The Zienkiewicz–Zhu error estimator for multiple material problems

Nambiar, R.V.; Lawrence, K. L.

doi:10.1002/cnm.1630080408

Cited by 10 publications

(2 citation statements)

References 2 publications

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“…falsifies-the true mechanical phenomenon of a discrete stress jump. In the context of error estimation based on intentionally improved nodal stresses in comparison to stress in the element interior, suchlike falsified nodal stresses yield a spoiled support for interpolation, a fact that was already recognized by Nambiar and Lawrence [1992].…”

Section: 22mentioning

confidence: 99%

Error analysis for quadtree-type mesh coarsening algorithms adapted to pixelized heterogeneous microstructures

Fischer

Eidel

2020

Comput Mech

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Pixel-and voxel-based representations of microstructures obtained from tomographic imaging methods is an established standard in computational materials science. The corresponding highly resolved, uniform discretitization in numerical analysis is adequate to accurately describe the geometry of interfaces and defects in microstructures and, therefore, to capture the physical processes in these regions of interest. For the defect-free interior of phases and grains however, the high resolution is in view of only weakly varying field properties not necessary such that mesh-coarsening in these regions can improve efficiency without severe losses of accuracy in simulations. The present work proposes two different variants of adaptive, quadtree-based meshcoarsening algorithms applied to pixelized images that serves the purpose of a preprocessor for consecutive finite element analyses, here, in the context of numerical homogenization. Error analysis is carried out on the microscale by error estimation which itself is assessed by true error computation. A modified stress recovery scheme for a superconvergent error estimator is proposed which overcomes the deficits of the standard recovery scheme for nodal stress computation in cases of interfaces with stiffness jump. By virtue of error analysis the improved efficiency by the reduction of unknowns is put into relation to the increase of the discretization error. This quantitative analysis sets a rational basis for decisions on favorable meshes having the best trade-off between accuracy and efficiency as will be underpinned by various examples.

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Section: 22mentioning

confidence: 99%

Error analysis for quadtree-type mesh coarsening algorithms adapted to pixelized heterogeneous microstructures

Fischer

Eidel

2020

Comput Mech

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“…In section 4, it will be coupled with the frictional contact LDC method. To avoid modifying the ZZ estimator to take into account contact contributions as in [WS98], the errors will be estimated in each body separately as proposed by [NL92] for multimaterial problems. Then Ω l = ∪K; K ∈ T l−1 ; {K ⊂ Ω 1 and e 1 K > α} or {K ⊂ Ω 2 and e 2 K > α} (19) where e 1 (respectively e 2 ) denotes the error estimator performed on Ω 1 (respectively Ω 2 ).…”

Section: Zienkiewicz and Zhu A Posteriori Error Estimatorsmentioning

confidence: 99%

On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics

Liu

Ramière

Lebon

2017

Computer Methods in Applied Mechanics and Engineering

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This paper addresses an efficient and robust automatic adaptive local multilevel mesh refinement strategy for unilateral frictional contact problems in elastostatics. The proposed strategy couples the Local Defect Correction multigrid method LDC (Hackbusch, 1984) with the ZZ (Zienkiewicz and Zhu, 1987) a posteriori error estimator. An extension of LDC method to frictional contact problems is introduced. An interesting feature of this extended LDC algorithm is that it still only lies on interpolations of displacement fields. Neither forces conservation nor exchange of contact status is required between the refinement levels. The ZZ a posteriori error estimator is exploited to automatically build the sub-grids of the LDC method. A criterion linked to the relative error in stress is used. The efficiency of the proposed strategy is analyzed on examples derived from nuclear engineering. Practical numerical choices are proposed and justified. The refinement process automatically varies and stops with respect to a given tolerance. Post-treatments show that the sub-grids focus around the contact areas and that the converged LDC solution always respects the prescribed tolerance.

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An algorithm for adaptive refinement of triangular element meshes

Nambiar

Valera

Lawrence

et al. 1993

Numerical Meth Engineering

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SUMMARYA simple algorithm is developed for adaptive and automatic h refinement of two-dimensional triangular finite element meshes. The algorithm is based on a n element refinement ratio that can be calculated from an a posteriori error indicator. The element subdivision algorithm is robust and recursive. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several example problems are presented to illustrate the utility of the approach.

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The Zienkiewicz–Zhu error estimator for multiple material problems

Cited by 10 publications

References 2 publications

Error analysis for quadtree-type mesh coarsening algorithms adapted to pixelized heterogeneous microstructures

Error analysis for quadtree-type mesh coarsening algorithms adapted to pixelized heterogeneous microstructures

On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics

An algorithm for adaptive refinement of triangular element meshes

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