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

Abstract

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.

An improved local error estimate in adaptive finite element analysis based on element energy projection technique

“…In recent years, a novel recovery-based technique named element energy projection (EEP) method was proposed by Yuan and Wang (2004), Yuan and Lin (2007). The method has been successfully applied to a series of linear (Yuan et al, 2018;Yuan and He, 2006;Dong et al, 2019) and nonlinear problems (Yuan et al, 2017Jiang et al, 2020;Sun and Yuan, 2021). The original EEP method in multi-dimensional problems (2D or 3D) poses strict restrictions on the mesh patterns, i.e.…”

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

“…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).…”