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

Abstract

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.

A Subdivision Transformation Method for Weakly Singular Boundary Integrals in Thin-structural Problem

“…However, as the singularities of the kernel function in the boundary integral equation (BIE), the computational accuracy of weak singular integral directly affects the final results, so additional processing schemes must be given to remove these singularities to ensure the successful solution of BEM. Especially for thin-structural problems, poorly shaped elements (such as elements with large angles or narrow lengths) will appear in the discrete geometric model of thin structure [12][13][14], which will seriously affect the accuracy of singular and near singular integrals. Many works have been demonstrated that the near singular integrals can be accurately evaluated by analytical and semi-analytical methods [15,16], Sinh and other nonlinear transformation methods [4,[17][18][19], etc.…”

“…To obtain high calculation accuracy, the integral element needs to be subdivided into several integral blocks with good shape. And there are many element subdivision methods, such as the singular points directly connected to the element vertexes [12], interval block method [12], tree structure methods [25], etc. When these subdivision methods are employed, some integral blocks with poor shape (such as the integral blocks with large angle and large aspect ratio) will be subdivided, which increases the difficulty in dealing with singular integral and decrease the computational accuracy.…”

BESLE: Boundary element software for 3D linear elasticity

“…To deal with this issue an MPI parallel algorithm described in [5,15] was incorporated allowing for an efficient solution. For this purpose, it is also possible to use adaptive cross approximation, also known as hierarchical matrices [16][17][18] and other high-performance BEM implementations, such as those in [19,20]. However, in BESLE we adopt the strategy of finding an adequate balance between the number of DOFs and subdomains under consideration and this mean that very large scale problems are now feasible to be treated without any additional arduous method.…”