A Study on Spline Quasi-interpolation Based Quadrature Rules for the Isogeometric Galerkin BEM

Falini, Antonella; Kanduč, Tadej

doi:10.1007/978-3-030-27331-6_6

Cited by 8 publications

(11 citation statements)

References 30 publications

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“…In order to increase the stability of the computation of the solution u ex , we improved the used quadrature formula in the evaluation of the integrals for the Galerkin method by setting the parameter qft = qf7pT or by using qft = qf9pT, which are Gaussian quadrature rules of order 8 and 10, respectively; see, e.g., [26], available with the FreeFem integration routines. In general, when the order increases, the Gaussian rule may become unstable; moreover, due to the triangular domains, suitable rules should be constructed; see, e.g., [27][28][29] and references therein. In our tests, we could obtain more stable results for the computation of the L 2 norm but, for the H 1 seminorm, we could still observe some divergent cases.…”

Section: Numerical Resultsmentioning

confidence: 99%

Adaptive Refinement in Advection–Diffusion Problems by Anomaly Detection: A Numerical Study

Falini

Sampoli

2021

Algorithms

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We consider advection–diffusion–reaction problems, where the advective or the reactive term is dominating with respect to the diffusive term. The solutions of these problems are characterized by the so-called layers, which represent localized regions where the gradients of the solutions are rather large or are subjected to abrupt changes. In order to improve the accuracy of the computed solution, it is fundamental to locally increase the number of degrees of freedom by limiting the computational costs. Thus, adaptive refinement, by a posteriori error estimators, is employed. The error estimators are then processed by an anomaly detection algorithm in order to identify those regions of the computational domain that should be marked and, hence, refined. The anomaly detection task is performed in an unsupervised fashion and the proposed strategy is tested on typical benchmarks. The present work shows a numerical study that highlights promising results obtained by bridging together standard techniques, i.e., the error estimators, and approaches typical of machine learning and artificial intelligence, such as the anomaly detection task.

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Section: Numerical Resultsmentioning

confidence: 99%

Adaptive Refinement in Advection–Diffusion Problems by Anomaly Detection: A Numerical Study

Falini

Sampoli

2021

Algorithms

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“…Typically it leads to lower computational costs than global fitting. Recently, the QI method has also been used to deal with the integrals arising in Isogeometric Boundary Element Methods (IGABEM), see [13,23,25].…”

Section: Quasi-interpolationmentioning

confidence: 99%

Efficient matrix assembly in isogeometric analysis with hierarchical B-splines

Pan

Jüttler

Mantzaflaris

2021

Journal of Computational and Applied Mathematics

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Hierarchical B-splines that allow local refinement have become a promising tool for developing adaptive isogeometric methods. Unfortunately, similar to tensor-product B-splines, the computational cost required for assembling the system matrices in isogeometric analysis with hierarchical B-splines is also high, particularly if the spline degree is increased. To address this issue, we propose an efficient matrix assembly approach for bivariate hierarchical B-splines based on the previous work [42]. The new algorithm consists of three stages: approximating the integrals by quasi-interpolation, building three compact look-up tables and assembling the matrices via sum-factorization. A detailed analysis shows that the complexity of our method has the order O(N p 3 ) under a mild assumption about mesh admissibility, where N and p denote the number of degrees of freedom and spline degree respectively. Finally, several experimental results are demonstrated to verify the theoretical results and to show the performance of the proposed method.

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“…In the preceding research to this paper, a promising approach for curved isoparametric boundaries for 2D problems combines an elegant singularity extraction and a local spline quasi-interpolation operators [2,7,17]. For this type of quadrature schemes the optimal convergence orders of the approximate solution can be recovered with a small numbers of quadrature nodes [19].…”

Section: Introductionmentioning

confidence: 99%

Isoparametric singularity extraction technique for 3D potential problems in BEM

Kanduc

2022

Preprint

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To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach, curved geometries that are represented by mapped rectangles or triangles from the parametric domain are considered. The singularity extraction on the governing singular integrals can be performed either as an operation of subtraction or division, each having some advantages.A particular series expansion of a singular kernel about a source point is investigated. The series in the intrinsic coordinates consists of functions of a type R p x q y r , where R is a square root of a quadratic bivariate homogeneous polynomial, corresponding to the first fundamental form of a smooth surface, and p, q, r are integers, satisfying p ≤ −1 and q, r ≥ 0. By extracting more terms from the series expansion of the singular kernel, the smoothness of the regularized kernel at the source point can be increased. Analytical formulae for integrals of such terms are obtained from antiderivatives of R p x q y r , using recurrence formulae, and by evaluating them at the edges of rectangular or triangular parametric domains.Numerical tests demonstrate that the singularity extraction technique can be a useful prerequisite for a numerical quadrature scheme to obtain accurate evaluations of the governing singular integrals in 3D collocation BEM.

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A Study on Spline Quasi-interpolation Based Quadrature Rules for the Isogeometric Galerkin BEM

Cited by 8 publications

References 30 publications

Adaptive Refinement in Advection–Diffusion Problems by Anomaly Detection: A Numerical Study

Adaptive Refinement in Advection–Diffusion Problems by Anomaly Detection: A Numerical Study

Efficient matrix assembly in isogeometric analysis with hierarchical B-splines

Isoparametric singularity extraction technique for 3D potential problems in BEM

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