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Unformatted text preview: ) 2 @ un = 0 D (x y) 2 @ N : (5.6.2b) Example 5.6.1. Consider the biharmonic equation
2 w = f (x y) (x y) 2 where ( ) := ( )xx + ( )yy
is the Laplacian and is a bounded two-dimensional region. Problems involving biharmonic operators arise in elastic plate deformation, slow viscous ow, combustion, etc.
Depending on the boundary conditions, this problem may be transformed to a system of
two second-order equations having the form (5.6.2). For example, it seems natural to let u1 = ; w
; u1 = f: Let w = ;u2 to obtain the vector system
; u1 = f ; u2 + u1 = 0 (x y) 2 : This system has the form (5.6.2) with
u = u1
Q= 0 0
The simplest boundary conditions to prescribe are w = ;u2 = 2 w = ;u1 = 1 f= f :
(x y) 2 @ : With these (Dirichlet) boundary conditions, the variational form of this problem is
(v f ) =
and A(v u) = ZZ (v1)x(u1)x + (v1)y (u1)y + (v2 )x(u2)x + (v2 )y (u2)y + v2u1]dxdy: The requirement that (5.6.1a) be satis ed for all vector functions v 2 H01 gives the two
scalar variational problems
ZZ (v1 )x(u1)x + (v1 )y (u1)y ; v1 f ]dxdy = 0 1
8v1 2 H0 38 Mesh Generation and Assembly
ZZ (v2)x(u2)x + (v2)y (u2)y + v2 u1]dxdy = 0 1
8v2 2 H0 : We may check that smooth solutions of these variational problems satisfy the pair of
second-order di erential equations listed above.
We note in passing, that the boundary conditions presented with this example are
not the only ones of interest. Other boundary conditions do not separate the system as
Following the procedures described in Section 5.5, we evaluate (5.6.1) in an elementby-element manner and transform the elemental strain energy and load vector to the
canonical element to obtain Ae(V U) = ZZ V0 G1 U0 + V0 G2 U0 + V0 G2 U0 +
T T e T e e 0 V0 G3 U0 + V0 QU0 ] det(J )d d
T T e (5.6.3a) e where G1 = P 2 + 2 ]
e x y G2 = P
(V f )e = xx ZZ + yy G3 = P 2 + 2 ] e V0 f det(J )d d :
T e x y (5.6.3b)
(5.6.3c) 0 The restriction of the piecewise-polynomial approximation U0 to element e is written
in terms of shape functions as U0 ( ) = np
X c N( )
ej j =1 j (5.6.4a) where np is the number of shape functions on elem...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14