RESEARCH ARTICLE
Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling
Shijie Wang,
Yidu Yang,
Hai Bi,
Shijie Wang
School of Mathematical Science, Guizhou Normal University, GuiYang, China
Search for more papers by this authorYidu Yang
School of Mathematical Science, Guizhou Normal University, GuiYang, China
Search for more papers by this authorCorresponding Author
Hai Bi
School of Mathematical Science, Guizhou Normal University, GuiYang, China
Correspondence
Hai Bi, School of Mathematical Science, Guizhou Normal University, GuiYang, 550001, China.
Email: [email protected]
Communicated by: S. Wise
Search for more papers by this authorShijie Wang,
Yidu Yang,
Hai Bi,
Shijie Wang
School of Mathematical Science, Guizhou Normal University, GuiYang, China
Search for more papers by this authorYidu Yang
School of Mathematical Science, Guizhou Normal University, GuiYang, China
Search for more papers by this authorCorresponding Author
Hai Bi
School of Mathematical Science, Guizhou Normal University, GuiYang, China
Correspondence
Hai Bi, School of Mathematical Science, Guizhou Normal University, GuiYang, 550001, China.
Email: [email protected]
Communicated by: S. Wise
Search for more papers by this authorAbstract
In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295-324), we establish a three-scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.
REFERENCES
- 1Bui TQ, Nguyen MN, Zhang C. Buckling analysis of Reissner-Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method. Eng Anal Bound Elem. 2011; 35: 1038-1053.
- 2Sadamoto S, Tanaka S, Taniguchi K, et al. Buckling analysis of stiffened plate structures by an Buckling analysis of stiffened plate structures. Thin-Walled Struct. 2017; 117: 303-313.
- 3Yu T, Bui TQ, Yin S, et al. On the thermal buckling analysis of functionally graded plates with internal defects using extended isogeometric analysis. Compos Struct. 2016; 136: 684-695.
- 4Yu T, Yin S, Bui TQ, Xia S, Tanaka S, Hirose S. NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method. Thin-Walled Struct. 2016; 101: 141-156.
- 5Yin S, Yu T, Bui TQ, Liu P, Hirose S. Buckling and vibration extended isogeometric analysis of imperfect graded Reissner-Mindlin plates with internal defects using NURBS and level sets. Comput Struct. 2016; 177: 23-38.
- 6Yu T, Yin S, Bui TQ, Liu C, Wattanasakulpong N. Buckling isogeometric analysis of functionally graded plates under combined thermal and mechanical loads. Compos Struct. 2017; 162: 54-69.
- 7Liu P, Bui TQ, Zhu D, et al. Buckling failure analysis of cracked functionally graded plates by a stabilized discrete shear gap extended 3-node triangular plate element. Composites Part B. 2015; 77: 179-193.
- 8Antunes PRS. On the buckling eigenvalue problem. J Phys A Math Theor. 2011; 44: 215205.
- 9Cheng Q-M, Yang HC. Universal bounds for eigenvalues of a buckling problem II. Trans Am Math Soc. 2012; 364(11): 6139-6158.
- 10Cheng Q-M, Qi X, Wang Q, Xia C. Inequalities for eigenvalues of the buckling problem of arbitrary order. Annali di Matematica. 2018; 197(1): 211-232.
- 11Du F, Mao J, Wang Q, Wu C. Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons. J Differential Equations. 2016; 260: 5533-5564.
- 12Ishihara K. On the mixed finite element approximation for the buckling of plates. Numer Math. 1979; 33: 195-210.
- 13Rannacher R. Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer Math. 1979; 33: 23-42.
- 14Brenner SC, Monk P, Sun J. C0 interior penalty galerkin method for biharmonic eigenvalue problems. Lect Notes Comput Sci Eng. 2015; 106: 3-15.
10.1007/978-3-319-19800-2_1 Google Scholar
- 15Brown BM, Davies EB, Jimack PK, Mihajlović M. D.. On the accurate finite element solution of a class of fourth order eigenvalue problems. arXiv: math/9905038 v1 [math. SP] 6 May; 1999.
- 16Cheng Q-M, Yang H. Universal bounds for eigenvalues of a buckling problem. Commun Math Phys. 2006; 262: 663-675.
- 17Jost J, Li-Jost X, Wang Q, Xia C. Universal inequalities for eigenvalues of the buckling problem of arbitrary order. Comm Part Diff Equ. 2010; 35: 1563-1589.
- 18Kathrin S, Wagner A. Optimality conditions for the buckling of a clamped plate. J Math Anal Appl. 2015; 432: 254-273.
