A Super-Convergence Strategy for Two-Dimensional FEM Based on Element Energy Projection Technique

Yuan, Songmei; Wu, Yue; Xu, Junjie; Yuan, Zhanbin; Qiu, Xing; Ye, Kangsheng

doi:10.1166/jno.2017.2272

Cited by 2 publications

(5 citation statements)

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“…However, it is found that the assumption does not hold true for multi-dimensional FEM and hence special treatments are made with the concept of recursive dimension-by-dimension (D-by-D) discretization and recovery. Viewing a traditional 2D FE discretization as a two-step 1D discretization, the D-by-D recovery technique is implemented by successive application of 1D-based EEP technique, which has been proved effective in 2D and even three-dimensional FE analysis (Yuan et al , 2017; Yuan et al , 2018).…”

Section: Element Energy Projection Solutionmentioning

confidence: 99%

“…However, it is also observed that to maintain the same convergence orders as the recovered solution u ˜˚ , it is not necessary to use the highest possible accuracy for { d }, { d ′} and { d ′}. Through a large amount of intuitive analyses and numerical experiments, it has been found by Yuan et al (2017, 2018) that, for element of degree m , as long as the convergence orders no less than those shown in Table 1, equation (17) can render super-convergent EEP displacement solution u ∗ , which is at least one order higher than the FE solution u h in accuracy and hence can be used as an error estimator. Table 1 also gives examples of some qualified quantities from FE solution.…”

Section: Element Energy Projection Solutionmentioning

confidence: 99%

“…Table 1 also gives examples of some qualified quantities from FE solution. For more detailed discussions of EEP super-convergent solutions for 2D FEM, please refer to Yuan et al (2017, 2018).…”

Section: Element Energy Projection Solutionmentioning

confidence: 99%

“…Usually, especially for 1D eigenproblems, higher-order modes exhibit more complicated variations and need finer meshes than lower-order ones. For the sake of efficiency, a mesh inheritance technique has been used in 1D case (Yuan et al , 2017), in which the initial FE mesh πk+1(0) for the ( k + 1)-th order eigenpair would simply inherit the final mesh of the previous order, i.e. πk+1(0)=πk.…”

Section: Mesh Refinementmentioning

confidence: 99%

“…In the framework of the EEP method, not only super-convergent derivatives but also displacements can be recovered. The method has been successfully applied to a series of linear (Yuan et al , 2017; Yuan et al , 2018; Yuan and He, 2006; Dong et al , 2019) and nonlinear problems (Yuan et al , 2017; Yuan et al , 2014; Jiang et al , 2020). All the numerical results in the above-mentioned applications show that the EEP solutions can achieve super-convergence at least one order higher than the corresponding FE solutions, which has been mathematically proved in one-dimensional (1D) cases (Yuan and Xing, 2014).…”

Section: Introductionmentioning

confidence: 99%

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Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique

Sun

Yuan

2021

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Purpose A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique. Design/methodology/approach By linearizing the free vibration problem of elastic membranes into a series of linear equivalent problems, reliable a posteriori point-wise error estimator is constructed via EEP super-convergent technique. Hierarchical local mesh refinement is incorporated to better deal with tough problems. Findings Several classical examples were analyzed, confirming the effectiveness of the EEP-based error estimation and overall adaptive procedure equipped with a local mesh refinement scheme. The computational results show that the adaptively-generated meshes reasonably catch the difficulties inherent in the problems and the procedure yields both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm. Originality/value By reasonable linearization, the linear-problem-based EEP technique is successfully transferred to two-dimensional eigenproblems with local mesh refinement incorporated to effectively and flexibly deal with singularity problems. The corresponding adaptive strategy can produce both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm and thus can be expected to apply to other types of eigenproblems.

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Section: Element Energy Projection Solutionmentioning

confidence: 99%

Section: Element Energy Projection Solutionmentioning

confidence: 99%

Section: Element Energy Projection Solutionmentioning

confidence: 99%

Section: Mesh Refinementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique

Sun

Yuan

2021

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An adaptive nonlinear finite element analysis of minimal surface problem based on element energy projection technique

Jiang

Yuan

Qiu

2020

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Purpose This paper aims to propose a new adaptive strategy for two-dimensional (2D) nonlinear finite element (FE) analysis of the minimal surface problem (MSP) based on the element energy projection (EEP) technique. Design/methodology/approach By linearizing nonlinear problems into a series of linear problems via the Newton method, the EEP technique, which is an effective and reliable point-wise super-convergent displacement recovery strategy for linear FE analysis, can be directly incorporated into the solution procedure. Accordingly, a posteriori error estimate in maximum norm was established and an adaptive 2D nonlinear FE strategy of h-version mesh refinement was developed. Findings Three classical known surfaces, including a singularity problem, were analysed. Moreover, an example whose analytic solution is unavailable was considered and a comparison was made between present results and those computed by the MATLAB PDE toolbox. The results show that the adaptively-generated meshes reflect the difficulties inherent in the problems and the proposed adaptive analysis can produce FE solutions satisfying the user-preset error tolerance in maximum norm with a fair adaptive convergence rate. Originality/value The EEP technique for linear FE analysis was extended to the nonlinear procedure of MSP and can be expected to apply to other 2D nonlinear problems. The employment of the maximum norm makes point-wisely error control on the sought surfaces possible and makes the proposed method distinguished from other adaptive FE analyses.

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A Super-Convergence Strategy for Two-Dimensional FEM Based on Element Energy Projection Technique

Cited by 2 publications

References 0 publications

Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique

Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique

An adaptive nonlinear finite element analysis of minimal surface problem based on element energy projection technique

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