Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique

Abstract

FindingsNumerical results for a number of representative and challenging SL problems are presented to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method. Research limitationsThe method is limited to regular SL problems, but it can also solve some singular SL problems in an indirect way. ValueComprehensive utilization of the EEP technique yields a simple, efficient and reliable adaptive FE procedure that finds sufficiently fine meshes for preset error tolerances on eigenvalues and eigenfunctions to be achieved, even on problems which proved troublesome to competing methods. The method can readily be extended to vector SL problems.

Adaptive finite element analysis for damage detection of non-uniform Euler–Bernoulli beams with multiple cracks based on natural frequencies

“…(3) The material properties (i.e. flexural rigidity and linear density) are reduced at the cracks using (28) where  and  are the exact crack depth and location or reliable results obtained through other methods. For all the examples, it was found that the present procedure produced satisfactory results, with both locations and sizes of cracks fully satisfying the pre-specified error tolerance.…”

“…As the exact solutions ( α i , β i ) are not usually available, the proposed procedure uses the following stop criterion instead: where ω k and ωkh are the actual and computed frequencies of cracked beams, respectively. The above stop criteria in absolute error estimation for eigensolutions in adaptive analysis show satisfying effect (Yuan et al ., 2013; Yuan et al ., 2014, 2017).…”

“…Using the above Sturm sequence property and the convectional bisection method (Clough and Penzien, 1993), the intervals of each eigenvalue can be determined, and the inverse iteration technique is successfully introduced to compute the eigenpairs (Yuan et al ., 2013; Yuan et al ., 2017). Based on these considerations, the following inverse iteration procedure is adopted: where i is the loop index.…”

Adaptive finite element–discrete element analysis for stratal movement and microseismic behaviours induced by multistage propagation of three-dimensional multiple hydraulic fractures

“…Recently, using the finite element method (FEM) for the treatment of continuous media and discrete element method (DEM) for discontinuous media, a combined finite element (FE) and discrete element (DE) method has been proposed to simulate the evolution of induced damages, fracture propagation and failure processes in geotechnical materials and concretes (Lisjak et al, 2018;Munjiza et al, 1995;Profit et al, 2016;Z arate et al, 2018). With the continued development of computational methods, several high-performance, adaptive analysis algorithms and techniques for mesh refinement have been used to overcome mathematical problems for certain structures and engineering applications (Wang et al, 2018b;Yuan et al, 2013;Yuan et al, 2017). In particular, dynamic fractures in solids have been simulated to provide accurate solutions by refining meshes around fracture tips (Azadi and Khoei, 2011;Cornejo et al, 2019;Profit et al, 2016).…”

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).…”