauer. The ADIFOR 2.0 system for the automatic differentiation of Fortran 77 programs. IEEE Comput. Sci. Eng., 3:18-32, 1996.
[13] M. Bonnet and S. Mukherjee. Implicit BEM formulations for usual and sensitivity analysis in elasto-plasticity using the consistent tangent operator concept. Int. J. Sołids Struct., 33:4461-4480, 1996.
[14] R.I. Borja. Cam-Clay plasticity, Part II: Implicit integration of constitutive eąuation based on a nonlinear elastic stress predictor. Comput. Methods Appl. Mech. Engng., 88:225-240, 1991.
[15] R.I. Borja and S.R. Lee. Cam-Clay plasticity, Part I: Implicit integration of elasto-plastic constitutive relations. Comput. Methods Appl. Mech. Engng., 78:49-72, 1990.
[16] R.I. Borja and C. Tamagnini. Cam-Clay plasticity, Part III: Extension of the in-finitesimal model to include finite strains. Comput. Methods Appl. Mech. Engng., 155:73-95, 1998.
[17] G. Bugeda and L. Gil. Shape sensitivity analysis for structural problems with non-linear materiał behaviour. Int. J. Numer. Methods Engng., 46:1385-1404, 1999.
[18] J. Casey. Approximate kinematical relations in plasticity. Int. J. Solids Struct, 21:671-682, 1985.
[19] J.L. Chaboche. Time independent constitutive theories for cyclic plasticity. Int. J. Plasticity, 2:149-188, 1986.
[20] G. Cheng and Y. Liu. A new computation scheme for sensitivity analysis. Engng. Optim., 12:219-235, 1987.
[21] S. Cho and K.K. Choi. Design sensitivity analysis and optimization of nonlinear transient dynamics. Part I: Sizing design. Int. J. Numer. Methods Engng., 48:351-373, 2000.
[22] K.K. Choi and N.-H. Kim. Structural Sensitivity Analysis and Optimization, Vol. 1 and 2. Springer, 2005.
[23] K.K. Choi and N.H. Kim. Design optimization of springback in a deepdrawing process. AIAA J., 40:147-153, 2002.
[24] CC. Chu and A. Needleman. Void nucleation effects in biaxially stretched sheets. J. Engng. Mater. Technol., 102:249-256, 1980.
[25] M.A. Crisfield. Nonlinear Finite Element Analysis of Solids and Structures, Vol. 1: Essentials, Vol. 2: Advanced Topics. J. Wiley & Sons, 1991.
[26] Y.F. Dafalias and E.P. Popov. Plastic internal variables formalism of cyclic plasticity. Trans. ASME, J. Appl. Mech., 98:645-651, 1976.
[27] K. Dems and Z. Mróz. Variational approach to first- and second-order sensitivity analysis. Int. J. Numer. Methods Engng., 21:637-646, 1985.
[28] V.S. Deshpande and N.A. Fleck. Isotropic constitutive model for metallic foams. J. Mech. Phys. Solids, 48:1253-1276, 2000.
[29] I. Doltsinis and T. Rodić. Process design and sensitivity analysis in metal forming. Int. J. Numer. Methods Engng., 45:661-692, 1999.
[30] D.C. Drucker and W. Prager. Soil mechanics and plastic analysis or limit design. Quart. Appl. Math., 10:157-165, 1952.
[31] W. Fellin and A. Ostermann. Parameter sensitivity in finite element analysis with constitutive models of ratę type. Int. J. Numer. Anal. Methods Geomech., 30:91-112, 2006.
[32] P.M. Frank. Introduction to System Sensitivity Theory. Academic Press, 1978.
[33] A. Griewank, D. Juedes, and J. Utke. ADOL-C, a package for the automatic differentiation of algorithms written in C/C++. ACM Trans, on Math. Software, 22:131-167, 1996.
[34] A.L. Gurson. Continuum theory of ductile rupture by void nucleation and growth: Part I — Yield criteria and flow rules for porous ductile media. Trans. ASME, J. Engng. Mater. Technol., 99:2-15, 1977.
[35] M.A. Gutierrez and R. de Borst. Studies in materiał parameter sensitivity of softening solids. Comput. Methods Appl. Mech. Engng., 162:337-350, 1998.
[36] R.T. Haftka. Semi-analytical static nonlinear structural sensitivity analysis. AIAA J., 31:1307-1312, 1993.
[37] R.T. Haftka and Z. Giirdal. Elements of Structural Optimization. Kluwer, 1992.
[38] R.T. Haftka and Z. Mróz. First- and second-order sensitivity analysis of linear and nonlinear structures. AIAA J., 24:1187-1192, 1986.
