Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error

Dı́ez, Pedro; Ródenas, Juan José; Zienkiewicz, O. C.

doi:10.1002/nme.1837

Cited by 50 publications

(86 citation statements)

References 31 publications

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“…al. [33] introduced an expression to evaluate an upper bound of the error in the energy norm with a correction term to account for the lack of internal equilibrium. A generalization of that expression is introduced in (15), which does not only include the lack of internal equilibrium but also the lack of boundary equilibrium making use of the two correction terms shown in (16):…”

Section: Error Estimation a Spr-based Upper Bounding Techniquementioning

confidence: 99%

“…The most conservative root provides the upper bound in the energy norm up to the constant C, see [33],…”

Section: Particular Case Using the Spr-c Recovery Techniquementioning

confidence: 99%

“…In both tables the second, third and fourth columns are the effectivity indices for the techniques as indicated in (36). The last column corresponds to the constant C * Ω evaluated using (33). An analytical value of the constant is also evaluated, for comparison purposes, from (18) since the analytical solution is known.…”

Section: Problem 1: 3 Rd Order Polynomial Displacements In a 2×2 Squarementioning

confidence: 99%

“…Díez et al [33] presented a methodology to obtain computable upper bounds of the error in the energy norm considering the stress recovered field provided by the SPR-C technique and taking into account the lack of internal equilibrium of the recovered stresses. This technique allowed to obtain one of the firsts procedures to get practical upper bounds for FEM and XFEM based on recovery techniques [33,34]. Since the SPR-C technique does not provide an equilibrated stress field at the global level, the upper bound property is obtained by adding a correction term to account for equilibrium defaults.…”

Section: Introductionmentioning

confidence: 99%

“…The technique presented is based on the previously mentioned recovery technique whose use is crucial in the process since the improved field provided by this technique is continuous and fulfils the boundary equilibrium. The main idea of the proposed technique is to obtain an upper bound of the error in the energy norm by evaluating an upper bound of the error correction term introduced in [33] using an explicit-type error evaluation technique, that requires the use of a problem-dependent constant. We will also propose a procedure to obtain an accurate evaluation of the constant required by the explicit-type error evaluation technique which allows to obtain error upper bounds in practical application, even for coarse meshes.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A recovery-explicit error estimator in energy norm for linear elasticity

Nadal

Dı́ez

Ródenas

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recoverybased estimates is proposed. The idea is 1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, 2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and 3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The the numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications.Error bounding, recovery techniques, explicit residual error estimator

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Section: Error Estimation a Spr-based Upper Bounding Techniquementioning

confidence: 99%

“…The most conservative root provides the upper bound in the energy norm up to the constant C, see [33],…”

Section: Particular Case Using the Spr-c Recovery Techniquementioning

confidence: 99%

Section: Problem 1: 3 Rd Order Polynomial Displacements In a 2×2 Squarementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A recovery-explicit error estimator in energy norm for linear elasticity

Nadal

Dı́ez

Ródenas

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Recovery‐Based Error Estimation and Bounding in XFEM

González‐Estrada,

Ródenas García,

Bordas

2023

Partition of Unity Methods

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Hybridizable discontinuous Galerkin p‐adaptivity for wave propagation problems

Giorgiani

Fernández‐Méndez

Huerta

2013

Numerical Methods in Fluids

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SUMMARYA p-adaptive Hybridizable Discontinuous Galerkin method for the solution of wave problems is presented in a challenging engineering problem. Moreover, its performance is compared with a high-order continuous Galerkin. The hybridization technique allows to reduce the coupled degrees of freedom to only those on the mesh element boundaries, while the particular choice of the numerical fluxes opens the path to a superconvergent post-processed solution. This super-convergent post-processed solution is used to construct a simple and inexpensive error estimator. The error estimator is employed to obtain solutions with the prescribed accuracy in the area (or areas) of interest and also drives a proposed iterative mesh adaptation procedure. The proposed method is applied to a non-homogeneous scattering problem in an unbounded domain. This is a challenging problem because, on one hand, for high frequencies numerical difficulties are an important issue due to the loss of the ellipticity and the oscillatory behavior of the solution. And on the other hand, it is applied to real harbor agitation problems. That is, the Mild Slope equation in frequency domain (Helmholtz equation with non-constant coefficients) is solved on real geometries with the corresponding perfectly matched layer to damp the diffracted waves. The performance of the method is studied on two practical examples. The adaptive Hybridizable Discontinuous Galerkin method exhibits better efficiency compared to a high-order continuous Galerkin method using static condensation of the interior nodes.

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Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error

Cited by 50 publications

References 31 publications

A recovery-explicit error estimator in energy norm for linear elasticity

A recovery-explicit error estimator in energy norm for linear elasticity

Recovery‐Based Error Estimation and Bounding in XFEM

Hybridizable discontinuous Galerkin p‐adaptivity for wave propagation problems

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