Gauss Legendre–Gauss Jacobi quadrature rules over a tetrahedral region

Rathod, H.T.; Venkatesudu, B.; Nagaraja, K.V.; Islam, Shafiqul

doi:10.1016/j.amc.2007.01.014

Cited by 14 publications

(7 citation statements)

References 19 publications

(28 reference statements)

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“…In Reference 34 a class of optimal Gauss‐based quadrature rules (Gauss–Radau rule) has been developed for spline spaces which usually occur in Galerkin‐based finite element discretizations for which the original spaces are

𝒞^{1}

quadratics or

𝒞^{2}

cubics. In the current work, we adopt the Gauss–Jacobi quadrature rule used also in References 35‐38 among others. It should also be noted that the Gauss‐based quadrature rules are known to exhibit fast convergence, and the full tensor product Gauss–Jacobi quadrature rule is proven to converge exponentially and it is exact for polynomials of degree up to

2 m - 1

, see for instance Reference 36.…”

Section: Isogeometric Modified Methods Of Characteristicsmentioning

confidence: 99%

High‐order isogeometric modified method of characteristics for two‐dimensional coupled Burgers' equations

Asmouh

El‐Amrani

Seaı̈d

et al. 2022

Numerical Methods in Fluids

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This paper presents a novel isogeometric modified method of characteristics for the numerical solution of the two‐dimensional nonlinear coupled Burgers' equations. The method combines the modified method of characteristics and the high‐order NURBS () elements to discretize the governing equations. The Lagrangian interpretation in this isogeometric analysis greatly reduces the time truncation errors in the Eulerian methods. A third‐order explicit Runge–Kutta scheme is used for the discretization in time. We present a detailed description of the algorithm used for the calculation of departure points and the interpolation stage. Our focus is on constructing highly accurate and stable solvers for the two‐dimensional nonlinear coupled Burgers' equations at high Reynolds numbers. A variety of benchmark tests and numerical examples are provided to show the effectiveness, accuracy, and performance of the proposed modified method of characteristics by virtue of potential advantages of isogeometric analysis. The method developed is anticipated to provide new research directions to the practical calculation of incompressible flows and to studies of their physical behavior.

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𝒞^{1}

quadratics or

𝒞^{2}

2 m - 1

, see for instance Reference 36.…”

Section: Isogeometric Modified Methods Of Characteristicsmentioning

confidence: 99%

High‐order isogeometric modified method of characteristics for two‐dimensional coupled Burgers' equations

Asmouh

El‐Amrani

Seaı̈d

et al. 2022

Numerical Methods in Fluids

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“…This needs a set of weights ω g , integration points x g on the reference element and the transformation T e and its spatial derivative J T e . The gauss weights and points are taken from the literature Rathod et al, 2007a,b) and the transformation T e , that maps the reference tetrahedron Ω ref to the element Ω e , is given by a linear function…”

Section: Tablementioning

confidence: 99%

Finite element method completely implemented for graphic processor units using parallel algorithm libraries

Pichler

Haase

2017

The International Journal of High Performance Computing Applica

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A finite element code is developed in which all computational expensive steps are performed on a graphics processing unit (GPU) via the THRUST and the PARALUTION library. The code is focused on simulation of transient problems where the repeated computations per time step create the computational cost. It is applied to solve partial and ordinary differential equations as they arise in thermal-runaway simulations of automotive batteries. The speedup obtained by utilizing the GPU for every critical step is compared against the single core and the multi-threading solution which is also supported by the chosen libraries. This way a high total speedup on the GPU is achieved without the need for programming a single classical Compute Unified Device Architecture (CUDA) kernel.

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“…The integrals cannot be evaluated exactly; the use of a quadrature method becomes a must. A useful mathematical algorithm for the derivation of quadrature points and corresponding weights over tetrahedral elements is provided in [16]. Assembling the governing matrices, the discrete elastodynamic form presented in (25) is obtained but with a much larger number of degrees of freedom.…”

Section: Fem Constructionmentioning

confidence: 99%

On the wave propagation in isotropic fractal media

Joumaa

Ostoja–Starzewski

2011

Z. Angew. Math. Phys.

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In this paper, we explore the wave propagation phenomenon in three-dimensional (3D) isotropic fractal media through analytical and computational means. We present the governing scalar wave equation, perform its eigenvalue decomposition, and discuss its corresponding modal solutions. The homogenization through which this fractal wave equation is derived makes its mathematical analysis and consequently the formulation of exact solutions possible if treated in the spherical coordinate system. From the computational perspective, we consider the finite element method and derive the corresponding weak formulation which can be implemented in the numerical scheme. The Newmark time-marching method solves the resulting elastodynamic system and captures the transient response. Two solvers capable of handling problems of arbitrary initial and boundary conditions for arbitrary domains are developed. They are validated in space and time, with particular problems considered on spherical shell domains. The first solver is elementary; it handles problems of purely radial dependence, effectively, 1D. However, the second one deals with general advanced 3D problems of arbitrary spatial dependence. (2000). 74H45 Vibration. Mathematics Subject Classification

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Gauss Legendre–Gauss Jacobi quadrature rules over a tetrahedral region

Cited by 14 publications

References 19 publications

High‐order isogeometric modified method of characteristics for two‐dimensional coupled Burgers' equations

High‐order isogeometric modified method of characteristics for two‐dimensional coupled Burgers' equations

Finite element method completely implemented for graphic processor units using parallel algorithm libraries

On the wave propagation in isotropic fractal media

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