Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM

Abstract

This paper deals with the discrete counterpart of 2D elliptic model problems rewritten in terms of Boundary Integral Equations. The study is done within the framework of Isogeometric Analysis based on B-splines. In such a context, the problem of constructing appropriate, accurate and efficient quadrature rules for the Symmetric Galerkin Boundary Element Method is here investigated. The new integration schemes, together with row assembly and sum factorization, are used to build a more efficient strategy to derive the final linear system of equations. Key ingredients are weighted quadrature rules tailored for B-splines, that are constructed to be exact in the whole test space, also with respect to the singular kernel. Several simulations are presented and discussed, showing accurate evaluation of the involved integrals and outlining the superiority of the new approach in terms of computational cost and elapsed time with respect to the standard element-by-element assembly.

An adaptive IgA‐BEM with hierarchical B‐splines based on quasi‐interpolation quadrature schemes

“…Remark Entries in β 2 are regular integrals if F ∈ C 2 ([ a , b ]) (see section 3.2 in the work of Aimi et al).…”

“…Secondly, the kernel splitting of U introduced in Section 4.1 allows us to separate the geometrical influence on the integrand function from the part related to the singularity of the kernel. The details on the splitting procedure for the case of open boundary curves were already described in the work of Aimi et al The splitting technique related to the closed curve geometries is new in this paper. Then, preliminaries of the utilized spline quasi‐interpolation operator are summarized in Section 4.2.…”

“…The limit case for δ ( s , t ) = | s − t | was already proven in Hence, K 1 is well defined on [ a , b ] 2 , when the boundary is an open curve. On the other hand, it can be checked analogously that when closed boundary curves are considered, the new definition of K 1 guarantees that the above limit holds as well.…”

“…A recurrence formula to obtain exact expressions for the modified moments is derived from the B‐spline recursive definition . The recurrence formula relies on given initial values, specifically on where is the characteristic function of the interval [ c 1 , c 2 ].…”

“…Recently, the novel combination of the IgA paradigm with symmetric Galerkin BEM (IgA‐SGBEM) has revealed to be very effective among BEM schemes. Moreover, the full potential of B‐splines over the more common Lagrangian basis has been recently exploited in the work of Aimi et al…”