Extension of the Zienkiewicz–zhu Error Estimator to Shape Sensitivity Analysis

Fuenmayor, F. J.; Oliver, Javier; Ródenas, Juan José

doi:10.1002/(sici)1097-0207(19970430)40:8<1413::aid-nme118>3.0.co;2-#

Cited by 17 publications

(8 citation statements)

References 12 publications

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“…Fuenmayor et al 4 extended the Zienkiewicz–Zhu error estimator 7 to shape sensitivity analysis in order to develop a discretization error estimator for shape sensitivity analysis, obtaining the expression shown in (9). This proves that the sensitivity of the error is equivalent to the error in sensitivities .…”

Section: Evolutionary Algorithms With Mesh Generation By Projectiomentioning

confidence: 99%

“…To solve this problem, in this paper we propose the use of the h ‐adaptive analysis technique for generations of individuals described in 1. The procedure consists of using the sensitivity analysis of the discretization error in energy norm with respect to the design variables 2–4 to project the error obtained from a representative individual of the generation into the different geometrical configurations to be analyzed. H ‐adapted meshes for the analysis of each particular individual can be created with this information, which in the vast majority of cases provides FE results with the prescribed accuracy; thus, avoiding the high computational cost associated with the full h ‐adaptive remeshing.…”

Section: Introductionmentioning

confidence: 99%

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On the need for the use of error‐controlled finite element analyses in structural shape optimization processes

Ródenas

Bugeda

Albelda

et al. 2011

Numerical Meth Engineering

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This is the peer reviewed version of the following article: Ródenas, J. J., Bugeda, G., Albelda, J. and Oñate, E. (2011), On the need for the use of error-controlled finite element analyses in structural shape optimization processes. SUMMARYThis work analyzes the influence of the discretization error associated to the Finite Element (FE) analyses of each design configuration proposed by the structural shape optimization algorithms over the behavior of the algorithm. The paper clearly shows that if FE analyses are not accurate enough, the final solution provided by the optimization algorithm will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper proposes the combination of two strategies to reduce the computational cost related to the use of mesh adaptivity in evolutionary optimization algorithms: (a) the use of an algorithm for the mesh generation by projection of the discretization error, which reduces the computational cost associated to the adaptive FE analysis of each geometrical configuration and, (b) the successive increase of the required accuracy of the FE analyses in order to obtain a considerable reduction of the computational cost in the early stages of the optimization process.

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Section: Evolutionary Algorithms With Mesh Generation By Projectiomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the need for the use of error‐controlled finite element analyses in structural shape optimization processes

Ródenas

Bugeda

Albelda

et al. 2011

Numerical Meth Engineering

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“…Remark 1 To simplify the evaluation of ∂σ * ∂a m we considered ∂σ * ∂a m = ∂σ ∂a m * . The numerical results will show that this approximation, previously used in [19], does not influence the results.…”

Section: Evaluation Of Derivativesmentioning

confidence: 84%

An extension of shape sensitivity analysis to an immersed boundary method based on Cartesian grids

et al. 2017

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Ródenas, JJ.; Fuenmayor Fernández, F.; Tur Valiente, M. (2018). An extension of shape sensitivity analysis to an immersed boundary method based on Cartesian grids. Computational Mechanics. 62(4):701-723. https://doi.Abstract Gradient-based shape optimization processes of mechanical components require the gradients (sensitivity) of the magnitudes of interest to be calculated with sufficient accuracy. The aim of this study was to develop algorithms for the calculation of shape sensitivities considering geometric representation by parametric surfaces (i.e. NURBS or T-splines) using 3D Cartesian h-adapted meshes independent of geometry. A formulation of shape sensitivities was developed for an environment based on Cartesian meshes independent of geometry, which implies, for instance, the need to take into account the special treatment of boundary conditions imposed in non body-fitted meshes. The immersed boundary framework required to implement new methods of velocity field generation, which have a primary role in the integration of both the theoretical concepts and the discretization tools in shape design optimization. Examples of elastic problems with three-dimensional components are given to demonstrate the efficiency of the algorithms.

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“…Speciÿcally, as noted recently by Fuenmayor et al [33], the success of adaptive reÿnement is highly sensitive to the initial mesh resolution. If the initial mesh is too coarse, the solution based on adaptive mesh reÿnement may never converge to the true solution.…”

Section: Two-dimensional Applicationsmentioning

confidence: 88%

“…Since the error estimator uses a smoothed stress ÿeld from the FE approximation rather than the true stress ÿeld, its e cacy depends on the chosen manner of smoothing. It is now recognized [33] that the L 2 projection method is asymptotically correct only when the mesh discretization error is modest. Recently, new smoothing techniques like Super-convergent Patch Recovery [34] and Recovery by Equilibrium Patches [35] have been developed to enhance the precision in estimating the discretization error.…”

Section: Voxel-based Meshing and Adaptive Reÿnementmentioning

confidence: 99%

Voxel‐based meshing and unit‐cell analysis of textile composites

Kim

Swan

2003

Numerical Meth Engineering

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SUMMARYUnit-cell homogenization techniques are frequently used together with the ÿnite element method to compute e ective mechanical properties for a wide range of di erent composites and heterogeneous materials systems. For systems with very complicated material arrangements, mesh generation can be a considerable obstacle to usage of these techniques. In this work, pixel-based (2D) and voxel-based (3D) meshing concepts borrowed from image processing are thus developed and employed to construct the ÿnite element models used in computing the micro-scale stress and strain ÿelds in the composite. The potential advantage of these techniques is that generation of unit-cell models can be automated, thus requiring far less human time than traditional ÿnite element models. Essential ideas and algorithms for implementation of proposed techniques are presented. In addition, a new error estimator based on sensitivity of virtual strain energy to mesh reÿnement is presented and applied. The computational costs and rate of convergence for the proposed methods are presented for three di erent mesh-reÿnement algorithms: uniform reÿnement; selective reÿnement based on material boundary resolution; and adaptive reÿnement based on error estimation.

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Extension of the Zienkiewicz–zhu Error Estimator to Shape Sensitivity Analysis

Cited by 17 publications

References 12 publications

On the need for the use of error‐controlled finite element analyses in structural shape optimization processes

On the need for the use of error‐controlled finite element analyses in structural shape optimization processes

An extension of shape sensitivity analysis to an immersed boundary method based on Cartesian grids

Voxel‐based meshing and unit‐cell analysis of textile composites

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