Automatic adaptive refinement for plate bending problems using Reissner-Mindlin plate bending elements

Lee, Chi King; Hobbs, R.E.

doi:10.1002/(sici)1097-0207(19980115)41:1<1::aid-nme264>3.0.co;2-o

Cited by 20 publications

(31 citation statements)

References 24 publications

(11 reference statements)

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“…From the experience obtained from adaptive FE analyses [28,31,[33][34][35][36][37] and EFGM analyses [8,9,14,23,24], an adaptive refinement procedure will be optimal only when both the global and local errors of the solution are controlled properly and distributed equally among all the degrees of freedom of the numerical model. Since the recovered stress field could be evaluated at any point inside , the notion of estimated error density,ē d can be introduced as…”

Section: Global and Local Refinement Indicators And New Node Spacing mentioning

confidence: 99%

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part II: Adaptive refinement

Lee

Shuai

2006

Comput Mech

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In Part II of this study, an automatic adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D linear boundary value problems is suggested. Based in the theoretical development and the numerical experiments done in Part I of this study, the Zienkiewicz and Zhu (Z-Z) error estimation scheme is combined with a new stress recovery procedure for the a posteriori error estimation of the adaptive refinement procedure. By considering the a priori convergence rate of the RKPM and the estimated error norm, an adaptive refinement strategy for the determination of optimal point distribution is proposed. In the suggested adaptive refinement scheme, the local refinement indicators used are computed by considering the partition of unity property of the RKPM shape functions. In addition, a simple but effective variable support size definition scheme is suggested to ensure the robustness of the adaptive RKPM procedure. The performance of the suggested adaptive procedure is tested by using it to solve several benchmark problems. Numerical results indicated that the suggested refinement scheme can lead to the generation of nearly optimal meshes for both smooth and singular problems. The optimal convergence rate of the RKPM is restored and thus the effectivity indices of the Z-Z

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Section: Global and Local Refinement Indicators And New Node Spacing mentioning

confidence: 99%

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part II: Adaptive refinement

Lee

Shuai

2006

Comput Mech

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“…From past experience [26,30,31], if the displacement (assumed strain or stress) "eld is interpolated in terms of a full¸agrangian polynomial (i.e. 4 polynomial terms 1, , , for linear interpolation) and the quality of the surface mesh generated is reasonably good and a suitable re"nement procedure is employed, can be neglected.…”

Section: Basic Notation and A Priori Error Estimationmentioning

confidence: 99%

“…The local smoothed stress "eld is constructed by a local patch least-squares "t procedure over those optimal sampling points of the patch similar to the one used in 2-D, 3-D and plate bending problems [2,21,25,31]. For the two quadrilateral elements tested in this paper, since complete linear Lagrangian polynomial terms (Table I and II) are used for natural strain and stress interpolation, the optimal sampling points are the 2;2 integration points of the element [35].…”

Section: A Pos¹eriori Error Estimation and Adaptive Refinementmentioning

confidence: 99%

“…(1) ¹he e+ectiveness of the adaptive re,nement procedure for the resolution of boundary layers and singularities: As demonstrated in previous studies [26,31], the presence of boundary layers and singularities inside the shell domain will seriously reduce the convergence rate of uniform re"nement, and only by using a suitable adaptive re"nement procedure can the optimal convergence rate be restored. In the numerical study, the performance of the adaptive re"nement procedure will be tested using di!erent elements for the solution of problems with singularities and strong boundary layers.…”

Section: Generalmentioning

confidence: 99%

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On using degenerated solid shell elements in adaptive refinement analysis

Lee

Sze

1999

Int. J. Numer. Meth. Engng.

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Three di!erent degenerated shell elements are studied in an adaptive re"nement procedure for the solution of shell problems. The stress recovery procedure expressed in a convective patch co-ordinate system is used for the construction of continuous smoothed stress "elds for the a posteriori error estimation. The performance of the stress recovery procedure, the error estimator and the adaptive re"nement strategy are tested by solving three benchmark shell problems. It is found that when adaptive re"nement is used, the adverse e!ects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates. It is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations. In addition, if complete Lagrangian polynomial terms are used, the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed. Figure 11. Contour plots for Problem 1, last uniform mesh: (Left column: Quada9s element, Central column: SQ9S element, Right column: ANS6S element) (a) contour plots for principal membrane force N ; (b) contour plots for shear force Q WWFigure 12. Contour plots for Problem 2. (Left column: Quada9s element, Central column: SQ9S element, Right column: ANS6S element): (a) contour plots for principal membrane force N ; (b) contour plots for shear force Q WW DEGENERATED SOLID SHELL ELEMENTS IN ADAPTIVE REFINEMENT ANALYSIS 643 Figure 13. Results for Problem 1 by adaptive re"nement: (a) last adaptive meshes; (b) shear force Q WW ; (c) membrane force Nwith width comparable to the thickness of the shell (Figure 6(b)), the shear error norm will hardly be reduced unless the size of the element is of the order of the thickness of the shell. Hence, in the uniform re"nement, the convergence rate is reduced as the shear error gradually dominates the total error. In adaptive re"nement, as the mesh is re"ned, more elements (Figure 13(a)) are automatically generated along the soft-simply supported and the free edges as the shear error become relative large. As a result, the convergence rate is improved and the shear error was reduced for all the elements tested in adaptive re"nement as shown in Figure Figure 15. (a) Convergence of Problem 1, total relative error; (b) convergence of Problem 1, shear relative error a value of 0)6}0)7. The number of polynomial terms used also explains the relative accuracy of these three elements. In the Quada9s element, at least six terms are used in the interpolation of all the strain components while at most "ve terms and three terms are used in the SQ9S and the ANS6S elements, respectively. Such an e!ect can also be seen by comparing the contour plots shown in Figures 13 and 14. 648 Figure 16. (a) Convergence of Problem 2, total relative error; (b) convergence of Problem 2, shear relative error Results of Problem 3For this problem, four meshes are used in the uniform re"nement and their characteris...

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“…For the thin plate problems one usually refers to the classical Zienkiewicz paper [12], such as, e.g. [13]. These papers focus on the treatment of the Reissner-Mindlin type plate, a model presumed to be much more easier for use in engineering practice, an assumption that still is widely discussed.…”

Section: Introductionmentioning

confidence: 99%