Proper orthogonal decomposition Pascal polynomial-based method for solving Sobolev equation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mehdi Dehghan ◽  
Baharak Hooshyarfarzin ◽  
Mostafa Abbaszadeh
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Meshfree Method ◽  
Orthogonal Decomposition ◽  
Sobolev Type ◽  
Computational Domain ◽  
Sobolev Equation ◽  
Content Type ◽  
Proper Orthogonal

Purpose This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain. Design/methodology/approach In this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering. Findings The authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method. Originality/value The developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity.

A Comparison of Classic and Snapshot Proper Orthogonal Decomposition on the Three Dimensional Wall Jet Flow Field

2014 ◽  
Author(s):  
Mahdi Hosseinali ◽  
Stephen Wilkins ◽  
Lhendup Namgyal ◽  
Joseph Hall
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Energy Content ◽  
Near Field ◽  
Three Dimensional ◽  
Orthogonal Decomposition ◽  
Wall Jet ◽  
Polar Coordinate ◽  
Wall Jet Flow ◽  
Turbulent Wall ◽  
Proper Orthogonal

In this paper, classic Proper Orthogonal Decomposition (POD) on a polar coordinate and snapshot POD on a Cartesian grid will be applied separately in the near field of a turbulent wall jet. Three-component stereoscopic PIV measurements are performed in the transverse plane of a wall jet formed using a round contoured nozzle with a Reynolds number of 250,000. Eigenfunctions and energy distributions of the two methods are compared. Reconstructions using same number of modes and same content of energy have been compared. The effect of grid resolution on the energy content of the classic method has also been studied.

scholarly journals Direct synthesis of equivalent circuits from reduced FE models using proper orthogonal decomposition

2016 ◽  
Vol 35 (6) ◽  
pp. 2035-2044 ◽  
Author(s):  
Toshihito Shimotani ◽  
Yuki Sato ◽  
Hajime Igarashi
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Fe Analysis ◽  
Electric Machines ◽  
Orthogonal Decomposition ◽  
Computational Time ◽  
Equivalent Circuits ◽  
Fe Model ◽  
Content Type ◽  
Electromagnetic Devices ◽  
Proper Orthogonal

Purpose The purpose of this paper is to propose a fast synthesis method of the equivalent circuits of electromagnetic devices using model order reduction. Finite element method (FEM) has been widely used to design electromagnetic devices. For FE analysis of these devices connected to control and deriving circuits, FE equations coupled with the circuit equations have to be solved for many times in their design processes. If the FE models are replaced by equivalent circuit models, computational time could be drastically reduced. Design/methodology/approach In the proposed method, a reduced FE model is obtained using proper orthogonal decomposition (POD) in which the size of FE equation is effectively reduced so that the computational time for FE analysis is shortened. Then, the equivalent circuits are directly synthesized from the admittance function of the reduced system. Findings Accuracy and computational efficiency of the proposed method are compared with those of another POD-based method in which the equivalent circuits are synthesized from fitting of frequency characteristics using optimization algorithm. There are no significant differences in the accuracy of both methods, while the speedup ratio of the former method is found larger than that for the latter method for the same sampling points. Originality/value The equivalent circuits of electric machines and devices have been synthesized on the basis of physical insight of engineers. This paper proposes a novel method by which the equivalent circuits are automatically synthesized from FE model of the electric machines and devices using POD.

Characteristics of Turbulent Three-Dimensional Wall Jets

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
M. Agelin-Chaab ◽  
M. F. Tachie
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Orthogonal Decomposition ◽  
The Self ◽  
Shear Stresses ◽  
Spread Rate ◽  
Similar Region ◽  
Self Similar ◽  
Proper Orthogonal

Three-dimensional turbulent wall jet was investigated using a particle image velocimetry technique. Three Reynolds numbers based on the jet exit velocity and diameter of 5000, 10,000, and 20,000 were studied. Profiles of the mean velocities, turbulence intensities, and Reynolds shear stresses as well as two-point velocity correlations and proper orthogonal decomposition analyses were used to document the salient features of the wall jets. The decay and spread rates are independent of Reynolds numbers in the self-similar region. The estimated values of 1.15, 0.054, and 0.255 for the decay rate, wall-normal spread rate, and lateral spread rate, respectively, are within the range of values reported in the literature. The two-point correlation analysis showed that the inclination of the streamwise velocity correlation contours in the inner layer is 11±3 deg in the wall region, which is similar to those of canonical turbulent boundary layers. The results from the proper orthogonal decomposition indicate that low-order modes contribute more to the turbulence statistics in the self-similar region than in the developing region. The Reynolds shear stresses are the biggest benefactors of the low-order mode contribution while the wall-normal turbulence intensities are the least.

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