Journal articles: 'Galerkin methods'

1

Adjerid, Slimane, and Mahboub Baccouch. "Galerkin methods." Scholarpedia 5, no. 10 (2010): 10056. http://dx.doi.org/10.4249/scholarpedia.10056.

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2

Cross, M. "Computational Galerkin methods." Applied Mathematical Modelling 9, no. 3 (June 1985): 226. http://dx.doi.org/10.1016/0307-904x(85)90012-5.

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3

Marion, Martine, and Roger Temam. "Nonlinear Galerkin Methods." SIAM Journal on Numerical Analysis 26, no. 5 (October 1989): 1139–57. http://dx.doi.org/10.1137/0726063.

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4

Nakazawa, Shohei. "Computational Galerkin methods." Computer Methods in Applied Mechanics and Engineering 50, no. 2 (August 1985): 199–200. http://dx.doi.org/10.1016/0045-7825(85)90091-x.

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Cockburn, B. "Discontinuous Galerkin methods." ZAMM 83, no. 11 (November 3, 2003): 731–54. http://dx.doi.org/10.1002/zamm.200310088.

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Warburton, T. C., I. Lomtev, Y. Du, S. J. Sherwin, and G. E. Karniadakis. "Galerkin and discontinuous Galerkin spectral/hp methods." Computer Methods in Applied Mechanics and Engineering 175, no. 3-4 (July 1999): 343–59. http://dx.doi.org/10.1016/s0045-7825(98)00360-0.

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7

Banks, J. W., and T. Hagstrom. "On Galerkin difference methods." Journal of Computational Physics 313 (May 2016): 310–27. http://dx.doi.org/10.1016/j.jcp.2016.02.042.

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Canuto, C., R. H. Nochetto, and M. Verani. "Adaptive Fourier-Galerkin methods." Mathematics of Computation 83, no. 288 (November 21, 2013): 1645–87. http://dx.doi.org/10.1090/s0025-5718-2013-02781-0.

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Di Pietro, Daniele A. "Cell centered Galerkin methods." Comptes Rendus Mathematique 348, no. 1-2 (January 2010): 31–34. http://dx.doi.org/10.1016/j.crma.2009.11.012.

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Belytschko, T., Y. Y. Lu, and L. Gu. "Element-free Galerkin methods." International Journal for Numerical Methods in Engineering 37, no. 2 (January 30, 1994): 229–56. http://dx.doi.org/10.1002/nme.1620370205.

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Sukumar, N., B. Moran, A. Yu Semenov, and V. V. Belikov. "Natural neighbour Galerkin methods." International Journal for Numerical Methods in Engineering 50, no. 1 (2000): 1–27. http://dx.doi.org/10.1002/1097-0207(20010110)50:1<1::aid-nme14>3.0.co;2-p.

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12

Zienkiewicz, O. C., R. L. Taylor, S. J. Sherwin, and J. Peiró. "On discontinuous Galerkin methods." International Journal for Numerical Methods in Engineering 58, no. 8 (August 6, 2003): 1119–48. http://dx.doi.org/10.1002/nme.884.

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13

Paulino, Glaucio H., and L. J. Gray. "Galerkin Residuals for Adaptive Symmetric-Galerkin Boundary Element Methods." Journal of Engineering Mechanics 125, no. 5 (May 1999): 575–85. http://dx.doi.org/10.1061/(asce)0733-9399(1999)125:5(575).

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14

Donea, J. "Generalized Galerkin Methods for Convection Dominated Transport Phenomena." Applied Mechanics Reviews 44, no. 5 (May 1, 1991): 205–14. http://dx.doi.org/10.1115/1.3119502.

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A brief survey is made of recent advances in the development of finite element methods for convection dominated transport phenomena. Because of the nonsymmetric character of convection operators, the standard Galerkin formulation of the method of weighted residuals does not possess optimal approximation properties in application to problems in this class. As a result, numerical solutions are often corrupted by spurious node-to-node oscillations. For steady problems describing convection and diffusion, spurious oscillations can be precluded by the use of upwind-type finite element approximations that are constructed through a proper Petrov-Galerkin weighted residual formulation. Various upwind finite element formulations are reviewed in this paper, with a special emphasis on the major breakthroughs represented by the so-called streamline upwind Petrov-Galerkin and Galerkin least-squares methods. The second part of the paper is devoted to a review of time-accurate finite element methods recently developed for the solution of unsteady problems governed by first-order hyperbolic equations. This includes Petrov-Galerkin, Taylor-Galerkin, least-squares, and various characteristic Galerkin methods. The extension of these methods to deal with unsteady convection-diffusion problems is also considered.

