Some quadrature rules for Galerkin methods using B-spline basis functions

Hopkins, Tim; Wait, R.

doi:10.1016/0045-7825(79)90067-7

Cited by 4 publications

(2 citation statements)

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“…The problem of deriving quadrature rules for integrals involving B-splines was first considered over 30 years ago. Indeed, in [10] the authors computed rules for the moments of (linear, quadratic and cubic) B-spline functions, in order to solve a parabolic PDE using Galerkin's method. The interest in the topic is revived lately, after the introduction of IGA.…”

Section: Introductionmentioning

confidence: 99%

Exploring Matrix Generation Strategies in Isogeometric Analysis

Mantzaflaris

Jüttler

2014

Mathematical Methods for Curves and Surfaces

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Abstract. An important step in simulation via isogeometric analysis (IGA) is the assembly step, where the coefficients of the final linear system are generated. Typically, these coefficients are integrals of products of shape functions and their derivatives. Similarly to the finite element analysis (FEA), the standard choice for integral evaluation in IGA is Gaussian quadrature. Recent developments propose different quadrature rules, that reduce the number of quadrature points and weights used. We experiment with the existing methods for matrix generation. Furthermore we propose a new, quadrature-free approach, based on interpolation of the geometry factor and fast look-up operations for values of B-spline integrals. Our method builds upon the observation that exact integration is not required to achieve the optimal convergence rate of the solution. In particular, it suffices to generate the linear system within the order of accuracy matching the approximation order of the discretization space. We demonstrate that the best strategy is one that follows the above principle, resulting in expected accuracy and improved computational time.

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Section: Introductionmentioning

confidence: 99%

Exploring Matrix Generation Strategies in Isogeometric Analysis

Mantzaflaris

Jüttler

2014

Mathematical Methods for Curves and Surfaces

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“…The differential equation is converted to its corresponding weak form. Then, it is approximated using B-spline functions thereby constructing the eigenvalue problem (Hopkins and Wait, 1979;Mittal and Jain, 2011;Gu et al, 2011;Yanan et al, 2011;Mohammadi, 2014;Xiaoyun and Weiyin, 2014). Finally, after applying essential boundary conditions, natural frequencies and corresponding mode shapes are calculated and compared with those obtained from finite element (FE) analysis.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of combined system of framed tube and shear walls by Galerkin method using B‐spline functions

Rahgozar

Mahmoudzadeh

Malekinejad

et al. 2014

Structural Design Tall Build

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SUMMARYIn this article, dynamic parameters (natural frequencies and mode shapes) of tall buildings that consist of framed tube and shear walls are obtained using a simple approximate method. The three-dimensional structure is replaced by an equivalent cantilever beam, considering both bending and shear deformations. On the basis of dynamic equilibrium, the governing differential equation of motion is obtained and converted to its corresponding weak form. B-spline functions are then utilized to approximate the weak form and to obtain the final matrix form of the problem. Finally, by applying essential boundary conditions, the natural frequencies and corresponding mode shapes are calculated. To demonstrate the accuracy of the proposed method, numerical examples are solved, and the results are compared with those obtained from SAP2000 computer analysis. The results show that the proposed method is efficient and accurate enough to be used in preliminary design.

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References

Micula

1999

Handbook of Splines

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Some quadrature rules for Galerkin methods using B-spline basis functions

Cited by 4 publications

References 1 publication

Exploring Matrix Generation Strategies in Isogeometric Analysis

Exploring Matrix Generation Strategies in Isogeometric Analysis

Dynamic analysis of combined system of framed tube and shear walls by Galerkin method using B‐spline functions

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