An Adaptive FEM for Buckling Analysis of Nonuniform Bernoulli-Euler Members via the Element Energy Projection Technique

Abstract

This paper presents a new adaptive finite element (FE) procedure for accurate, efficient, and reliable computation of the critical buckling loads and the associated modes of nonuniform Bernoulli-Euler members. After the conventional FE solution on a given mesh has been obtained, a novel conceptual and practical strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the recently developed element energy projection (EEP) technique to be readily used to calculate super-convergent FE solutions for buckling modes, which are subsequently used to estimate the FE solution errors and to further guide mesh refinements, until the accuracy of the FE solution matches the user-preset error tolerance. Numerical examples are given to show the effectiveness, efficiency, accuracy, and reliability of the proposed method, which also paves the way for development of an adaptive FE solver for more general and complicated eigenproblems.

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

“…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.…”

“…Furthermore, adaptive FEM has been applied to successfully solve structure eigenvalue problems: e.g. buckling problems for non-uniform Euler–Bernoulli beam members (Yuan et al ., 2013) and free vibration problems for two-dimensional structures (Yuan et al ., 2014).…”

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).…”

An hp-version adaptive finite element algorithm for eigensolutions of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation

“…Moreover, the accuracy of the FEM as a whole depends on the particular meshing method employed, as well as on user experience. In contrast, the adaptive FEM automatically adjusts the algorithm to seek the optimal solution to achieve the targeted accuracy (Gerasimov et al , 2015; Yuan et al , 2013), reduce the computation cost, improve the process of conventional finite element (FE) mesh design and improve the efficiency and accuracy of the solution (Zienkiewicz, 2006; Bespalov et al , 2017). It mainly includes the h -version adaptive method for increasing the mesh density (Zienkiewic and Zhu, 1987; Wang et al , 2019), the p -version adaptive method for increasing the element order (Douglas and Dupont, 1974; Arthurs et al , 2013) and the hp -version adaptive method (Bardell et al , 1997; Tews and Rachowica, 2009; Toth, 2019).…”