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15

Wang, Li, W. Kyle Anderson, J. Taylor Erwin, and Sagar Kapadia. "Discontinuous Galerkin and Petrov Galerkin methods for compressible viscous flows." Computers & Fluids 100 (September 2014): 13–29. http://dx.doi.org/10.1016/j.compfluid.2014.04.035.

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Bonnet, Marc, Giulio Maier, and Castrenze Polizzotto. "Symmetric Galerkin Boundary Element Methods." Applied Mechanics Reviews 51, no. 11 (November 1, 1998): 669–704. http://dx.doi.org/10.1115/1.3098983.

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This review article concerns a methodology for solving numerically, for engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form. The discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager’s sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions; some quadratic forms have a clear energy meaning; variational properties characterize the solutions and other results, invalid in traditional boundary element methods enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, time-dependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, and computer implementations. Areas and aspects which at present require further research are identified, and comparative assessments are attempted with respect to traditional boundary integral-elements. This article includes 176 references.

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17

Schnepp, S., and T. Weiland. "Discontinuous Galerkin methods with transienthpadaptation." Radio Science 46, no. 5 (June 8, 2011): n/a. http://dx.doi.org/10.1029/2010rs004639.

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18

Carstensen, C., P. Bringmann, F. Hellwig, and P. Wriggers. "Nonlinear discontinuous Petrov–Galerkin methods." Numerische Mathematik 139, no. 3 (March 6, 2018): 529–61. http://dx.doi.org/10.1007/s00211-018-0947-5.

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19

Kessler, Manuel. "Viscosity in discontinuous Galerkin methods." PAMM 7, no. 1 (December 2007): 4100039–40. http://dx.doi.org/10.1002/pamm.200700897.

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20

Faermann, Birgit. "Adaptive galerkin boundary element methods." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 78, S3 (1998): 909–10. http://dx.doi.org/10.1002/zamm.19980781527.

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21

Busch, K., M. König, and J. Niegemann. "Discontinuous Galerkin methods in nanophotonics." Laser & Photonics Reviews 5, no. 6 (May 2, 2011): 773–809. http://dx.doi.org/10.1002/lpor.201000045.

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22

Schmidtmann, Olaf, Fred Feudel, and Norbert Seehafer. "Nonlinear Galerkin Methods for 3D Magnetohydrodynamic Equations." International Journal of Bifurcation and Chaos 07, no. 07 (July 1997): 1497–507. http://dx.doi.org/10.1142/s0218127497001187.

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The usage of nonlinear Galerkin methods for the numerical solution of partial differential equations is demonstrated by treating an example. We describe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan–Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions.

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23

Lu, Hao. "Galerkin and weighted Galerkin methods for a forward-backward heat equation." Numerische Mathematik 75, no. 3 (January 1, 1997): 339–56. http://dx.doi.org/10.1007/s002110050242.

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24

Hou, Dianming, Mohammad Tanzil Hasan, and Chuanju Xu. "Müntz Spectral Methods for the Time-Fractional Diffusion Equation." Computational Methods in Applied Mathematics 18, no. 1 (January 1, 2018): 43–62. http://dx.doi.org/10.1515/cmam-2017-0027.

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AbstractIn this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov–Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than {H^{1}}). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov–Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods.

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25

Efendiev, Yalchin, Juan Galvis, Guanglian Li, and Michael Presho. "Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations." Communications in Computational Physics 15, no. 3 (March 2014): 733–55. http://dx.doi.org/10.4208/cicp.020313.041013a.

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AbstractIn this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.

