Local quality of smoothening based a-posteriori error estimators for laminated plates under transverse loading
TL;DR: In this paper, the local and global quality of various smoothening based a-posteriori error estimators is tested for symmetric laminated composite plates subjected to transverse loads.
Abstract: The local and global quality of various smoothening based a-posteriori error estimators is tested in this paper, for symmetric laminated composite plates subjected to transverse loads. Smoothening based on strain recovery and displacement-field recovery is studied here. Effect of ply orientation, laminate thickness, boundary conditions, mesh topology, and plate model is studied for a rectangular plate. It is observed that for interior patches of elements, both the estimators based on strain or displacement smoothening are reliable. For element patches at the boundary of the domain, all estimators tend to be unreliable (especially for angle-ply laminates). However, the strain recovery based estimator is clearly more robust for element patches at the boundary, as compared to displacement-recovery based error estimators. Globally, all the estimators tested here were found to be very robust.
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TL;DR: In this paper, a new quadrilateral four-node finite element is developed from a hybrid stress formulation involving, as primary variables, compatible displacements and element-wise equilibrated stress resultants.
Abstract: This paper presents a hybrid stress approach for the analysis of laminated composite plates. The plate mechanical model is based on the so called First-order Shear Deformation Theory, rationally deduced from the parent three-dimensional theory. Within this framework, a new quadrilateral four-node finite element is developed from a hybrid stress formulation involving, as primary variables, compatible displacements and elementwise equilibrated stress resultants. The element is designed to be simple, stable and locking-free. The displacement interpolation is enhanced by linking the transverse displacement to the nodal rotations and a suitable approximation for stress resultants is selected, ruled by the minimum number of parameters. The transverse stresses through the laminate thickness are reconstructed a posteriori by simply using three-dimensional equilibrium. To improve the results, the stress resultants entering the reconstruction process are first recovered using a superconvergent patch-based procedure called Recovery by Compatibility in Patches, that is properly extended here for laminated plates. This preliminary recovery is very efficient from the computational point of view and generally useful either to accurately evaluate the stress resultants or to estimate the discretization error. Indeed, in the present context, it plays also a key role in effectively predicting the shear stress profiles, since it guarantees the global convergence of the whole reconstruction strategy, that does not need any correction to accommodate equilibrium defects. Actually, this strategy can be adopted together with any plate finite element. Numerical testing demonstrates the excellent performance of both the finite element and the reconstruction strategy.
37 citations
TL;DR: In this paper, a detailed study of the quality of the point-wise stresses obtained using higher-order shear deformable, hierarchic and layerwise theories is done for a plate under transverse loading.
Abstract: The design of laminated composite based components requires a detailed analysis of the response of the structure when subjected to external loads. For the analysis of laminated composite plates, several plate theories have been proposed in the literature. Generally, these plate theories are used to obtain certain global response quantities like the buckling load. However, the use of these theories to obtain local response quantities, i.e. point-wise stresses; interlaminar stresses and strains, can lead to significant errors. In this paper, a detailed study of the quality of the point-wise stresses obtained using higher-order shear deformable, hierarchic and layerwise theories is done for a plate under transverse loading. The effect of equilibrium based post-processing on the transverse stress quantities is also studied. From the detailed study it is observed that the layerwise theory is very accurate. However, for all the models proper mesh design is required to capture boundary layer effects, discretization error, etc. Using focussed adaptivity, and post-processed state of stress, accurate representation of the local state of stress can be obtained, even with the higher-order shear deformable theories. Using this approach, the first-ply failure load is obtained with the Tsai-Wu criterion. It is observed that use of an adaptive procedure leads to significantly lower failure loads as compared to those given in the literature.
17 citations
Cites background or methods from "Local quality of smoothening based ..."
...An approximation to the error can be given as e* = u* uh where u 2 Sxy s is obtained for each element s as described below (see [35] for details)....
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...Various smoothening based a posteriori error estimation techniques for laminated composites have been proposed by the authors for the local quantity of interest [35]....
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...This definition is called the L2 projection based error estimator (see [35] for details)....
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...with NDOF = (pxy + 1 + k)(pxy + 2 + k)/2; qj(x,y) as the monomials of order 6pxy + k (see [35] for details) defined in terms of the local coordinates x̂ 1⁄4 x xc; ŷ 1⁄4 y yc....
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TL;DR: In this paper, a reliable shape optimization for laminated plate structures has been attempted for a fixed higher order plate model, a simple a-posteriori strain recovery algorithm, following ZZ type patch recovery technique, has been developed.
Abstract: ZZ type discretization error estimator for composite laminatesShape optimizationEffect of discretization error control on final shape and weight minimizationFirst-ply failure as a constraint In this study a reliable shape optimization for laminated plate structures has been attempted For a fixed higher order plate model, a simple a-posteriori strain recovery algorithm, following ZZ type patch recovery technique, has been developed The recovery is seen to be accurate The effect of higher approximation order and mesh refinement on the quality of the obtained solution quantities like stress components and displacements, is studied in detail The shape of the cutout is optimized with weight minimization as the objective function and the first-ply failure criterion as the constraint It is observed that control of the discretization error (via adaptive mesh refinements) leads to vastly different final designs, as compared to those obtained using reasonably refined meshes, but without adaptivity It is seen that without adaptivity, the design obtained is unsafe, as either more material removal is predicted or failure is predicted at higher loads, as compared to that obtained using adaptivity
16 citations
Cites result from "Local quality of smoothening based ..."