[39] E.W. Hart. Constitutive relations for the nonelastic deformations of metals. Trans. ASME, J. Engng. Mater. Technol, 98:193-202, 1976.
[40] E.J. Haug. Second-order design sensitivity analysis of structural systems. AIAA ]., 19:1087-1088, 1981.
[41] E.J. Haug, K.K. Choi, and V. Komkov. Design Sensitwity Analysis of Structural Systems. Series in Math. Sci. Engng. Academic Press, 1986.
[42] R. Hill. Continuum micromechanics of elastoplastic polycrystals. J. Mech. Phys. Solids, 13:89-101, 1965.
[43] R. Hill. Generalized constitutive relations for incremental deformation of metal crystals by multislip. J. Mech. Phys. Solids, 14:95-102, 1966.
[44] R. Hill. The Mathematical Theory of Plasticity. Oxford Univ. Press, 1998.
[45] R. Hill and J.R. Rice. Constitutive analysis of elastic/plastic crystals at arbitrary strain. J. Mech. Phys. Solids, 20:401-413, 1972.
[46] E. Hinton, editor. Introduction to Nonlinear Finite Element Analysis. NAFEMS, 1992.
[47] M.T. Huber. Właściwa praca odkształcenia jako miara wytężenia materyału. Przyczynek do podstaw teoryi wytrzymałości. Czasopismo Techniczne, XXII, 1904.
[48] T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and nonlinear media. Int. J. Numer. Methods Engng., 15:1413-1418, 1980.
[49] T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamie Finite Element Analysis. Dover, 2000.
[50] J.W. Hutchinson. Elastic-plastic behavior of polycrystalline metals and compos-ites. Proc. Roy. Soc. London Ser. A, 319:247-272, 1970.
[51] S.Y. Jao and J.S. Arora. Design sensitivity analysis of nonlinear structures using endochronic constitutive model. Part I: General development; Part II: Discretiza-tion and applications. Comput. Mech., 10:39-57, 59-72, 1992.
[52] B. Jeremić, K. Runneson, and S. Sture. A model for elastic-plastic pressure sensitive materiał subjected to large deformations. Int. J. Solids Struct., 36:4901-4918, 1999.
[53] P. Jetteur. Implicit integration algorithm for elastoplasticity in piane stress analysis. Engng. Comput, 3:251-253, 1986.
[54] M.S. Joun and S.M. Hwang. Die shape optimal design in three-dimensional shape metal extrusion by the finite element method. Int. J. Numer. Methods Engng., 41:311-335, 1998.
[55] F. van Keulen and H. de Boer. Rigorous improvement of semianalytical design sensitivities by exact differentiation of rigid body motions. Int. J. Numer. Methods Engng., 42:71-91, 1997.
[56] F. van Keulen, R.T. Haftka, and N.H. Kim. Review of options for structural design sensitivity analysis. Part 1: Linear systems. Comput. Meth. Appl. Mech. Engng., 194:3213-3243, 2005.
[57] A.S. Khan and S. Huang. Continuum Theory of Plasticity. J. Wiley & Sons 1995.
[58] N.H. Kim and K.K. Choi. Design sensitivity analysis and optimization of non-linear transient dynamics. Mech. Struct. Mach., 29:351-371, 2001.
[59] N.H. Kim, K.K. Choi, and J.S. Chen. Shape design sensitivity analysis and optimization of elasto-plasticity with frictional contact. AIAA J., 38*1742-1753 2000.
[60] N.H. Kim, K.K. Choi, and J.S. Chen. Structural optimization of finite deforma-tion elastoplasticity using continuum-based shape design sensitivity formulation. Comput. Struct, 79:1959-1976, 2001.
[61] M. Kleiber. Incremental Finite Element Modelling in Non-Linear Solid Mechan-ics. Ellis Horwood - PWN, 1989.
[62] M. Kleiber. Shape and non-shape sensitivity analysis for problems with any materiał and kinematic nonlinearity. Comput. Methods Appl Mech. Engng 108*73-97, 1993.
[63] M. Kleiber and T.D. Hien. Parameter sensitivity of inelastic buckling and post-buckling response. Comput. Methods Appl. Mech. Engng., 145:239-262, 1998.
[64] M. Kleiber, T.D. Hien, H. Antunez, and P. Kowalczyk. Parameter sensitivity of elasto-plastic response. Engng. Comput, 12:263-280, 1995.
[65] M. Kleiber, T.D. Hien, H. Antunez, and P. Kowalczyk. Parameter Sensitwity in Nonlinear Mechanics. J. Wiley k Sons, 1997.
