efficient and memory saving method based on ...research article efficient and memory saving method...

Research ArticleEfficient and Memory Saving Method Based on PseudoskeletonApproximation for Analysis of Finite Periodic Structures

Chunbei Luo , Yong Zhang, and Hai Lin

State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310058, China

Correspondence should be addressed to Hai Lin; [email protected]

Received 5 April 2018; Revised 11 June 2018; Accepted 24 June 2018; Published 22 July 2018

Academic Editor: Paolo Baccarelli

Copyright © 2018 Chunbei Luo et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An efficient and memory saving method based on pseudoskeleton approximation (PSA) is presented for the effective and accurateanalysis of finite periodic structures. Different from the macro basis function analysis model, our proposed method uses theformulations derived by the local Rao-Wilton-Glisson basis functions. PSA is not only used to accelerate the matrix-vectorproduct (MVP) inside the single unit but also adopted to decrease the calculation burden of the coupling between the differentcells. Moreover, the number of decomposed coupling matrices is minimized due to the displacement invariance of the periodicproperty. Consequently, even compared with the multilevel fast multipole algorithm (MLFMA), the new method saves muchmore memory resources and computation time, which is also demonstrated by the numerical examples.

1. Introduction

Periodic structures have recently found wide applications inthe electromagnetic engineering such as antenna arrays andmetamaterials with negative permittivity and negativepermeability. Hence, the accurate and efficient analysis ofperiodic structures becomes quite essential. If the periodicstructure is an infinite array, simple methods can be appliedbased on Floquet’s theorem [1] or periodic Green’s function[2], where only a single cell of the periodic structure is thedomain of interest.

However, all periodic structures have finite size in real-life problems, although the size may be very large. Therefore,the numerical algorithms which accurately consider themutual coupling between all cells should be used if the accu-rate results are required. The method of moments (MoM) [3]and its fast algorithms such as fast multipole method (FMM)[4], adaptive cross approximation (ACA) [5], and FFT-basedmethods [6] are flexible approaches to study the surfaceproblems. However, the efficiency of numerical methodsis still limited since the periodic property is not used inthe algorithm framework. Recently, a hybrid method com-bining the accurate MoM and periodic method of moment

(PMM) [7] has been proposed which can gain the balancebetween the two methods. Moreover, some physicallybased entire-domain basis functions [8] have been devel-oped to reduce the number of unknowns. Further, theFMM and FFT techniques are integrated to acceleratethe calculation [9].

Compared to the ACA method, pseudoskeleton approxi-mation (PSA) [10] is also an efficient low-rank-basedalgebraic fast algorithm which makes it a really competitivealternative. In this paper, we propose an efficient methodwith low-memory requirement based on PSA to performthe analysis for finite periodic structures effectively and accu-rately. In consideration of the accuracy of the mutual interac-tions [8] and the simplicity of implementation, our proposedmethod uses the formulations derived from the local basisfunctions instead of macro basis function (MBF) [11, 12].In this paper, PSA is not only used to accelerate the matrix-vector product (MVP) inside the single unit but also adoptedto decrease the calculation burden of the coupling betweenthe different cells. Moreover, the number of decomposedcoupling matrices is minimized due to the displacementinvariance of the periodic property. With these improve-ments, an efficient method with low-memory usage of finite

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 1612498, 6 pageshttps://doi.org/10.1155/2018/1612498

http://orcid.org/0000-0002-6645-3462http://orcid.org/0000-0002-1682-8465https://doi.org/10.1155/2018/1612498

periodic objects can be achieved. Several numerical examplesare given to show the priority of the proposed methodcompared to the conventional multilevel fast multipolealgorithm (MLFMA) [13] for periodic structures.

2. MoM and PSA Formulation

In this section, the basis principle of MoM and PSA is brieflyintroduced at first. Then, the choice of arguments in PSAis discussed.