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26

Dazel, Olivier, and Gwenael Gabard. "Discontinuous Galerkin Methods for poroelastic materials." Journal of the Acoustical Society of America 133, no. 5 (May 2013): 3242. http://dx.doi.org/10.1121/1.4805189.

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27

Koivu, Matti, and Teemu Pennanen. "Galerkin methods in dynamic stochastic programming." Optimization 59, no. 3 (April 2010): 339–54. http://dx.doi.org/10.1080/02331931003696368.

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28

Heumann, Holger, and Ralf Hiptmair. "Stabilized Galerkin methods for magnetic advection." ESAIM: Mathematical Modelling and Numerical Analysis 47, no. 6 (October 7, 2013): 1713–32. http://dx.doi.org/10.1051/m2an/2013085.

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29

Kulkarni, Rekha P., and N. Gnaneshwar. "Iterated discrete polynomially based Galerkin methods." Applied Mathematics and Computation 146, no. 1 (December 2003): 153–65. http://dx.doi.org/10.1016/s0096-3003(02)00533-7.

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30

Hale, Jack K., and Geneviève Raugel. "Regularity, determining modes and Galerkin methods." Journal de Mathématiques Pures et Appliquées 82, no. 9 (September 2003): 1075–136. http://dx.doi.org/10.1016/s0021-7824(03)00045-x.

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31

Grigorieff, R. D., and I. H. Solan. "Spline petrov-galerkin methods with quadrature." Numerical Functional Analysis and Optimization 17, no. 7-8 (January 1996): 755–84. http://dx.doi.org/10.1080/01630569608816723.

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32

Kretzschmar, Fritz, Sascha M. Schnepp, Igor Tsukerman, and Thomas Weiland. "Discontinuous Galerkin methods with Trefftz approximations." Journal of Computational and Applied Mathematics 270 (November 2014): 211–22. http://dx.doi.org/10.1016/j.cam.2014.01.033.

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33

Donoghue, Geoff, and Masayuki Yano. "Spatio-stochastic adaptive discontinuous Galerkin methods." Computer Methods in Applied Mechanics and Engineering 374 (February 2021): 113570. http://dx.doi.org/10.1016/j.cma.2020.113570.

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34

McLachlan, Robert I., and Ari Stern. "Multisymplecticity of Hybridizable Discontinuous Galerkin Methods." Foundations of Computational Mathematics 20, no. 1 (April 22, 2019): 35–69. http://dx.doi.org/10.1007/s10208-019-09415-1.

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35

Lin, Yanping. "Galerkin methods for nonlinear Sobolev equations." Aequationes Mathematicae 40, no. 1 (December 1990): 54–66. http://dx.doi.org/10.1007/bf02112280.

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36

Morton, K. W. "Generalised galerkin methods for hyperbolic problems." Computer Methods in Applied Mechanics and Engineering 52, no. 1-3 (September 1985): 847–71. http://dx.doi.org/10.1016/0045-7825(85)90017-9.

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37

Kreuzer, Christian, and Emmanuil H. Georgoulis. "Convergence of adaptive discontinuous Galerkin methods." Mathematics of Computation 87, no. 314 (February 26, 2018): 2611–40. http://dx.doi.org/10.1090/mcom/3318.

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38

Morel, Jim E., James S. Warsa, Brian C. Franke, and Anil K. Prinja. "Comparison of Two Galerkin Quadrature Methods." Nuclear Science and Engineering 185, no. 2 (February 2017): 325–34. http://dx.doi.org/10.1080/00295639.2016.1272383.

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39

Dedner, Andreas, and Pravin Madhavan. "Adaptive discontinuous Galerkin methods on surfaces." Numerische Mathematik 132, no. 2 (April 3, 2015): 369–98. http://dx.doi.org/10.1007/s00211-015-0719-4.

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40

Morton, K. W., and E. Süli. "Evolution-Galerkin methods and their supraconvergence." Numerische Mathematik 71, no. 3 (September 1, 1995): 331–55. http://dx.doi.org/10.1007/s002110050148.

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41

Antonietti, Paola F., Franco Brezzi, and L. Donatella Marini. "Bubble stabilization of discontinuous Galerkin methods." Computer Methods in Applied Mechanics and Engineering 198, no. 21-26 (May 2009): 1651–59. http://dx.doi.org/10.1016/j.cma.2008.12.033.

