Multigrid methods for Prandtl-Reuss plasticity

Wieners, Christian

doi:10.1002/(sici)1099-1506(199909)6:6<457::aid-nla173>3.0.co;2-p

Cited by 17 publications

(11 citation statements)

References 19 publications

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“…Although this discretization is not fully stable for the Stokes problem, it improves the quality of finite element solutions for problems in elasticity and plasticity; cf. [17].…”

Section: Prolongation Letĩmentioning

confidence: 99%

Criteria for the approximation property for multigrid methods in nonnested spaces

Neuss‐Radu

Wieners

2004

Math. Comp.

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Abstract. We extend the abstract frameworks for the multigrid analysis for nonconforming finite elements to the case where the assumptions of the second Strang lemma are violated. The consistency error is studied in detail for finite element discretizations on domains with curved boundaries. This is applied to prove the approximation property for conforming elements, stabilized Q 1 /P 0 -elements, and nonconforming elements for linear elasticity on nonpolygonal domains.Proving the approximation property for the multigrid analysis for nonconforming finite element discretizations is formalized in [7,4,15] for many cases: it suffices to verify criteria on the approximation quality and the consistency error. In these papers, it is required that a continuous bilinear form can be extended to a nonconforming finite element space, which is not valid for many interesting applications.The purpose of this paper is to establish a full set of criteria which guarantees the approximation property for a wide range of nonnested discretizations, where we do not assume that the discrete bilinear form coincides with the continuous bilinear form for all conforming functions. In the notation, we follow Bramble [5, Chap. 4], and our results can be applied directly to the multigrid theory described there. The results extend known results by Brenner [7] and Stevenson [15], and they provide a systematic and constructive way of studying nonnested multigrid algorithms for more general nonnested spaces and varying forms.The paper is organized as follows. First, we introduce an abstract setting describing a multigrid hierarchy for nonconforming discretizations of an elliptic partial differential equation without full regularity. As usual, the multigrid approximation property is derived by comparison with the finite element approximation property, which we formulate using an interpolation operator and its adjoint with respect to the energy scalar product. In a second step (Section 1.9), we derive the approximation property from consistency assumptions on a conforming comparison space, similar to the approach in [7].In Section 2, we consider the case of conforming finite elements on a polygonal approximation of the computational domain. Here, we choose a comparison space consisting of curved finite elements. After introducing a suitable interpolation operator, we use the equivalence of the operator norm scale (used throughout Section 1)

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“…Although this discretization is not fully stable for the Stokes problem, it improves the quality of finite element solutions for problems in elasticity and plasticity; cf. [17].…”

Section: Prolongation Letĩmentioning

confidence: 99%

Criteria for the approximation property for multigrid methods in nonnested spaces

Neuss‐Radu

Wieners

2004

Math. Comp.

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“…We will see in the next section that problem (35) can be implemented with the same interface designed for the implicit Euler method. Moreover, if the Runge-Kutta scheme is algebraically stable, B-stability of this scheme follows for the differential algebraic equations for g > 0 and for the corresponding dissipative evolution equation in the rate independent case g ¼ 0 [23]. Furthermore, the application of embedded Runge-Kutta schemes allows for an adaptive time stepping [14,22].…”

Section: Incremental Modelmentioning

confidence: 99%

Parallel 3-d simulations for porous media models in soil mechanics

Wieners

Ammann²,

Diebels

et al. 2002

Computational Mechanics

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Abstract:We introduce a general parallel model for solving coupled nonlinear and time-dependent problems in soil mechanics. In particular, we discuss the application to a triphasic porous media model, where we compute the deformation of unsaturated soil together with the porefluid flow of water and air in the soil, and where the material behaviour of the skeleton is assumed to be elasto-viscoplastic. In two large-scale numerical experiments we finally present an extended evaluation of our parallel model for demanding 3-d configurations.References:

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“…The details are outlined in [1]. Compare [8] for a prove in a weak setting of plasticity for algebraically stable DIRK-methods.…”

Section: Contractivitymentioning

confidence: 99%

“…Chapter 4 describes the return mapping strategy [6] which we generalize to implicit Runge-Kutta methods. Afterwards in Chapter 5, we show, how algebraically stable Runge-Kutta methods preserve a contractivity property of the constitutive equations [2,3,8].…”

Section: Introductionmentioning

confidence: 99%

Index 2 DAEs in Plasticity Theory

Büttner¹,

Simeon²

2002

Proc. Appl. Math. Mech.

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The equations of rate-independent elastoplasticity form a differential-algebraic equation (DAE) with discontinuities. For the numerical solution, implicit Runge-Kutta methods are applied and combined with the return mapping strategy of computational plasticity. It turns out that the convergence order depends crucially on the switching point detection. Further, it is shown that algebraically stable Runge-Kutta methods preserve the contractivity of the elastoplastic flow.

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Multigrid methods for Prandtl-Reuss plasticity

Cited by 17 publications

References 19 publications

Criteria for the approximation property for multigrid methods in nonnested spaces

Criteria for the approximation property for multigrid methods in nonnested spaces

Parallel 3-d simulations for porous media models in soil mechanics

Index 2 DAEs in Plasticity Theory

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