2.1. MoM Equations and Its Fast Algorithms. Consider atime-harmonic e−jωt electromagnetic wave scattering orradiation problem of an arbitrary perfect electrically con-ducting (PEC) object. The object is excited by an incidentelectric field Einc r , then the electric field integral equation(EFIE) associated with the surface equivalent currents J rcan be expressed by

−jωμ

t̂ ⋅ Einc r = t̂ ⋅S′

1 + 1k2

∇∇ J r′ G r, r′ dr′, 1

where t̂ is the tangential unit vector on the surface S′ of theobject and ω and μ stand for the angular frequency and per-

meability, respectively. G r, r′ = e−jk r−r′ / 4π r − r′ is 3Dscalar Green’s function. The linear system of MoM isobtained by discretizing the unknown vector J r with Rao-Wilton-Glisson (RWG) [14] basis functions and applyingGalerkin’s testing method. Let ZI =V represent the aboveEFIE matrix system. The MVP process can be acceleratedby fast algorithms which can be written as follows:

ZI = ZnearI + ZfarI, 2where Znear is the matrix of near field interactions which aredirectly computed and stored and Zfar stands for couplingsbetween the far-field interactions which will be acceleratedtogether with ZfarI in the iterative solving process.

2.2. Basic PSA Frameworks. According to the low-rankdecomposition, the far-field interaction matrix Zfar withrank-deficient property can be approximated by a productof two much smaller submatrices U and V:

Zm×nfar ≈Um×rVr×n, 3

where r is the effective rank and satisfies r≪m, n. While inthe skeleton approximation (SA) theory [10], there is a non-singular r × r submatrix Ẑ in Zfar. Denote the rectangularmatrices as C and R which contain the columns and rowsof Ẑ, respectively, then Zfar is expressed as

Zfar =CẐ−1R, 4

where C and R have dimensions with m × r and r × n,respectively. In the PSA method, Zfar is reevaluated as

Zfar ≈ CGR, 5

where G is the pseudoinverse of Ẑ with dimensions ofp × p p > r , then p columns and p rows are chosen fromZfar to get the C and R. Three aspects need to be noted

here: (i) Ẑ−1 is the inverse of Ẑ and the computation ofinverse is very expensive; (ii) the determination of the valueof p is a balance between accuracy and efficiency, where p isa number large enough so that r most important bases willbe embedded; (iii) how to choose those working columnsand rows of C and R.

For the (ii) and (iii) problems, we will discuss them in thenext subsection. For the first problem, assuming that Ẑ can bedecomposed via singular value decomposition (SVD) as

Ẑ = PΣQ∗, 6

where P and Q are p × p unitary matrices, Σ is a diagonalp × p matrix with nonnegative real numbers, and Q∗ is thecomplex conjugate transpose of Q. In the actual imple-mentation, the dimension of P, Q, and Σ can be furtherdecreased by a preset threshold [15]. Let P̂, Σ, and Q̂represent the reduced submatrix of P, Σ, and Q, respec-tively. Then, calculation of the pseudoinverse of Ẑ (i.e., G)is straightforward:

Ẑ−1 ≈G = Q̂Σ−1P̂∗ 7

By combining (5) and (7), the original far-fieldinteraction matrix Zfar can be decomposed as

Zfar ≈ CQ̂Σ−1P̂∗R 8

2.3. Choice of Arguments in PSA. As mentioned in the previ-ous subsection, the choice of the value of p and the specificsampling rows and columns determine the performance ofPSA. In the randomized PSA (RPSA) [15], p is equal to 2r.Then, the problem of (ii) transforms into how to estimatethe rank r of the original matrix. In [16], Chai and Jiao givethe approximation of rank of the 3D EM problem byRank3D~O k0 , where k0 is the studying wave number. Inthis paper, the rank is approximated by

Rank3D ≈ k0d + α ln π + k0d , 9

where d is the diameter of the bounding box correspondingto the octree structure and α is a preset positive parameter.The larger the α is, the more accurate the matrix decomposi-tion is. In this paper, when α is set as 3, the satisfactoryaccuracy can be guaranteed.

For the problem of (iii), instead of using randomnumbers in RPSA, we use a strategy analogous to ACA in thispaper. Firstly, initialize from the 0th row as the first rowindex. Then, find the largest entry in this row, and thecorresponding column value in which this entry is locatedis chosen as the next column index. Similarly, find the largestentry in the current column and get the next row index whichshould be different from all previous row indexes. This pro-cess is carried out iteratively until p rows and columns arefound and stored. Please refer to [17] for more information.