[66] M. Kleiber and T. Hisada, editors. Design Sensitwity Analysis. Atlanta Tech Publ., 1992.
[67] M. Kleiber and P. Kowalczyk. Constitutive parameter sensitivity in elasto-plasticity. Comput. Mech., 17:36-48, 1995.
[68] M. Kleiber and P. Kowalczyk. Sensitivity analysis in piane stress elasto-plasticity and elasto-viscoplasticity. Comput. Methods Appl. Mech. Engng., 137*395-409 1996.
[69] M. Kleiber, A. Siemaszko, and R. Stocki. Interactive reliability-based design optimization of geometrically nonlinear structures. In Srd World Congress of Structural and Multidisciplinary Optimization, pages 538-540, Buffalo NY Mav 17-21, 1999.
[70] J. Korelc. Symbolic methods in numerical analysis. http://www fcnr uni-Ij.si/Symech/, 2000.
[71] P. Kowalczyk. Design sensitivity analysis in large deformation elasto-plastic and elasto-viscoplastic problems. Int. J. Numer. Methods Engng., 66:1234-1270, 2006.
[72] P. Kowalczyk and M. Kleiber. Parameter sensitivity for large deformation inelas-tic problems. Comput. Assisted Mech. Engng. Sci., 4:209-228, 1997.
[73] P. Kowalczyk and M. Kleiber. Shape sensitivity in elasto-plastic computations. Comput. Methods Appl. Mech. Engng., 171:371-386, 1999.
[74] R.D. Krieg and S.W. Key. Implementation of a time-dependent plasticity theory into structural computer programs. In J.A. Stricklin and K.J. Saczalski, edi-tors, Constitutwe Eąuations in Viscoplasticity: Computational and Engineering Aspects, AMD-20. ASME, New York, 1976.
[75] E.H. Lee. Elastic-plastic deformations at finite strains. Trans. ASME, J. Appl. Mech., 36:1-6, 1969.
[76] E.H. Lee and D.T. Liu. Finite-strain elatic-plastic theory with applications to plane-wave analysis. J. Appl. Phys., 38:19-27, 1967.
[77] T.H. Lee and J.S. Arora. A computational method for design sensitivity analysis of elasto-plastic structures. Comput. Methods Appl. Mech. Engng., 122:27-50, 1995.
[78] T.H. Lee, J.S. Arora, and V. Kumar. Shape design sensitivity analysis of vis-coplastic structures. Comput. Methods Appl. Mech. Engng., 108:237-259, 1993.
[79] J. Lemaitre and J.L. Chaboche. Mechanics of Solid Materials. Cambridge Univ. Press, 1990.
[80] L.-J. Leu and S. Mukherjee. Sensitivity analysis of hyperelastic-viscoplastic solids undergoing large deformations. Comput. Mech., 15:101-116, 1994.
[81] J. Lubliner. Plasticity Theory. Macmillan, 1990.
[82] A.M. Maniatty and M.-F. Chen. Shape sensitivity analysis for steady metal-forming processes. Int. J. Numer. Methods Engng., 39:1199-1217, 1996.
[83] K. Maute and E. Ramm. Adaptive topology optimization. Struct. Optim., 10:100-112, 1995.
[84] H. Menrath, K. Maute, and E. Ramm. Topology optimization including elasto-plasticity. In D.R.J. Owen, E. Ońate, and E. Hinton, editors, Computational Plasticity. Fundamentals and Applications, pages 817-822. CIMNE, Barcelona, 1997.
[85] P. Michaleris, D.A. Tortorelli, and CA. Vidal. Tangent operators and design sen-sitivity formulations for transient nonlinear coupled problems with applications to elastoplasticity. Int. J. Numer. Methods Engng., 37:2471-2499, 1994.
[86] R. von Mises. Mechanik des festen Kórpers im plastisch deformablen Zustand. Nachr. Kónigl. Ges. Wiss. Góttingen, Math. Phys. KI., pages 582-592, 1913.
[87] G.P. Mitchell and D.R.J. Owen. Numerical solutions for elasto-plastic problems. Engng. Comput, 5:274-284, 1988.
[88] B. Moran, M. Ortiz, and C.F. Shih. Formulation of implicit finite element methods for multiplicative finite deformation plasticity. Int. J. Numer. Methods En-gng., 29:483-514, 1990.
[89] Z. Mróz. On the description of anisotropic hardening. J. Mech. Phys. Solids, 15:163-175, 1967.
