Block method for problems on L-shaped domains

Dosiyev, A. A.; Mazhar, Zeka; Buranay, Suzan Cival

doi:10.1016/j.cam.2010.07.007

Cited by 4 publications

(6 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our final example consists of solving the Burgers' system (1) in an L‐shape computational domain defined by

Ω = [- 2, 2] \times [- 2, 2] ∖ [0, 2] \times [0, 2]

subject to initial and Dirichlet boundary conditions obtained from the following analytical solution

\begin{align} u false(x, y, t false) & = prefix- \frac{4 π \exp ((), prefix- \frac{5 π^{2} t}{R e}) \cos false(2 π x false) \sin false(π y false)}{R e ((), 2 + \exp ((), prefix- \frac{5 π^{2} t}{R e}) \sin false(2 π x false) \sin false(π y false))}, \\ v false(x, y, t false) & = prefix- \frac{2 π \exp ((), prefix- \frac{5 π^{2} t}{R e}) \sin false(2 π x false) \cos false(π y false)}{R e ((), 2 + \exp ((), prefix- \frac{5 π^{2} t}{R e}) \sin false(2 π x false) \sin false(π y false))} . \end{align}

A similar geometry has been considered in Reference 45 among others. This problem has also been widely used in the literature to analyze the performance of numerical methods for solving coupled Burgers' equations, see for instance References 18,20.…”

Section: Numerical Resultsmentioning

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

View full text Add to dashboard Cite

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.

show abstract

“…Our final example consists of solving the Burgers' system (1) in an L‐shape computational domain defined by

Ω = [- 2, 2] \times [- 2, 2] ∖ [0, 2] \times [0, 2]

subject to initial and Dirichlet boundary conditions obtained from the following analytical solution

\begin{align} u false(x, y, t false) & = prefix- \frac{4 π \exp ((), prefix- \frac{5 π^{2} t}{R e}) \cos false(2 π x false) \sin false(π y false)}{R e ((), 2 + \exp ((), prefix- \frac{5 π^{2} t}{R e}) \sin false(2 π x false) \sin false(π y false))}, \\ v false(x, y, t false) & = prefix- \frac{2 π \exp ((), prefix- \frac{5 π^{2} t}{R e}) \sin false(2 π x false) \cos false(π y false)}{R e ((), 2 + \exp ((), prefix- \frac{5 π^{2} t}{R e}) \sin false(2 π x false) \sin false(π y false))} . \end{align}

Section: Numerical Resultsmentioning

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

View full text Add to dashboard Cite

show abstract

“…Moreover, the classic Jacobi or Gauss-Seidel iterations converge as geometrical progression with the ratio independent of n (see [12,13]). Let u ε m be an approximate value of the solution u m of system (21), with an accuracy of ε = 5 × 10 −16 in double, and ε = 5 × 10 −34 in quadruple precisions.…”

Section: Slit Problemmentioning

confidence: 99%

“…We use the harmonic extension of the solution of problem (27) to the sector (3), with r 0 ∈  √ 2, 2  and α = 3 4 as in [13]. On the arc of sector T , consider the quadrature nodes P 0 Fig.…”

Section: L-shaped Problemmentioning

confidence: 99%

“…Finally, we mention the papers [11][12][13], in which by using the one block version of the block method (BM) given in [25,26] the highly accurate results are obtained for the solution of the Motz problem, cracked-beam problem and problems in L shaped domains respectively.…”

Section: Introductionmentioning

confidence: 99%

“…These coefficients have great importance in many applications, especially in fracture mechanics. The methods for the approximation of the coefficients a j can be divided into two groups: (i) the post-processing approach in the finite element or finite difference methods (see [5][6][7][8][9][10]), in the block method (see [11][12][13]) and in the block-grid method [14][15][16] (ii) directly calculated methods, without finding approximate solution of the boundary value problem (see [4,1,[17][18][19][20] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

One-block method for computing the generalized stress intensity factors for Laplace’s equation on a square with a slit and on an L-shaped domain

Dosiyev

Buranay

2015

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

a b s t r a c tA highly accurate approximation for the coefficients of the series expansion of the solution of Laplace's equation around the singular vertex, which are called the generalized stress intensity factors (GSIFs), is obtained by One-Block Method. The method is demonstrated for the slit problem, which has a strong singularity, and for the popular problem on an L-shaped domain. Furthermore, the extremely accurate series segment solution for each of the problems is obtained by taking an appropriate number of calculated GSIFs. Numerical results are presented through tables and figures.

show abstract

A method of harmonic extension for computing the generalized stress intensity factors for Laplace’s equation with singularities

Dosiyev¹

2018

Computers & Mathematics with Applications

View full text Add to dashboard Cite

Block method for problems on L-shaped domains

Cited by 4 publications

References 16 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

One-block method for computing the generalized stress intensity factors for Laplace’s equation on a square with a slit and on an L-shaped domain

A method of harmonic extension for computing the generalized stress intensity factors for Laplace’s equation with singularities

Contact Info

Product

Resources

About