3. Proposed Method for Periodic Structures

We consider a case of arbitrarily shaped PEC patch, forexample, refer in Figure 1. The total impedance matrix

2 International Journal of Antennas and Propagation

contains two parts: self-coupling blocks and mutualcoupling blocks. Therefore, both the blocks are analyzedand decomposed by PSA to gain better efficiency andlow-memory usage.

3.1. PSA for Decomposition of Self-Coupling Matrix. In theprevious mentioned periodic algorithms, the self-couplingblock elements are calculated by a direct MoMmethod whichis not efficient for a large array unit. Therefore, PSA is used todecompose the self-coupling matrix. As the same to all fastalgorithms of MoM, the low-rank decompositions based onPSA are performed on the far-field groups while the near-field elements are directly computed and saved. Moreover,some operations are taken out to further improve theefficiency of PSA. Firstly, the pseudoinverse of the Ẑ (i.e., G)will not be directly calculated since the complexity of SVDis very high. Instead, the LU decomposition will be per-formed when the dimension grows up. Different from thepreviously proposed PSA, C, R, and the LU decompositionof Ẑ will be stored. Moreover, C and R in (8) are furtherdecomposed by ACA technique. Finally, the original far-field coupling matrix can be decomposed into six subma-trices which is showed by the following formula:

Zfar =UCVC LẐUẐ −1URVR, 10

where UCVC (or URVR) is the ACA decomposition of C(or R) and LẐUẐ is the LU decomposition of Ẑ. Note thatthe inverse of LU matrix of Ẑ is not directly computed.Actually, the LU back substitute is performed after theMVP based on the matrix URVR in each MVP process.

3.2. PSA for Decomposition of Mutual Coupling Matrices. Inthe traditional fast algorithms, the matrix decompositionsare only performed on the far-field groups especially for thephysically related methods such as MLFMA. However, themutual interactions between two different cells (off-diagonalblocks) are all computed by PSA in our implementation. Thistreatment has been also used in [18] and demonstrated to bemore efficient but losing little accuracy. Moreover, thedisplacement invariance property is explored in ourimplementation. Since the mutual coupling remains the samewhen the distance between two groups’ centers is not

changed, the corresponding matrix will be calculated andstored only once under the index of relative distance. In theMVP process, the stored coupling matrices may be usedseveral times when the relative distances are the same. Byusing this scheme, the number of mutual couplingmatrices can be decreased dramatically. For example, theoriginal number of off-diagonal matrix blocks ofFigure 1 is 20 × 20 − 20 = 380 while the reduced numberwill be 62 if the displacement invariance is used.

3.3. Preconditioner Considerations. The preconditioningstage should be also considered carefully while dealing withcomplex structures. In this paper, the preconditioning matrixM for impedance matrix Z is built based on the self-couplingmatrix M0.

M = diag M0,M0,… ,M0 11

Moreover, the inner-outer iteration scheme can be alsoapplied when facing ill-conditioned matrix systems such asEFIE for radiation analysis. In our implementation, the inneriteration is the solution of the self-coupling matrices.Certainly, the inner solution area can be also extended toimprove the convergence rate of outer iteration.

4. Numerical Results

In this section, several numerical experiments are presentedto demonstrate the efficiency and low-memory requirementof the proposed method. For all the simulations, the meshsizes are no less than 0.1λ (λ represents the wave length).The biconjugate gradient stabilized method (BiCGSTAB) isadopted as the iterative solver for the matrix equations, andthe threshold of the iterative stopping criterion for residuumis set to 0.001. The sparse approximate inverse precondi-tioner is used in all simulations. All the computations werecarried out on a workstation with four Xeon E5-4620 CPUand 256GB of RAM with OpenMP technique, and the digitswere stored in double precision.