[90] Z. Mróz and R.T. Haftka. Design sensitivity analysis of non-linear structures in regular and critical states. Int. J. Solids Struci, 31:2071-2098, 1994.
[91] Z. Mróz and J. Piekarski. Sensitivity analysis and optimal design of non-linear structures. Int. 3. Numer. Methods Engng., 42:1231-1262, 1998.
[92] S. Mukherjee and A. Chandra. A boundary element formulation for design sen-sitivity in materially nonlinear problems. Acta Mech., 78:243-253, 1989.
[93] S. Nemat-Nasser. On finite strain elasto-plasticity. Int. J. Solids Struct, 18:857-872, 1982.
[94] J.T. Odeń. Finite Elements of Nonlinear Continua. McGraw-Hill, 1972.
[95] M. Ohsaki. Sensitivity analysis of elasto-plastic structures by using explicit in-tegration method. Appl. Mech. Rev., 50:S156-S161, 1997.
[96] M. Ohsaki and J.S. Arora. Design sensitivity analysis of elasto-plastic structures. Int. J. Numer. Methods Engng., 37:737-762, 1994.
[97] N. Olhoff and J. Rasmussen. Study of inaccuracy in semi-analytical sensitivity analysis — a model problem. Struct. Optim., 3:203,213, 1991.
[98] N. Olhoff, J. Rasmussen, and E. Lund. A method of "exact" numerical differentia-tion for error elimination in finite-element-based semi-analytical shape sensitivity analyses. Mech. Struct. Mach., 21:1-66, 1993.
[99] I. Ozaki, F. Kimura, and M. Berz. Higher-order sensitivity analysis of finite element method by automatic differentiation. Comput. Mech., 16:223-234, 1995.
[100] E. Parente Jr. and L.E. Vaz. Improvement of semi-analytical design sensitivi-ties of nonlinear structures using eąuilibrium relations. Int. J. Numer. Methods Engng., 50:2127-2142, 2001.
[101] P. Pedersen, G. Cheng, and J. Rasmussen. On accuracy problems for semi-analytical sensitivity analysis. Mech. Struct. Mach., 17:373-384, 1989.
[102] P. Perzyna. The constitutive eąuation for rate-sensitive plastic materials. Quart. Appl. Math., 20:321-332, 1963.
[103] D. Pierce, C.F. Shih, and A. Needleman. A tangent modulus method for rate-dependent solids. Comput. Struct, 18:875-887, 1984.
[104] E. Pramono and K. Wiliam. Implicit integration of composite yield surfaces with corners. Engng. Comput, 6:186-197, 1989.
[105] J.N. Reddy. An Introduction to the Finite Element Method. McGraw-Hill, 2nd edition, 1993.
[106] E. Rohan and J.R. Whiteman. Shape optimization of elasto-plastic structures and continua. Comput. Methods Appl. Mech. Engng., 187:261-288, 2000.
[107] Y.S. Ryu, M. Haririan, CC. Wu, and J.S. Arora. Structural design sensitivity analysis of nonlinear response. Comput. Struct., 21:245-255, 1985.
[108] A. Saltelli, K. Chan, and E.M. Scott, editors. Sensitivity Analysis. J. Wiley & Sons, 2000.
[109] A. Schofield and CP. Wroth. Critical State Soil Mechanics. McGraw-Hill, 1968.
[110] J.C. Simo. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part I. Continuum for-mulation. Comput. Methods Appl. Mech. Engng., 66:199-219, 1988.
[111] J.C. Simo. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II. Computational aspects. Comput. Methods Appl. Mech. Engng., 68:1-31, 1988.
[112] J.C. Simo and T.J.R. Hughes. General return mapping algorithms for ratę independent plasticity. In CS. Desai, editor, Constitutiue Eąuations for Engineering Materials, pages 221-231. Elsevier, 1987.
[113] J.C. Simo and T.J.R. Hughes. Computational Inelasticity. Springer, 1998.
[114] J.C. Simo, J.G. Kennedy, and S. Govindjee. Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. Int. J. Numer. Methods Engng., 26:2161-2185, 1988.
[115] J.C. Simo and R.L. Taylor. Consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Engng., 48:101-118, 1985.
[116] J.C. Simo and R.L. Taylor. A return mapping algorithm for piane stress elastoplasticity. Int. J. Numer. Methods Engng., 22:649-670, 1986.
[117] W. Sosnowski. Finite Element Simułation of Industrial Sheet Metal Forming Processes. Prace IPPT PAN, Warszawa, 1995, habilitation thesis.
