New developments on BE/BE multi-zone algorithms based on krylov solvers: applications to 3D frequency-dependent problems

Abstract

In this paper, new developments concerning the use of BE/BE coupling algorithms for solving 3D time-harmonic problems are reported. The algorithms are derived by considering different iterative solvers. Their chief idea is to work with the global sparse matrix of the coupled system, however without considering the many zero blocks associated with the non-coupled nodes of different subregions. The use of iterative solvers makes it possible to store and manipulate only the block matrices with non-zero coefficients. Preconditioned iterative solvers based on the Lanczos, bi-conjugate gradient, and GMRES (generalized minimal residual) methods are used to derive the BE/BE coupling algorithms. A description of the formulation of these solvers, which are completely general and can be applied to any non-singular, non-hermitian systems of equations, is provided. The performance of the algorithms is verified by solving some foundation-soil interaction problems. Important parameters for estimating the efficiency of the algorithms as required CPU times, matrix sparsity, and accuracy of the obtained responses are presented in the results of the paper

Application of a generic domain‐decomposition strategy to solve shell‐like problems through 3D BE models

“…In this case, no additional interfaces (and corresponding domains) have to be generated to complete the system of equations; i.e. the number of equations and the number of unknowns is already the same [23]. Thus, the only conditions to be imposed at the interfaces i j are…”

“…Note that the strategies above can be promptly employed for parallelizing FE and BE codes. Among the Krylov solvers applicable to the solution of the BE systems of equations, PetrovGalerkin-type schemes, such as Lanczos and bi-conjugate gradient (Bi-CG), minimal residual approaches as GMRES, and hybrid procedures like CGS and Bi-CGSTAB, have all been considered before [23,[31][32][33][34][35][36][37][38][39][40]. Here, the Jacobi-preconditioned biconjugate gradient solver (J-BiCG) [31], which in a number of numerical experiments has proven to be reliable and efficient, is the one selected.…”

“…In effect, this corresponds to an optimal matrixstorage format for BE models, only possible because of the use of iterative solvers. In addition, discontinuous boundary elements, which avoid imposing additional traction continuity conditions at interface nodes [23], are available. Thus, inner corners and edges, present on a problem discretized with a generic number of coupled BE models, may be easily simulated.…”

Generic domain decomposition and iterative solvers for 3D BEM problems

“…Many works have pointed out the efficiency of Krylov's iterative schemes in BE formulations [7,[31][32][33][34][35][36][37][38][39]. In References [28][29][30][31][40][41][42], the most important Krylov's algorithms applied to solve BE systems, such as BiCG, Lanczos, GMRES, CGS e BiCGSTAB, can be found in detail.…”

“…The boundary-element substructuring strategy based on iterative solvers has also been addressed in several papers [7][8][9]42]. In fact, it is believed that the iterative-solver-based substructuring technique proposed in these works, by the reasons given above, is much more efficient than those based on direct solvers [43][44][45][46].…”