Runge–Kutta methods in elastoplasticity

Büttner, Jörg; Simeon, Bernd

doi:10.1016/s0168-9274(01)00133-7

Cited by 38 publications

(40 citation statements)

References 7 publications

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“…where the normal tensor n n+ = R n+ / R n+ at the midpoint instant is computed observing that Equation (30) 4 induces the following co-alignment relation:…”

Section: Double-step Midpoint Methodsmentioning

confidence: 99%

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Generalized midpoint integration algorithms for J₂ plasticity with linear hardening

Artioli

Auricchio

Veiga

2007

Numerical Meth Engineering

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SUMMARYWe consider four schemes based on generalized midpoint rule and return map algorithm for the integration of the classical J 2 plasticity model with linear hardening. The comparison, aiming to establish which is the preferable scheme among the four considered, is both theoretical and numerical. On one side, extending and completing the existing results in the literature, we investigate the four schemes from the theoretical viewpoint, addressing in particular the existence of solution, long-term behaviour, accuracy and stability. On the other hand, we develop an extensive set of numerical tests, based on pointwise stress-strain loading histories, iso-error maps and initial boundary-value problems.

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“…where the normal tensor n n+ = R n+ / R n+ at the midpoint instant is computed observing that Equation (30) 4 induces the following co-alignment relation:…”

Section: Double-step Midpoint Methodsmentioning

confidence: 99%

“…Runge-Kutta methods [4], generalized Runge-Kutta methods [5] and multi-step methods [6], for instance, share…”

Section: Introductionmentioning

confidence: 99%

Generalized midpoint integration algorithms for J₂ plasticity with linear hardening

Artioli

Auricchio

Veiga

2007

Numerical Meth Engineering

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“…The backward Euler method is one of the most popular integration methods for inelastic problems due to relative simplicity and good accuracy for large step sizes [14,15,37]. However, other more complex integration methods are also applied in return mapping algorithms (see [15,[38][39][40]). …”

Section: Implicit Integrationmentioning

confidence: 99%

Return mapping algorithms and consistent tangent operators in ferroelectroelasticity

Семенов

Liskowsky

Balke

2009

Numerical Meth Engineering

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SUMMARYReturn mapping algorithms for a rather general class of phenomenological rate-independent models for ferroelectroelastic materials are presented. The fully coupled thermodynamically consistent threedimensional constitutive model with two internal variables (remanent polarization vector and remanent strain tensor) proposed by C. M. Landis in 2002 is used for the simulation of electromechanical hysteresis effects in polycrystalline ferroelectric ceramics. Based on the operator splitting methodology, the return mapping algorithm employs the closest point projection scheme to obtain an efficient and robust integration of the constitutive model. The consistent tangent operator is obtained in closed form by linearizing the return mapping algorithm, and is found to be non-symmetric in the general case due to the dependence of the switching criterion on internal variables. Conditions that provide the symmetry of the consistent tangent matrix are analyzed. The compactness and generality of the received relations are achieved by means of using the thermodynamically based compact notations combining mechanical and electrical values. Both the cases scalar potential finite element (FE) formulation (primary variables: strain and electric field) and vector potential FE formulation (primary variables: strain and electric displacement) are considered. The accuracy and robustness of the algorithms are assessed through numerical examples.

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“…How to avoid the order reduction phenomena is still an open issue, see, for example, [60]. In [10] the case of small strain elastoplasticity is investigated regarding order reduction. The experience shows that almost the order of two is reached even in the case of constitutive models with case distinction, see model of Table 3.…”

Section: F(t Y(t)ẏ(t)) := G(t U(t) Q(t)) Q(t) − R(t U(t)u(t) Qmentioning

confidence: 99%

“…Additionally, stability problems of the structure are frequently solved by means of the arc-length method on global level, see Eq. (10). Further problems might occur in constitutive equations with softening behavior.…”

Section: F(t Y(t)ẏ(t)) := G(t U(t) Q(t)) Q(t) − R(t U(t)u(t) Qmentioning

confidence: 99%

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Hartmann¹,

Tebbens

Quint

et al. 2009

Z Angew Math Mech

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This article treats the computation of discretized constitutive models of evolutionary-type (like models of viscoelasticity, plasticity, and viscoplasticity) with quasi-static finite elements using diagonally implicit Runge-Kutta methods (DIRK) combined with the Multilevel-Newton algorithm (MLNA). The main emphasis is on promoting iterative methods, as opposed to the more traditional direct methods, for solving the non-symmetric systems which occur within the DIRK/MLNA. It is shown that iterative solution of the arising sequences of linear systems can be substantially accelerated by various techniques that aim at sharing part of the computational effort throughout the sequence. In this way iterative solution becomes attractive at clearly lower dimensions than the dimensions where direct solvers start to fail for memory reasons. The applications are related to small strain viscoplasticity of a smooth constitutive model for plastics and a finite strain plasticity model with non-linear kinematic hardening developed for metals.

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Runge–Kutta methods in elastoplasticity

Cited by 38 publications

References 7 publications

Generalized midpoint integration algorithms for J₂ plasticity with linear hardening

Generalized midpoint integration algorithms for J₂ plasticity with linear hardening

Return mapping algorithms and consistent tangent operators in ferroelectroelasticity

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

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Runge–Kutta methods in elastoplasticity

Cited by 38 publications

References 7 publications

Generalized midpoint integration algorithms for J2 plasticity with linear hardening

Generalized midpoint integration algorithms for J2 plasticity with linear hardening

Return mapping algorithms and consistent tangent operators in ferroelectroelasticity

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Contact Info

Product

Resources

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Generalized midpoint integration algorithms for J₂ plasticity with linear hardening

Generalized midpoint integration algorithms for J₂ plasticity with linear hardening