Efficient quadrature rules based on spline quasi-interpolation for application to IGA-BEMs

Efficient quadrature rules based on spline quasi-interpolation for application to IGA-BEMs We propose a new class of quadrature rules for the approximation of weakly and strongly singular integrals, based on the spline quasi-interpolation scheme introduced in Mazzia and Sestini (2009). These integrals in particular occur in the entries of the stiffness matrix coming from Isogeometric Boundary Element Methods (IgA-BEMs). The presented formulas are efficient, since they combine the locality of any spline quasi-interpolation scheme with the capability to compute the modified moments for B-splines, i.e. the weakly or strongly singular integrals of such functions. No global linear system has to be solved to determine the quadrature weights, but just local systems, whose size linearly depends on the adopted spline degree. The rules are preliminarily tested in their basic formulation, i.e. when the integrand is defined as the product of a singular kernel and a continuous function g . Then, such basic formulation is compared with a new one, specific for the approximation of the singular integrals appearing in the IgA-BEM context, where a B-spline factor is explicitly included in g . Such a variant requires the usage of the recursive spline product formula given in Mørken (1991), and it is useful when the ratio between g and its B-spline factor is smooth enough. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Computational and Applied Mathematics Elsevier

Efficient quadrature rules based on spline quasi-interpolation for application to IGA-BEMs

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Publisher
Elsevier
Copyright
Copyright © 2018 Elsevier Ltd
ISSN
0377-0427
eISSN
1879-1778
D.O.I.
10.1016/j.cam.2018.02.005
Publisher site
See Article on Publisher Site

Abstract

We propose a new class of quadrature rules for the approximation of weakly and strongly singular integrals, based on the spline quasi-interpolation scheme introduced in Mazzia and Sestini (2009). These integrals in particular occur in the entries of the stiffness matrix coming from Isogeometric Boundary Element Methods (IgA-BEMs). The presented formulas are efficient, since they combine the locality of any spline quasi-interpolation scheme with the capability to compute the modified moments for B-splines, i.e. the weakly or strongly singular integrals of such functions. No global linear system has to be solved to determine the quadrature weights, but just local systems, whose size linearly depends on the adopted spline degree. The rules are preliminarily tested in their basic formulation, i.e. when the integrand is defined as the product of a singular kernel and a continuous function g . Then, such basic formulation is compared with a new one, specific for the approximation of the singular integrals appearing in the IgA-BEM context, where a B-spline factor is explicitly included in g . Such a variant requires the usage of the recursive spline product formula given in Mørken (1991), and it is useful when the ratio between g and its B-spline factor is smooth enough.

Journal

Journal of Computational and Applied MathematicsElsevier

Published: Aug 15, 2018

References

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