Generic domain decomposition and iterative solvers for 3D BEM problems

Abstract

SUMMARYIn the past two decades, considerable improvements concerning integration algorithms and solvers involved in boundary-element formulations have been obtained. First, a great deal of efficient techniques for evaluating singular and quasi-singular boundary-element integrals have been, definitely, established, and second, iterative Krylov solvers have proven to be advantageous when compared to direct ones also including non-Hermitian matrices. The former fact has implied in CPU-time reduction during the assembling of the system of equations and the latter fact in its faster solution. In this paper, a triangle-polar-co-ordinate transformation and the Telles co-ordinate transformation, applied in previous works independently for evaluating singular and quasi-singular integrals, are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrixassembly routines. In addition, the Jacobi-preconditioned biconjugate gradient (J-BiCG) solver is used to develop a generic substructuring boundary-element algorithm. In this way, it is not only the system solution accelerated but also the computer memory optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted.

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

“…The latter procedure, which should represent an improved integration scheme, is here firstly verified for solving shell-like problems through 3D analysis. A comparison of the benefits of the individual procedures, here merely applied to solve a complete problem, is reported in Reference [26]. Note that the presence of nearly-weakly-singular and nearly-strongly-singular integrals is unavoidable when simulating thin-walled elements.…”

Analysis of thin-walled structural elements via 3D standard BEM with generic substructuring

“…In general, these techniques avail themselves of iterative solvers, which allow working with smaller parts of the system matrix. In the case of the SBS technique, matrix-vector and vector-vector products within the built-in solver are computed according to the coefficient matrices and boundary and interface values stored independently per subregion [1]. Nevertheless, contrary to FE models, where coefficients belonging to different elements do overlap, for coupled BE models no overlapping of coefficients belonging to different subregions takes place [1].…”

“…Second, to evaluate nearly-strongly-singular integrals, he applied a line-integral approach based on employing the Stokes' theorem to convert surface integrals to 1D integrals. In addition, well-known coordinate-transformation-based numerical quadratures [15,23] have been combined to generate more efficient algorithms for nearly singular integrals [1,7].…”

“…In the case of the SBS technique, matrix-vector and vector-vector products within the built-in solver are computed according to the coefficient matrices and boundary and interface values stored independently per subregion [1]. Nevertheless, contrary to FE models, where coefficients belonging to different elements do overlap, for coupled BE models no overlapping of coefficients belonging to different subregions takes place [1]. Thus, in this respect, this technique is optimal, requiring no additional design of special data structures to reduce memory and solver CPU time as required for FE models [8,10,25].…”

“…In [1], a robust generic substructuring technique for Boundary Element Methods, here termed the subregionby-subregion (SBS) method, has been proposed. It was inspired, in effect, by the element-by-element technique, largely employed in finite-element formulations to solve large-scale engineering problems [11][12][13].…”