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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 twodimensional 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 secondorder equations having the form (5.6.2). For example, it seems natural to let u1 = ; w
then
; 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
P=I
Q= 0 0
u2
10
The simplest boundary conditions to prescribe are w = ;u2 = 2 w = ;u1 = 1 f= f :
0
(x y) 2 @ : With these (Dirichlet) boundary conditions, the variational form of this problem is
(5.6.1a) with
ZZ
(v f ) =
v1fdxdy
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
secondorder 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
neatly.
Following the procedures described in Section 5.5, we evaluate (5.6.1) in an elementbyelement 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
e and
(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 piecewisepolynomial 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
 JosephE.Flaherty

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