[118] W. Sosnowski, I. Marczewska, and A. Marczewski. Sensitivity based optimization of sheet metal forming tools. J. Mater. Proc. TechnoL, 124:319-328, 2002.
[119] A. Srikanth and N. Zabaras. An updated Lagrangian finite element sensitiv-ity analysis of large deformations using ąuadrilateral elements. Int. J. Numer. Methods Engng., 52:1131-1163, 2001.
[120] D.W. Stillman. Design sensitivity analysis for structures using explicit time in-tegration. In 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2000. Sept. 6-8, 2000, Long Beach, CA, AIAA-2000-4906.
[121] S. Stupkiewicz. Approximate response sensitivities for nonlinear problems in explicit dynamie formulation. Struct. Multidisc. Optim., 21:283-291, 2001.
[122] S. Stupkiewicz, J. Korelc, M. Dutko, and T. Rodić. Shape sensitivity analysis of large deformation frictional contact problems. Comput. Methods Appl. Mech. Engng., 191:3555-3581, 2002.
[123] H. Tresca. Sur 1'ecoulement des corps solides soumis a de fortes pressions. Comptes Rendus Acad. Sci. Paris, 59:754, 1864.
[124] J.J. Tsay and J.S. Arora. Nonlinear structural design sensitivity analysis for path dependent problems. Part 1: General theory. Comput. Methods Appl. Mech. Engng., 81:183-208, 1990.
[125] J.J. Tsay, J.E.B. Cardoso, and J.S. Arora. Nonlinear structural design sensitivity analysis for path dependent problems. Part 2: Analytical examples. Comput. Methods Appl. Mech. Engng., 81:209-228, 1990.
[126] V. Tvergaard. Materiał failure by void coalescence in localized shear bands. Int. J. Solids Struct, 18:659-672, 1982.
[127] K.C. Valanis. A theory of viscoplasticity without a yield surface. Arch. Mech., 23:517-533, 1971.
[128] CA. Vidal and R.B. Haber. Design sensitivity analysis for rate-independent elasto-plasticity. Comput. Methods Appl Mech. Engng., 107:393-431, 1993.
[129] CA. Vidal, H.S. Lee, and R.B. Haber. The consistent tangent operator for design sensitivity analysis of history-dependent response. Computing Systems in Engng., 2:509-523, 1991.
[130] G. Weber and L. Anand. Finite deformation constitutive eąuations and a time in-tegration procedurę for isotropic hyperelastic-viscoplastic solids. Comput. Meth-ods Appl Mech. Engng., 79:173-202, 1990.
[131] M.L. Wilkins. Calculation of elastic-plastic flow. In B. Adler, S. Fernbach, and M. Rottenberg, editors, Methods in Computational Physics. Academic Press, 1964.
[132] K. Wiśniewski, P. Kowalczyk, and E. Turska. On the computation of design derivatives for Huber-Mises plasticity with nonlinear hardening. Int. J. Numer. Methods Engng., 57:271-300, 2003.
[133] K. Wiśniewski, P. Kowalczyk, and E. Turska. Analytical dsa for explicit dy-namics of elastic-plastic shells. Comput. Mech., 2006 (in print; on linę at DOI 10.1007/s00466-006-0068-3).
[134] Q. Zhang, S. Mukherjee, and A. Chandra. Shape design sensithdty analysis for geometrically and materially nonlinear problems by the boundary element method. Int. J. Solids Struci, 29:2503-2525, 1992.
[135] Y. Zhang and A. Der Kiureghian. Dynamie response sensitivity of inelastic struc-tures. Comput. Methods Appl. Mech. Engng., 108:23-36, 1993.
[136] G. Zhao, X. Ma, X. Zhao, and R.V. Grandhi. Studies on optimization of metal forming processes using sensitivity analysis methods. J. Mater. Proc. TechnoL, 147:217-228, 2004.
[137] G. Zhao, E. Wright, and R.V. Grandhi. Preform die shape design using an optimization method. Int. J. Numer. Methods Engng., 40:1213-1230, 1997.
[138] H. Ziegler. A modification of Prager's hardening rule. Quart. Appl. Math., 17:55-61, 1959.
[139] O.C. Zienkiewicz, A.H.C. Chen, M. Pastor, BA. Schrefler, and T. Shiomi. Computational Geomechanics with Special Reference to Earthąuake Engineering. J. Wiley & Sons, 1999.
[140] O.C. Zienkiewicz, P. Jain, and E. Ońate. Flow of solids during forming and extrusion: some aspects of numerical solutions. Int. J. Solids Struct., 14:15-38, 1978.
[141] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method. Butterworth-Heinemann, 5th edition, 2000.
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