Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

Taus, Matthias; Rodin, Gregory J.; Hughes, Thomas J.R.

doi:10.1142/s0218202516500354

Cited by 25 publications

(16 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…IGA couples the basis functions defining the surface geometry with the analytic approaches for the finite element scheme. Most relevant to this work, IGA has recently been applied to singular and hypersingular boundary integral equations with a collocation discretization [TRH16] with great success. A Nyström IGA method along with a regularized quadrature scheme is detailed in [ZMBF16].…”

Section: Related Workmentioning

confidence: 99%

A robust solver for elliptic PDEs in 3D complex geometries

Morse,

Rahimian,

Zorin

2020

Preprint

View full text Add to dashboard Cite

We develop a boundary integral equation solver for elliptic partial differential equations on complex 3D geometries. Our method is high-order accurate with optimal O(N) complexity and robustly handles complex geometries. A key component is our singular and near-singular layer potential evaluation scheme, hedgehog : a simple extrapolation of the solution along a line to the boundary. We present a series of geometry-processing algorithms required for hedgehog to run efficiently with accuracy guarantees on arbitrary geometries and an adaptive upsampling scheme based on a iteration-free heuristic for quadrature error that incorporates surface curvature. We validate the accuracy and performance with a series of numerical tests and compare our approach to a competing local evaluation method.

show abstract

Section: Related Workmentioning

confidence: 99%

A robust solver for elliptic PDEs in 3D complex geometries

Morse,

Rahimian,

Zorin

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…For example in [14] where the 3D Stokes problem is considered, the singularity is removed by exploiting carefully chosen known solutions of the analyzed partial differential equation. In other papers these integrals are reformulated by using a suitable coordinate transformation, see for example [23] for Duffy and [24] for polar transformations. In these cases the additional emerging transformation term approximately cancels out the singularity of the kernel and the resulting integrals become regular.…”

Section: Introductionmentioning

confidence: 99%

Cubature rules based on bivariate spline quasi-interpolation for weakly singular integrals

Falini,

Kanduč,

Sampoli

et al. 2020

Preprint

View full text Add to dashboard Cite

In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D Laplace boundary value problems, using a collocation method within the Isogeometric Analysis paradigm. In such setting the regular part of the integrand can be defined as the product of a tensor product B-spline and a general function. The rules are derived by using first the spline quasi-interpolation approach to approximate such function and then the extension of a well known algorithm for spline product to the bivariate setting. In this way efficiency is ensured, since the locality of any spline quasi-interpolation scheme is combined with the capability of an ad-hoc treatment of the B-spline factor. The numerical integration is performed on the whole support of the B-spline factor by exploiting inter-element continuity of the integrands.

show abstract

“…The isogeometric formulation of boundary element method (IgA-BEM) has been successfully applied to 2D and 3D problems, such as linear elasticity [4], fracture mechanics [5], acoustic [6] and Stokes flows [7].…”

Section: Introductionmentioning

confidence: 99%

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

Falini

Kanduč

2019

Advanced Methods for Geometric Modeling and Numerical Simulation

View full text Add to dashboard Cite

Two recently introduced quadrature schemes for weakly singular integrals [1] are investigated in the context of boundary integral equations arising in the isogeometric formulation of Galerkin Boundary Element Method (BEM). In the first scheme, the regular part of the integrand is approximated by a suitable quasi-interpolation spline. In the second scheme the regular part is approximated by a product of two spline functions. The two schemes are tested and compared against other standard and novel methods available in literature to evaluate different types of integrals arising in the Galerkin formulation. Numerical tests reveal that under reasonable assumptions the second scheme convergences with the optimal order in the Galerkin method, when performing h-refinement, even with a small amount of quadrature nodes. The quadrature schemes are validated also in numerical examples to solve 2D Laplace problems with Dirichlet boundary conditions.

show abstract

Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

Cited by 25 publications

References 43 publications

A robust solver for elliptic PDEs in 3D complex geometries

A robust solver for elliptic PDEs in 3D complex geometries

Cubature rules based on bivariate spline quasi-interpolation for weakly singular integrals

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

Contact Info

Product

Resources

About