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

Abstract

SUMMARYEfficient integration algorithms and solvers specially devised for boundary-element procedures have been established over the last two decades. A good deal of quadrature techniques for singular and quasisingular boundary-element integrals have been developed and reliable Krylov solvers have proven to be advantageous when compared to direct ones, also in case of non-Hermitian matrices. The former has implied in CPU-time reduction during the assembling of the system of equations and the latter in its faster solution. Here, a triangular polar co-ordinate transformation and the Telles co-ordinate transformation are employed separately and combined to develop the matrix-assembly routines (integration routines). In addition, the Jacobi-preconditioned Biconjugate Gradient solver (J-BiCG) is used along with a generic substructuring boundary element algorithm. Thus, solution CPU time and computer memory can be considerably reduced. Discontinuous boundary elements are also included to simplify the coupling of the BE models (substructures). Numerical experiments involving 3D thin-walled domains (shell-like structural elements) are carried out to show the performance of the computer code with respect to accuracy and efficiency of the system solution. Precision, CPU-time and potential applications of the BE code developed are commented upon.

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

“…For the off-diagonal terms (Fig. 4b), a more critical convergence behaviour is observed; the I (2) 3 procedure converges only for integration orders superior to 15, and the I (2) 1 procedure again does not converge. To obtain convergence with the line-integral approach, the number of integration points adopted for the line integrals has to be increased, and the polynomial transformation somewhat improves the performance of this procedure.…”

“…1 procedure, convergence is actually not reached for n ≤ 20; for I (2) 3 , convergence is obtained only for integration orders superior to 10. For the off-diagonal terms (Fig.…”

“…In addition, several improvements to the line-integral approach [7] for evaluating quasi-singular integrals are proposed; the primary modification is the application of cubic polynomial transformations. Note that quadratures based on cubic polynomial transformation, I (d) 2 [23], have not been applied to evaluate surface integrals (d = 2), as an independent quadrature or within the line-integral approach; as established in [2], this approach is somewhat less efficient than the procedures I (2) 1 and I (2) 3 . However, this procedure is implicitly used here to generate the I (2) 3 quadrature [1], designed to increase the accuracy of the quasi-singular surface integral computations arising from the line-integral approach.…”

“…In the modelling of thin-walled solids, such as steel beams, or composites containing thin reinforcement elements or layers, a challenge is, besides the evaluation of singular integrals, as usual, the efficient computation of quasi-singular (near singular) integrals. Weakly-singular and nearly-weakly-singular integrals may be efficiently evaluated by applying the coordinate-transformation-based quadratures discussed in references [1,2,7]. It is also known, as shown by Liu [16], that for zero-thickness solids (shell-like/needle-like ones) non-singular systems of algebraic equations will arise provided they are not entirely constrained at their boundary.…”

“…Thus, considering a shape function h q associated with the qth node of a boundary element and a generic fundamental kernel, f * ik (χ ; ξ) (i.e. the displacement or the traction), the integration rules for procedures I (2) 1 and I (2) 3 can be expressed as: Triangle-polar-coordinate-based quadrature (I (2) 1 )…”