4.1. Scattering Simulation. In order to verify the accuracy andefficiency of the proposed method, we first consider aperiodic structure consisting of N0 = 4 × 5 = 20 element cells(also shown in Figure 1), where the unit cell is a 0.5m PECsphere. The working frequency is set to be 1GHz which leadsto 20286 unknowns (degrees of freedom) in each patch.Hence, the total number of unknowns is N = 20 × 20286 =416520. The gap between two unit cells is set to be 0.5m.The radar cross sections (RCS) computed by MLFMA, theproposed method (periodic PSA), and the commercial soft-ware (FEKO) are illustrated in Figure 2(a). It could be clearlyseen that the numerical results from the proposed methodhave excellent agreement with both the conventionalMLFMA and FEKO. While as in Table 1, both the computa-tion time and peak memory usage by periodic PSA are lessthan the MLFMA method. What is more, another case isconsidered when the gap between two unit cells is decreasedto be 0.1m. The RCS results and the use of computerresources are shown in Figure 2(b) and Table 1, respectively.Although the rank of mutual coupling matrix gets bigger due

11 12 13 14 15

21 22 23 24 25

31 32 33 34 35

41 42 43 44 45

Figure 1: Periodic structure with a finite size of 4 × 5.

3International Journal of Antennas and Propagation

to the tight coupling interactions, the proposed method stillshows the good efficiency and low-memory usage with losinglittle accuracy.

Furthermore, the scattering of a large-scale periodicstructure consisting of N0 = 32 × 32 = 1024 element cells isconsidered. The working frequency is set to be 1GHz.Different radii of the sphere are considered which lead toabout 0.81, 3.38, and 5.46 million numbers of unknowns cor-responding to 0.1m, 0.2m, and 0.25m models, respectively.The distance between two centers of neighboring sphere isset to be a fixed value of 1.0m. Table 2 shows the informationof models and the numerical performance between MLFMAand the periodic PSA. It can be seen that for a large-scale

problem, the proposed PSA method still needs less computa-tion time and much less memory requirement. It should benoted that in the 3rd example in Table 2 when the sphereradius is set to 0.25m, the finest box in MLFMA has to beset to 0.20λ to avoid low-frequency breakdown. Figure 3shows that the RCS results of 0.25m sphere modelscomputed from the periodic PSA still agree well with FEKOand MLFMA.

4.2. Radiation Simulation. Lastly, we consider a radiationproblem, which contains N0 = 10 × 10 = 100 antenna cellswith delta-gap excitations. Each cell has 1145 unknowns,and the total number is N = 100 × 1145 = 114500. The gap

FEKOMLFMAPeriodic PSA

60

40

20

0RCS

(dBs

m)

−20

−400 60 120 180

Angle (deg)240 300 360

(a)

FEKOMLFMAPeriodic PSA

−30

−20

−10

0

0 60 120 180Angle (deg)

240 300 360

10

RCS

(dBs

m) 20

30

40

50

(b)

Figure 2: Radar cross sections (V-V polarization) of the 4 × 5 sphere array with (a) 0.5m and (b) 0.1m gap computed by FEKO, MLFMA,and periodic PSA. The incident plane wave angles satisfy θ = 0° and φ = 0°. The scattering angles work with θ varies from 0° to 360° and φ = 0°.

Table 1: Computation time and memory usage of MLFMA andperiodic PSA for scattering of 4 × 5 elements.

Gap Methods Computation time Memory usage

0.5mMLFMA 210 s 8726MB

Periodic PSA 109 s 2145MB

0.1mMLFMA 222 s 8801MB


Table 2: Computation time and memory usage of MLFMA andperiodic PSA for scattering of 32 × 32 elements.

Radius Gap Methods Computation time Memory usage

0.1m 0.8mMLFMA 369 s 18229MB


0.2m 0.6mMLFMA 1551 s 71455MB


0.25m 0.5mMLFMA 2341 s 96906MB


FEKO

70

60

50

40

30

20

10

0

0 60 120 180Angle (deg)

240 300 360

−10

PCS

(dBs

m)

MLFMAPeriodic PSA

Figure 3: Radar cross sections (V-V polarization) of the 32 × 32sphere array with 0.5m gap computed by FEKO, MLFMA, andperiodic PSA. The incident plane wave angles satisfy θ = 0° andφ = 0°. The scattering angles work with θ varies from 0° to 360°and φ = 0°.


between two cells in this model is 0.005m which is smallerthan 0.1λ. Since EFIE is the only choice for radiationproblems, the inner-outer iteration scheme is adopted toaccelerate the calculation process. Figure 4 shows the currentdensity distribution of the antenna array. The far-field gainpatterns computed by the proposed method and MLFMAare shown in Figure 5.

