Zienkiewicz-Zhu Error Estimator Research Articles

Polygonal finite element modeling of crack propagation via automatic adaptive mesh refinement

Polygonal finite element provides a great flexibility in mesh generation of crack propagation problems where the topology of the domain changes significantly. However, the control of the discretization error in such problems is a main concern. In this paper, a polygonal-FEM is presented in modeling of crack propagation problems via an automatic adaptive mesh refinement procedure. The adaptive mesh refinement is accomplished based on the Zienkiewicz–Zhu error estimator in conjunction with a weighted SPR technique. Adaptive mesh refinement is employed in some steps for reduction of the discretization error and not for tracking the crack. In the steps that no adaptive mesh refinement is required, local modifications are applied on the mesh to prevent poor polygonal element shapes. Finally, several numerical examples are analyzed to demonstrate the efficiency, accuracy and robustness of the proposed computational algorithm in crack propagation problems.

Implementation and performance of adaptive mesh refinement in the Ice Sheet System Model (ISSM v4.14)

Abstract. Accurate projections of the evolution of ice sheets in a changing climate require a fine mesh/grid resolution in ice sheet models to correctly capture fundamental physical processes, such as the evolution of the grounding line, the region where grounded ice starts to float. The evolution of the grounding line indeed plays a major role in ice sheet dynamics, as it is a fundamental control on marine ice sheet stability. Numerical modeling of a grounding line requires significant computational resources since the accuracy of its position depends on grid or mesh resolution. A technique that improves accuracy with reduced computational cost is the adaptive mesh refinement (AMR) approach. We present here the implementation of the AMR technique in the finite element Ice Sheet System Model (ISSM) to simulate grounding line dynamics under two different benchmarks: MISMIP3d and MISMIP+. We test different refinement criteria: (a) distance around the grounding line, (b) a posteriori error estimator, the Zienkiewicz–Zhu (ZZ) error estimator, and (c) different combinations of (a) and (b). In both benchmarks, the ZZ error estimator presents high values around the grounding line. In the MISMIP+ setup, this estimator also presents high values in the grounded part of the ice sheet, following the complex shape of the bedrock geometry. The ZZ estimator helps guide the refinement procedure such that AMR performance is improved. Our results show that computational time with AMR depends on the required accuracy, but in all cases, it is significantly shorter than for uniformly refined meshes. We conclude that AMR without an associated error estimator should be avoided, especially for real glaciers that have a complex bed geometry.

Element Free Galerkin Post-processing Technique Based Error Estimator for Elasticity Problems

The study present a Mesh Free based post-processing technique for asymptotically (upper) bounded error estimator for Finite Element Analyses of elastic problems. The proposed technique uses Galerkin Element Free procedure for recovery of the displacement derivatives over a patch of nodes in radial domains. The radial nodes patches are used to construct the trial shape functions utilizing the moving least-squares (MLS) techniques. The proposed technique has been tested on three benchmark elastic problems discretized using 4-node quadrilateral elements. The recovered nodal stresses are utilized to calculate the error in finite element solution in energy norm. The study also demonstrates adaptive analysis application of proposed error estimator. The performance of proposed error estimator based on mesh independent node patches has been compared with that of mesh dependent node patches based Zienkiewicz-Zhu (ZZ) error estimator on structured and unstructured mesh. The improved results of the proposed error estimator in terms of convergence rate and effectivity are obtained. It is shown that present study incorporates the superiority of the Mesh Free Galerkin method into finite element analysis environment.

A New Adaptive Mesh Refinement Method in FEA Based on Magnetic Field Conservation at Elements Interfaces and Non-Conforming Mesh Refinement Technique

Mesh quality strongly affects the solution accuracy in electromagnetic finite-element analysis. Hence, the realization of adequate mesh generation becomes a very important task. Several adaptive meshing methods for automatic adjustments of the mesh density in accordance with the shape and complexity of the analyzed problem have been proposed. However, the most of them are not enough robust, some are quite laborious and could not be universally used for adaptive meshing of complex analysis models. In this paper, a new adaptive mesh refinement method based on magnetic field conservation at the border between finite elements is proposed. The proposed error estimation method provides easy mesh refinements, and generates smaller element within regions with large curvature of the magnetic flux lines. The proposed adaptive mesh refinement method based on non-conforming edge finite elements, which could avoid generation of flat or ill-shaped elements, was applied to a simple magnetostatic permanent magnet model. To confirm the validity and accuracy, the obtained results were compared with those obtained by means of the Zienkiewicz-Zhu (ZZ) error estimator. The results show that the computational error using the proposed method was reduced down to 1% compared with that of the ZZ method, which yields error of 8.6%, for the same model.