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42

Sherwin, S. J., R. M. Kirby, J. Peiró, R. L. Taylor, and O. C. Zienkiewicz. "On 2D elliptic discontinuous Galerkin methods." International Journal for Numerical Methods in Engineering 65, no. 5 (2005): 752–84. http://dx.doi.org/10.1002/nme.1466.

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43

Richter, Andreas, Jörg Stiller, and Roger Grundmann. "Discontinuous Galerkin Methods for Aeroacoustical Investigations." PAMM 8, no. 1 (December 2008): 10697–98. http://dx.doi.org/10.1002/pamm.200810697.

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44

Bringmann, P., C. Carstensen, D. Gallistl, F. Hellwig, D. Peterseim, S. Puttkammer, H. Rabus, and J. Storn. "Towards adaptive discontinuous Petrov-Galerkin methods." PAMM 16, no. 1 (October 2016): 741–42. http://dx.doi.org/10.1002/pamm.201610359.

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45

Bosco, Anthony, and Vincent Perrier. "Discontinuous Galerkin methods for axisymmetric flows." Computers & Fluids 270 (February 2024): 106139. http://dx.doi.org/10.1016/j.compfluid.2023.106139.

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46

Hoskin, Dominique S., R. Loek Van Heyningen, Ngoc Cuong Nguyen, Jordi Vila-Pérez, Wesley L. Harris, and Jaime Peraire. "Discontinuous Galerkin methods for hypersonic flows." Progress in Aerospace Sciences 146 (April 2024): 100999. http://dx.doi.org/10.1016/j.paerosci.2024.100999.

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47

Palitta, Davide, and Valeria Simoncini. "Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations." Vietnam Journal of Mathematics 48, no. 4 (March 5, 2020): 791–807. http://dx.doi.org/10.1007/s10013-020-00390-7.

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48

He, Yinnian, and Kaitai Li. "Stability analysis of nonlinear galerkin & galerkin methods for nonlinear evolution equations." Applied Mathematics 11, no. 2 (June 1996): 137–52. http://dx.doi.org/10.1007/bf02662007.

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49

Carey, Graham F., A. L. Pardhanani, and S. W. Bova. "Advanced Numerical Methods and Software Approaches for Semiconductor Device Simulation." VLSI Design 10, no. 4 (January 1, 2000): 391–414. http://dx.doi.org/10.1155/2000/43903.

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In this article we concisely present several modern strategies that are applicable to driftdominated carrier transport in higher-order deterministic models such as the driftdiffusion, hydrodynamic, and quantum hydrodynamic systems. The approaches include extensions of “upwind” and artificial dissipation schemes, generalization of the traditional Scharfetter – Gummel approach, Petrov – Galerkin and streamline-upwind Petrov Galerkin (SUPG), “entropy” variables, transformations, least-squares mixed methods and other stabilized Galerkin schemes such as Galerkin least squares and discontinuous Galerkin schemes. The treatment is representative rather than an exhaustive review and several schemes are mentioned only briefly with appropriate reference to the literature. Some of the methods have been applied to the semiconductor device problem while others are still in the early stages of development for this class of applications. We have included numerical examples from our recent research tests with some of the methods. A second aspect of the work deals with algorithms that employ unstructured grids in conjunction with adaptive refinement strategies. The full benefits of such approaches have not yet been developed in this application area and we emphasize the need for further work on analysis, data structures and software to support adaptivity. Finally, we briefly consider some aspects of software frameworks. These include dial-an-operator approaches such as that used in the industrial simulator PROPHET, and object-oriented software support such as those in the SANDIA National Laboratory framework SIERRA.

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50

Zhang, Xinxia, Jihan Wang, Zhongshu Wu, Zheyi Tang, and Xiaoyan Zeng. "Spectral Galerkin Methods for Riesz Space-Fractional Convection–Diffusion Equations." Fractal and Fractional 8, no. 7 (July 22, 2024): 431. http://dx.doi.org/10.3390/fractalfract8070431.

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This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods.

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Journal articles on the topic 'Galerkin methods'

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