Obviously, the two methods give nearly the same results.However, as in Table 3, the computation time has beenreduced from 419 seconds to 172 seconds in solving matrixequations for the radiation problem. Moreover, the memoryusage for MLFMA is 8841MB while the value is only 358MBfor the proposed method. Again, the proposed method per-forms much better thanMLFMA. The first reason is that only360 off-diagonal blocks are need to be calculated and storedin our method while the original value is 100 × 100 − 100 =9900. Secondly, the mesh size is about 0.01λ for the antennacell (dense mesh). As it is known, the lowest-level box size islimited to be no less than 0.20λ since MLFMA suffers fromthe low-frequency breakdown, which results in a heavynear-field matrix consumption.

5. Conclusion

In this paper, an efficient and memory saving method basedon pseudoskeleton approximation (PSA) for the effectiveand accurate analysis of finite periodic structures ispresented. The appropriate choice of sampling rows andcolumns as well as sampling number in PSA is discussed

and confirmed. Based on the algebraic fast algorithm PSA,we have carefully handled the self-coupling and mutual-coupling blocks of the original impedance matrix. Moreover,the number of decomposed coupling matrices is minimizedby the employment of the displacement invariance of theperiodic property. The simulation results of large-scalescattering and radiation examples show that the proposedperiodic PSA method needs less computation time and muchless memory compared to conventional MLFMA. Hence,numerical results demonstrate the efficiency and superiorityof the proposed method.

Data Availability

No additional data are available.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant 61171035.

x

y

0.00045 0.0016 0.00566 0.0201 0.0712 0.253 0.896 3.18 11.3 40 142

Figure 4: Current density distribution of 10 × 10 patch array.

MLFMAPSA

30

20

10

Gai

n pa

ttern

(dBi

)

0

0 50 100 150 200Angle (deg)

250 300 350

−10

−20

−30

−40

Figure 5: Radiation patterns of the 10 × 10 antenna array with0.005m gap computed by MLFMA and periodic PSA. The far-field angles work with θ varies from 0° to 360° and φ = 0°.

Table 3: Computation time and memory usage of MLFMA andperiodic PSA for radiation of 10 × 10 antenna array.

Methods Computation time Memory usage

MLFMA 419 s 8841MB


5International Journal of Antennas and Propagation

References

[1] F. Xu, Y. Zhang, W. Hong, K. Wu, and T. J. Cui, “Finite-difference frequency-domain algorithm for modelingguided-wave properties of substrate integrated waveguide,”IEEE Transactions on Microwave Theory and Techniques,vol. 51, no. 11, pp. 2221–2227, 2003.

[2] K. Yasumoto and K. Yoshitomi, “Efficient calculation oflattice sums for free-space periodic Green’s function,” IEEETransactions on Antennas and Propagation, vol. 47, no. 6,pp. 1050–1055, 1999.

[3] R. F. Harrington, Field Computation by Moment Methods,Wiley-IEEE Press, 1993.

[4] C.-C. Lu and W. C. Chew, “A multilevel algorithm for solvinga boundary integral equation of wave scattering,” Microwaveand Optical Technology Letters, vol. 7, no. 10, pp. 466–470,1994.

[5] M. Bebendorf, “Approximation of boundary element matri-ces,” Numerische Mathematik, vol. 86, no. 4, pp. 565–589,2000.

[6] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Aim:adaptive integral method for solving large-scale electromag-netic scattering and radiation problems,” Radio Science,vol. 31, no. 5, pp. 1225–1251, 1996.

[7] J. Su, X. Xu, and B. Hu, “Hybrid PMM-MoM method for theanalysis of finite periodic structures,” Journal of Electromag-netic Waves and Applications, vol. 25, no. 2-3, pp. 267–282,2011.