The backward euler anisotropic a posteriori error analysis for parabolic integro-differential equations

We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro-differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz-Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016

Three-dimensional cohesive fracture modeling of non-planar crack growth using adaptive FE technique

In this paper, the three-dimensional adaptive finite element modeling is presented for cohesive fracture analysis of non-planer crack growth. The technique is performed based on the Zienkiewicz–Zhu error estimator by employing the modified superconvergent patch recovery procedure for the stress recovery. The Espinosa–Zavattieri bilinear constitutive equation is used to describe the cohesive tractions and displacement jumps. The 3D cohesive fracture element is employed to simulate the crack growth in a non-planar curved pattern. The crack growth criterion is proposed in terms of the principal stress and its direction. Finally, several numerical examples are analyzed to demonstrate the validity and capability of proposed computational algorithm. The predicted crack growth simulation and corresponding load-displacement curves are compared with the experimental and other numerical results reported in literature.

Numerical simulation of multiple crack growth in brittle materials with adaptive remeshing

AbstractIn this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in multiple cracked bodies. The Zienkiewicz–Zhu error estimator is employed in conjunction with the modified superconvergent patch recovery (SPR) technique to obtain more accurate estimation of error. A stability analysis is performed to determine active cracks from a set of competitive cracks. Various crack growth criteria together with the respective crack trajectory prediction are compared. Several numerical examples are illustrated to demonstrate the efficiency, robustness and accuracy of computational algorithm in the simulation of multiple crack growth. Copyright © 2010 John Wiley & Sons, Ltd.

Modeling of cohesive crack growth using an adaptive mesh refinement via the modified-SPR technique

In this paper, an adaptive finite element procedure is presented in modeling of mixed-mode cohesive crack propagation via the modified superconvergent path recovery technique. The adaptive mesh refinement is performed based on the Zienkiewicz–Zhu error estimator. The weighted-SPR recovery technique is employed to improve the accuracy of error estimation. The Espinosa–Zavattieri bilinear cohesive zone model is applied to implement the traction-separation law. It is worth mentioning that no previous information is necessary for the path of crack growth and no region of the domain is necessary to be filled by the cohesive elements. The maximum principal stress criterion is employed for predicting the direction of extension of the cohesive crack in order to implement the cohesive elements. Several numerical examples are analyzed numerically to demonstrate the capability and efficiency of proposed computational algorithm.

3D adaptive finite element modeling of non-planar curved crack growth using the weighted superconvergent patch recovery method

In this paper, an adaptive finite element analysis is presented for 3D modeling of non-planar curved crack growth. The fracture mechanical evaluation is performed based on a general technique for non-planar curved cracks. The Schollmann’s crack kinking criterion is used for the process of crack propagation in 3D problems. The Zienkiewicz–Zhu error estimator is employed in conjunction with a weighted SPR technique at each patch to improve the accuracy of error estimation. Applying the proposed technique to 3D non-planar curved crack growth problems shows significant improvements particularly at the boundaries and near crack tip regions. Several numerical examples are presented to illustrate the robustness of the proposed technique.

Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique

In this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in a 2D finite element mesh. The Zienkiewicz–Zhu error estimator is employed in conjunction with a modified SPR technique based on the recovery of gradients using analytical crack-tip fields in order to obtain more accurate estimation of errors. The optimization of crack-tip singular finite element size is achieved through the adaptive mesh strategy. Finally, several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of computational algorithm in calculation of fracture parameters and prediction of crack path pattern.

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

AbstractAn s‐adaptive finite element procedure is developed for the transient analysis of 2‐D solid mechanics problems with material non‐linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user‐specified tolerances. The spatial error is quantified by the Zienkiewicz–Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third‐order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s‐adaptive procedure is the use of finite element mesh superposition (s‐refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non‐linear transient problems since it is faster, simpler and more efficient than traditional h‐refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s‐adaptive method for quasi‐static and transient problems with material non‐linearity. Copyright © 2007 John Wiley & Sons, Ltd.

Adaptive finite element analysis of multi-phase problems in geotechnics

Adaptive FE solution strategies for coupled two-phase and three-phase problems involving porous soils with the pore space filled with water and/or (compressed) air are presented. The errors are estimated by applying the well-known Zienkiewicz–Zhu error estimator to each phase. The mesh is improved by mesh regeneration from scratch employing the advancing front method. Two different adaptive strategies, based on the error estimation either of incremental or total quantities, are pursued. The developed adaptive strategies are applied to the numerical simulation of a laboratory test and a full-scale in situ test, considering the flow of compressed air in an elastic deformable soil skeleton.

