Research progress of high-performance BEM and investigation on convergence of GMRES in local stress analysis of slender real thin-plate beams

• Published in: Engineering computations Vol. 36; no. 8; pp. 2530 - 2556
• Main Authors: Yao, Zhenhan, Zheng, Xiaoping, Yuan, Han, Feng, Jinlong
• Format: Journal Article
• Language: English
• Published: Bradford Emerald Publishing Limited 15.10.2019; Emerald Group Publishing Limited
• Subjects: Accuracy; Adaptive algorithms; Algebra; Algorithms; Boundary conditions; Boundary element method; Cantilever beams; Cantilever plates; Computational efficiency; Computing time; Convergence; Deformation; Engineering; Error analysis; Integral equations; Iterative methods; Mathematical analysis; Nodes; Partial differential equations; Preconditioning; Shells (structural forms); Solvers; Stress analysis; Two dimensional analysis; High performance BEM; Convergence of GMRES; Preconditioning; Real beam, Plate and shell; High accuracy BEM; Local stress analysis
• ISSN: 0264-4401; 1758-7077
• DOI: 10.1108/EC-10-2018-0477

Purpose Based on the error analysis, the authors proposed a new kind of high accuracy boundary element method (BEM) (HABEM), and for the large-scale problems, the fast algorithm, such as adaptive cross approximation (ACA) with generalized minimal residual (GMRES) is introduced to develop the high performance BEM (HPBEM). It is found that for slender beams, the stress analysis using iterative solver GMRES will difficult to converge. For the analysis of slender beams and thin structures, to enhance the efficiency of GMRES solver becomes a key problem in the development of the HPBEM. The purpose of this paper is study on the preconditioning method to solve this convergence problem, and it is started from the 2D BE analysis of slender beams. Design/methodology/approach The conventional sparse approximate inverse (SAI) based on adjacent nodes is modified to that based on adjacent nodes along the boundary line. In addition, the authors proposed a dual node variable merging (DNVM) preprocessing for slender thin-plate beams. As benchmark problems, the pure bending of thin-plate beam and the local stress analysis (LSA) of real thin-plate cantilever beam are applied to verify the effect of these two preconditioning method. Findings For the LSA of real thin-plate cantilever beams, as GMRES (m) without preconditioning applied, it is difficult to converge provided the length to height ratio greater than 50. Even with the preconditioner SAI or DNVM, it is also difficult to obtain the converged results. For the slender real beams, the iteration of GMRES (m) with SAI or DNVM stopped at wrong deformation state, and the computation failed. By changing zero initial solution to the analytical displacement solution of conventional beam theory, GMRES (m) with SAI or DNVM will not be stopped at wrong deformation state, but the stress error is still difficult to converge. However, by GMRES (m) combined with both SAI and DNVM preconditioning, the computation efficiency enhanced significantly. Originality/value This paper presents two preconditioners: DNVM and a modified SAI based on adjacent nodes along the boundary line of slender thin-plate beam. In the LSA, by using GMRES (m) combined with both DNVM and SAI, the computation efficiency enhanced significantly. It provides a reference for the further development of the 3D HPBEM in the LSA of real beam, plate and shell structures.