...More details of this estimator can be seen in earlier works of authors [46,47]....
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TL;DR: In this article, a region-by-region modeling strategy is presented for a chosen general family of equivalent, intermediate and layerwise models, which allows the user to put any model (of any order in the thickness direction) in any desired region of interest.
Abstract: Several plate models have been proposed in the literature for the analysis of laminated plates. These are based either on an equivalent through-thickness formulation or a layerwise formulation. It is shown in the literature that while the equivalent models are economical, the layerwise models are expensive but are also more accurate, especially with respect to the transverse stresses. Generally, the same model is used throughout the domain. The current study addresses the issue of economical and accurate computation of local stresses, strains and displacements (as well as global quantities) using combinations of layerwise, equivalent or intermediate models in various regions of the domain. A region-by-region modeling strategy is presented for a chosen general family of equivalent, intermediate and layerwise models. The proposed strategy allows the user to put any model (of any order in the thickness direction) in any desired region of interest. The effectiveness of the strategy is demonstrated through numerical examples. It is shown that this approach can significantly reduce computational cost and can also lead to good resolution of the local stress and displacement fields for domains with unsymmetric laminae, cut-outs, local damage, corner edges, sudden transition of boundary conditions and material.
10 citations
Cites background from "Local quality of smoothening based ..."
...This simple problem also highlights one of the major drawbacks of ad-hoc global–local computations, where the model and mesh are suitably refined only in the vicinity of the region of interest (for details see [39,40])....
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TL;DR: A novel adaptive procedure is presented, based on a posteriori estimation of the error in the quantity of interest, for a fixed plate model, which is very effective in controlling the local error to within the specified tolerances.
Abstract: Accurate computation of the critical response quantities for laminated composite structures has become essential, especially from the design and design certification point of view. Worst case scenario analysis (corresponding to the load envelope) of the structure require computation of local quantities of interest. In such a situation, control of both modelling error and discretisation error for the quantities of interest is required. In this study, for a fixed plate model, a novel adaptive procedure is presented, based on a posteriori estimation of the error in the quantity of interest. This focussed adaptive procedure involves prediction of the desired optimal mesh sizes in the neighborhood of the region of interest and away from the region of interest, based on an a priori estimate of the error in the quantity of interest. The final desired mesh is obtained in one shot. It is found that the error estimator, for the quantity of interest is reasonably robust. Further, the adaptive procedure is very effective in controlling the local error to within the specified tolerances.
9 citations
Cites background or methods from "Local quality of smoothening based ..."
...This definition is called the L2 projection based error estimator (see [4] for details)....
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...An approximation to the error can be given as e 1⁄4 u uh where u 2 Spþk s is obtained for each element s as described below (see [4] for details)....
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...with NDOF 1⁄4 ðp þ 1þ kÞðp þ 2þ kÞ=2; qjðx; yÞ as the monomials of order 6 p þ k (see [4] for details) defined in terms of the local coordinates x̂ 1⁄4 x xc, ŷ 1⁄4 y y c ....
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References
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TL;DR: A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes.
Abstract: A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes.
The estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials.
When combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable.
Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.
2,449 citations
TL;DR: The main theorem gives an error estimate in terms of localized quantities which can be computed approximately, and the estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same.
Abstract: A mathematical theory is developed for a class of a-posteriors error estimates of finite element solutions. It is based on a general formulation of the finite element method in terms of certain bilinear forms on suitable Hilbert spaces. The main theorem gives an error estimate in terms of localized quantities which can be computed approximately. The estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same. The theoretical results also lead to a heuristic characterization of optimal meshes, which in turn suggests a strategy for adaptive mesh refinement. Some numerical examples show the approach to be very effective.
1,431 citations
TL;DR: In this article, a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements.
Abstract: Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.
1,211 citations
Book•
01 Jan 1997
TL;DR: The What and Why of Fibrous Composites as mentioned in this paperibrous composites have been studied extensively in the literature, e.g., in the context of solid mechanics and 3-dimensional constitutive equations.
Abstract: The What and the Why of Fibrous Composites. Concepts of Solid Mechanics. 3--D Constitutive Equations. Plane Stress Constitutive Equations. Lamination Theory. Test Methods. Material Response. Interlaminar Stresses. Failure and Damage. Laminated Tubes. Micromechanics. Appendices. Indexes.
896 citations
"Local quality of smoothening based ..." refers background in this paper
...For a given lamina ‘l’, the generalised Hooke’s law (see [10,11]) gives:...
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TL;DR: Three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations are presented and it is proved that as the mesh size decreases, under suitable assumptions, two of the error estimator approach upper bounds on the norm of the true error.
Abstract: We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.
815 citations