The polynomial-preserving recovery for higher order finite element methods in 2D and 3D

The Polynomial-Preserving Recovery (PPR) technique is extended to recover continuous gradients from $C^0$ finite element solutions of an arbitrary order in 2D and 3D problems. The stability of the PPR is theoretically investigated in a general framework. In 2D, the stability is established under a simple geometric condition. The numerical experiments demonstrated that the PPR-recovered gradient enjoys superconvergence, and the Zienkiewicz-Zhu error estimator based on the PPR-recovered gradient is asymptotically exact.

Smart element method I. The Zienkiewicz–Zhu feedback

AbstractA new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz–Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz–Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self‐adaptive and a self‐adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions.The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz–Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd.

An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio

An adaptive algorithm is proposed for solving the Stokes problem with continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio. Anisotropic, a posteriori error estimates in the H 1 × L 2 norm are derived, the constant being independent from the mesh aspect ratio. The estimates involve the first derivatives of the velocity error and the velocity of a dual problem having the pressure error in the right side. An explicit error indicator is then proposed. The first derivatives of the velocity error are approximated using Zienkiewicz–Zhu error estimator (post-processing). The pressure error in the right side of the dual problem is approximated by smoothing. Numerical results on meshes with large aspect ratio show that the error indicator is robust whenever the pressure error in the dual problem is approximated with sufficient accuracy. Finally, an adaptive algorithm is proposed, with goal to build an anisotropic triangulation such that the relative estimated error is close to a preset tolerance.

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral nite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz{Zhu (ZZ) type error estimators are derived which originate from dierent backgrounds. In the course of the analysis, the rst estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two dierent analyses of the estimators is that they allow dierent and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and eciency of the ZZ error estimation is obtained. The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and conrmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diusion problem.

Anisotropic, adaptative finite elements for the computation of a solutal dendrite

We compute solutions of solutal phase-field models for dendritic growth of an isothermal binary alloy using anisotropic mesh refinement techniques. The adaptive strategy is based on anisotropic a posteriori estimators using a superconvergent recovery technique in the form of the Zienkiewicz-Zhu error estimator. The phase-field model contains an anisotropic strongly nonlinear second order operator modelling the dendritic branches, this strong nonlinearity is included in the a posteriori error estimators by using a monotonicity result. The monotonicity holds for phase-field anisotropy below a certain threshold value beyond which there are no known well-posedness results. We present computational results for both regimes showing the performance of the proposed method.

Adaptive Meshing for Deep-drawing Process

This paper incorporates the concept of adaptive meshing for finite-element analysis of the deep-drawing process. In adaptive meshing, the mesh is automatically refined both in the areas of insufficient accuracy and sharp stress gradients. The Zienkiewicz-Zhu error estimator based upon the difference between the finite-element solution and the corresponding smoothened solution is used to judge the accuracy both at the element and the global levels. The post-processing for determining more accurate solutions is done by fitting a higher order polynomial expansion to the fmite-element solution in nodal patches. An illustrative problem is solved and the adaptive refmement at different stages of deformation is presented.

Dynamic loading of polycrystalline shape memory alloy rods

Shape memory alloys (SMAs) have recently been considered for dynamic loading applications for energy absorbing and vibration damping devices. An SMA body subjected to external dynamic loading will experience large inelastic deformations that will propagate through the body as phase transformation and/or detwinning shock waves. The wave propagation problem in a cylindrical polycrystalline SMA rod induced by an impact loading is considered in this paper. Numerical solutions for various boundary conditions are presented for stress induced martensite and detwinning of martensite. The numerical simulations utilize an adaptive finite element method (FEM) based on the Zienkiewicz–Zhu error estimator. Selected results are compared to known analytical solutions to verify the adaptive FEM approach. The energy dissipation in an SMA rod is evaluated for a square pulse stress input applied at various temperatures involving both stress induced martensite and detwinning of martensite. The dynamic response of a NiTi SMA rod is also studied experimentally in a split Hopkinson bar apparatus under detwinning conditions. Strain history records obtained by strain gauges placed at different locations along the SMA rod are compared with numerical simulations for a square pulse stress input. The quasi-static and dynamic stress–strain hysteretic response of the SMA, both due to detwinning, are found to be nearly identical. The quasi-static tests are used to calibrate the rate independent constitutive model used for the numerical simulations, which are found to match the experimental observations reasonably well.

Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz–Zhu error estimator

AbstractThe framework of Formaggia and Perotto (Numerische Mathematik 2001; 89: 641–667) is considered to derive a new anisotropic error indicator for a Laplace problem in the energy norm. The matrix containing the error gradient is approached using a Zienkiewicz–Zhu error estimator. A numerical study of the effectivity index is proposed for anisotropic unstructured meshes, showing that our indicator is sharp. An anisotropic adaptive algorithm is implemented, aiming at controlling the estimated relative error. Copyright © 2003 John Wiley & Sons, Ltd.