- 19Xu J, Zhou A. Local and parallel finite element algorithms based on two-grid discretizations. Math Comp. 2000; 69(231): 881-909.
- 20Dai X, Zhou A. Three-scale finite element discretizations for quantum eigenvalue problems. SIAM J Numer Anal. 2008; 46(1): 295-324.
- 21He Y, Xu J, Zhou A. Local and parallel finite element algorithms for the Navier-Stokes problem. J Comput Math. 2006; 24(3): 227-238.
- 22He Y, Xu J, Zhou A. Local and parallel finite element algorithms for the Stokes problem. Numer Math. 2008; 109: 415-434.
- 23He Y, Mei L, Shang Y, Cui J. Newton iterative parallel finite element algorithm for the steady Navier-Stokes equations. J Sci Comput. 2010; 44: 92-106.
- 24Ma Y, Zhang Z, Ren C. Local and parallel finite element algorithms based on two-grid discretization for the stream function form of Navier-Stokes equations. Appl Math Comput. 2006; 175: 786-813.
- 25Ma F, Ma Y, Wo W. Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations. Appl Math Mech. 2007; 28(1): 27-35.
- 26Shang Y, Wang K. Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations. Numer Algor. 2010; 54: 195-218.
- 27Liu Q, Hou Y. Local and parallel finite element algorithms for time-dependent convection-diffusion equations. Appl Math Mech -Engl Ed. 2009; 30(6): 787-794.
- 28Shang Y, He Y, Luo Z. A comparison of three kinds of local and parallel finite element algorithms based on two-grid discretizations for the stationary Navier-Stokes equations. Comput Fluids. 2011; 40: 249-257.
- 29Yu J, Shi F, Zheng H. Local and parallel finite element algorithms based on the partition of unity for the Stokes problem. SIAM J Sci Comput. 2014; 36(5): C547-C567.
- 30Zheng H, Yu J, Shi F. Local and parallel finite element algorithm based on the partition of unity for incompressible flows. J Sci Comput. 2015; 65(2): 512-532.
- 31Han X, Li Y, Xie H, You C. Local and parallel finite element algorithm based on multilevel discretization for eigenvalue problem. Int J Numer Anal Model. 2016; 13(1): 73-89.
- 32Tang Q, Huang Y. Local and parallel finite element algorithm based on Oseen-type iteration for the stationary incompressible MHD flow. J Sci Comput. 2017; 70(1): 149-174.
- 33Zheng H, Shi F, Hou Y, Zhao J, Cao Y, Zhao R. New local and parallel finite element algorithm based on the partition of unity. J Math Anal Appl. 2016; 435(1): 1-19.
- 34Du G, Zuo L. Local and parallel finite element post-processing scheme for the Stokes problem. Comput Math Appl. 2017; 73: 129-140.
- 35Du G, Zuo L. Local and parallel finite element method for the mixed Navier-Stokes/Darcy model with Beavers-Joseph interface conditions. Acta Math Sci. 2017; 37(5): 1331-1347.
- 36Yang Y, Han J. The multilevel mixed finite element discretizations based on local defect-correction for the Stokes eigenvalue problem. Comput Methods Appl Mech Engrg. 2015; 289: 249-266.
- 37Bi H, Yang Y, Li H. Local and parallel finite element discretizations for eigenvalue problems. SIAM J Sci Comput. 2013; 35(6): A2575-A2597.
- 38Wang M, Zhang S. Local a priori and a posteriori errores of finite elements for bibarmonic equation. Research Report, School of Mathematical Sciences and Institute of Mathematics. Beijing: Peking University; 2006. 13.
- 39Babuska I, Osborn JE. Eigenvalue problems. In: PG Ciarlet, ed. Handbook of Numerical Analysis, Finite Element Methods (Part 1), vol. 2. North-Holand: Elsevier Science Publishers; 1991: 640-787.
10.1016/S1570-8659(05)80042-0 Google Scholar
- 40Yang Y, Han J. Multilevel finite element discretizations based on local defect correction for nonsymmetric eigenvalue problems. Comput Math Appl. 2015; 70: 1799-1816.
- 41Anthonissen MJH, Bennett BAV, Smooke MD. An adaptive multilevel local defect correction technique with application to combustion. Combust Theor Model. 2005; 9: 273-299.
- 42Chen L. iFEM: An Integrated Finite Element Method Package in MATLAB. Technical Report, Irvine, University of California; 2009.
- 43Blum H, Rannacher R. On the boundary value problem of the biharmonic operator on domains with angular corners. Math Meth in the Appl Sci. 1980; 2: 556-581.
10.1002/mma.1670020416 Google Scholar
