High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations

Isaev, V. I.; Shapeev, V. P.

doi:10.1134/s0965542510100040

Cited by 29 publications

(6 citation statements)

References 13 publications

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“…Various versions of the CLS and CLR methods have been tested against such benchmarks [9][10][11][12][13][14]. For simplicity, the following discussion is confined to a square cavity…”

Section: Versions Of the Clr Methods For Solving The Navier-stokes Equmentioning

confidence: 99%

“…The fact that the approximate solution of the global problem consists of local analytical pieces enables a relatively simple construction of the method versions on irregular and adaptive grids [5,8,10,11].…”

Section: On Other Capabilities Of the Clr Methodsmentioning

confidence: 99%

“…For different versions of the CLR method for solving the 2D and 3D Navier-Stokes equations, the acceleration algorithm of the iteration processes based on Krylov subspaces [10,11,14,20] was also used with the aim at reducing the CPU time. Its application on a grid of the 256 × 256 size in problems with the Reynolds number Re = 1000 sometimes resulted in an acceleration of the CPU time by a factor of up to 25.…”

Section: -P10mentioning

confidence: 99%

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Collocation and Least Residuals Method and Its Applications

Shapeev

2016

EPJ Web of Conferences

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Abstract. The collocation and least residuals (CLR) method combines the methods of collocations (CM) and least residuals. Unlike the CM, in the CLR method an approximate solution of the problem is found from an overdetermined system of linear algebraic equations (SLAE). The solution of this system is sought under the requirement of minimizing a functional involving the residuals of all its equations. On the one hand, this added complication of the numerical algorithm expands the capabilities of the CM for solving boundary value problems with singularities. On the other hand, the CLR method inherits to a considerable extent some convenient features of the CM. In the present paper, the CLR capabilities are illustrated on benchmark problems for 2D and 3D NavierStokes equations, the modeling of the laser welding of metal plates of similar and different metals, problems investigating strength of loaded parts made of composite materials, boundary-value problems for hyperbolic equations.

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“…Various versions of the CLS and CLR methods have been tested against such benchmarks [9][10][11][12][13][14]. For simplicity, the following discussion is confined to a square cavity…”

Section: Versions Of the Clr Methods For Solving The Navier-stokes Equmentioning

confidence: 99%

Section: On Other Capabilities Of the Clr Methodsmentioning

confidence: 99%

Section: -P10mentioning

confidence: 99%

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Collocation and Least Residuals Method and Its Applications

Shapeev

2016

EPJ Web of Conferences

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“…During this process, coefficients c in neighbor cells can be taken from the previous global iteration (Gauss-Jacobi process) or current iteration (Gauss-Seidel process). Implemented versions of LSC are further detailed in [1,2,[7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

New Possibilities and Applications of the Least Squares Collocation Method

et al. 2018

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Abstract. Least squares collocation method (LSC) is a versatile numerical method for solving boundary value problems for PDE. The present article demonstrates the abilities of LSC to solve various problems -in particular, calculations of bending of isotropic irregular shaped plates and multi-layered anisotropic plates. In order to achieve higher accuracy, new versions of the method utilize high-degree polynomial spaces. The numerical experiments demonstrate high accuracy of the solutions.

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“…Numerical simulations were performed in this study by means of the collocations and least residuals (CLR) method which was successfully applied for various problems before [21][22][23]. Here, we use the conservative version of CLR method for the calculation of the temperature fields and positions of the internal phase boundaries.…”

Section: Numerical Algorithmmentioning

confidence: 99%

Modeling of laser welding of steel and titanium plates with a composite insert

Isaev

Черепанов

Shapeev

2017

AIP Conference Proceedings

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Abstract. A 3D model of laser welding proposed before by the authors was extended to the case of welding of metallic plates made of dissimilar materials with a composite multilayer intermediate insert. The model simulates heat transfer in the welded plates and takes into account phase transitions. It was proposed to select the composition of several metals and dimensions of the insert to avoid the formation of brittle intermetallic phases in the weld joint negatively affecting its strength properties. The model accounts for key physical phenomena occurring during the complex process of laser welding. It is capable to calculate temperature regimes at each point of the plates. The model can be used to select the welding parameters reducing the risk of formation of intermetallic plates. It can forecast the dimensions and crystalline structure of the solidified melt. Based on the proposed model a numerical algorithm was constructed. Simulations were carried out for the welding of titanium and steel plates with a composite insert comprising four different metals: copper and niobium (intermediate plates) with steel and titanium (outer plates). The insert is produced by explosion welding. Temperature fields and the processes of melting, evaporation, and solidification were studied.

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High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations

Cited by 29 publications

References 13 publications

Collocation and Least Residuals Method and Its Applications

Collocation and Least Residuals Method and Its Applications

New Possibilities and Applications of the Least Squares Collocation Method

Modeling of laser welding of steel and titanium plates with a composite insert

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