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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 50
INNOVATION IN COMPUTER METHODS FOR CIVIL AND STRUCTURAL ENGINEERING
Edited by: B.H.V. Topping and M.B. Leeming
Paper II.1

A Sixth Order Accuracy Nine-Point Scheme for the Biharmonic Equation and its Application to the Deflection of Loaded Plates

M. Arad, A. Yakhot and G. Ben-Dor

Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beersheva, Israel

Full Bibliographic Reference for this paper
M. Arad, A. Yakhot, G. Ben-Dor, "A Sixth Order Accuracy Nine-Point Scheme for the Biharmonic Equation and its Application to the Deflection of Loaded Plates", in B.H.V. Topping, M.B. Leeming, (Editors), "Innovation in Computer Methods for Civil and Structural Engineering", Civil-Comp Press, Edinburgh, UK, pp 31-37, 1997. doi:10.4203/ccp.50.2.1
Abstract
A two-dimensional numerical formula on a nine-point scheme for the biharmonic equation Nabla4 = f(x,y) (Nabla2 is the two-dimensional Laplacian operator) is derived. The truncation error for the suggested scheme is of the sixth-order O(h6) on a square mesh (hx = hy = h) and of the fourth-order O(hx4,hx2hy2,hy4) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The difference in the accuracy as compared to that of the 13-point standard finite difference formula having an accuracy of the second-order is significant. For example, using the presently proposed formula on a grid of 50 x 50 points yields the same order of accuracy as using the standard finite difference formula on a grid of 125000 x 125000 points. The advantage of the suggested scheme is demonstrated by solving the deflection of plates under various loads for different boundary conditions, e.g., simply supported plates and plates with built-in edges. In order to demonstrate the high-order accuracy of the method and to examine the convergence of the error, the numerical results are compared with problems having analytical solutions.

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