[8] W. B. Lu, T. J. Cui, Z. G. Qian, X. X. Yin, andW. Hong, “Accu-rate analysis of large-scale periodic structures using an efficientsub-entire-domain basis function method,” IEEE Transactionson Antennas and Propagation, vol. 52, no. 11, pp. 3078–3085,2004.

[9] W. B. Lu, T. J. Cui, and H. Zhao, “Acceleration of fastmultipole method for large-scale periodic structures with finitesizes using sub-entire-domain basis functions,” IEEE Transac-tions on Antennas and Propagation, vol. 55, no. 2, pp. 414–421,2007.

[10] S. A. Goreinov, N. L. Zamarashkin, and E. E. Tyrtyshnikov,“Pseudo-skeleton approximations by matrices of maximal vol-ume,” Mathematical Notes, vol. 62, no. 4, pp. 515–519, 1997.

[11] E. Suter and J. R. Mosig, “A subdomain multilevel approachfor the efficient MoM analysis of large planar antennas,”Microwave and Optical Technology Letters, vol. 26, no. 4,pp. 270–277, 2000.

[12] D. Gonzalez-Ovejero, F. Mesa, and C. Craeye, “Acceleratedmacro basis functions analysis of finite printed antenna arraysthrough 2D and 3D multipole expansions,” IEEE Transactionson Antennas and Propagation, vol. 61, no. 2, pp. 707–717,2013.

[13] J. Song, C.-C. Lu, and W. C. Chew, “Multilevel fast multipolealgorithm for electromagnetic scattering by large complexobjects,” IEEE Transactions on Antennas and Propagation,vol. 45, no. 10, pp. 1488–1493, 1997.

[14] S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scatteringby surfaces of arbitrary shape,” IEEE Transactions on Anten-nas and Propagation, vol. 30, no. 3, pp. 409–418, 1982.

[15] X. Zhu andW. Lin, “Randomised pseudo-skeleton approxima-tion and its application in electromagnetics,” ElectronicsLetters, vol. 47, no. 10, pp. 590–592, 2011.

[16] W. Chai and D. Jiao, “Theoretical study on the rank of integraloperators for broadband electromagnetic modeling from staticto electrodynamic frequencies,” IEEE Transactions on Compo-nents, Packaging andManufacturing Technology, vol. 3, no. 12,pp. 2113–2126, 2013.

[17] Y. Zhang and H. Lin, “Localized pseudo-skeleton approxi-mation method for electromagnetic analysis on electricallylarge objects,” Progress In Electromagnetics Research Letters,vol. 57, pp. 103–109, 2015.

[18] A. Heldring, J. M. Tamayo, C. Simon, E. Ubeda, and J. M. Rius,“Sparsified adaptive cross approximation algorithm for accel-erated method of moments computations,” IEEE Transactionson Antennas and Propagation, vol. 61, no. 1, pp. 240–246,2013.


International Journal of

AerospaceEngineeringHindawiwww.hindawi.com Volume 2018

RoboticsJournal of

Hindawiwww.hindawi.com Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwww.hindawi.com

Volume 2018

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of


Hindawiwww.hindawi.com

Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling &Simulationin EngineeringHindawiwww.hindawi.com Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

www.hindawi.com Volume 2018

Advances in

Multimedia

Submit your manuscripts atwww.hindawi.com
https://www.hindawi.com/journals/ijae/https://www.hindawi.com/journals/jr/https://www.hindawi.com/journals/apec/https://www.hindawi.com/journals/vlsi/https://www.hindawi.com/journals/sv/https://www.hindawi.com/journals/ace/https://www.hindawi.com/journals/aav/https://www.hindawi.com/journals/jece/https://www.hindawi.com/journals/aoe/https://www.hindawi.com/journals/tswj/https://www.hindawi.com/journals/jcse/https://www.hindawi.com/journals/je/https://www.hindawi.com/journals/js/https://www.hindawi.com/journals/ijrm/https://www.hindawi.com/journals/mse/https://www.hindawi.com/journals/ijce/https://www.hindawi.com/journals/ijap/https://www.hindawi.com/journals/ijno/https://www.hindawi.com/journals/am/https://www.hindawi.com/https://www.hindawi.com/

efficient and memory saving method based on ...research article efficient and memory saving method...

Documents