Unformatted text preview: .6) has the same form as the one-dimensional
problem (2.3.3). Indeed, the theory and extremal principles developed in Chapter 2 apply
to multi-dimensional problems of this form.
Theorem 3.1.1. The function w 2 HE that minimizes
I w] = A(w w) ; 2(w f ) ; 2 < w > : (3.1.7) is the one that satis es (3.1.6), and conversely.
Proof. The proof is similar to that of Theorem 2.2.1 and appears as Problem 1 at the
end of this section. 4 Multi-Dimensional Variational Principles 1
Corollary 3.1.1. Smooth functions u 2 HE satisfying (3.1.6) or minimizing (3.1.7) also satisfy (3.1.1). Proof. Again, the proof is left as an exercise.
Example 3.1.1. Suppose that the Neumann boundary conditions (3.1.1c) are changed
to Robin boundary conditions (x y) 2 @ pun + u = N: (3.1.8a) Very little changes in the variational statement of the problem (3.1.1a,b), (3.1.8). Instead
of replacing pun by in the boundary inner product (3.1.5c), we replace it by ; u.
Thus, the Galerkin form of the problem is: nd u 2 HE satisfying A(v u) = (v f )+ < v ; u > 8v 2 H01: (3.1.8b) Example 3.1.2. Variational principles for nonlinear problems and vector systems
of partial di erential equations are constructed in the same manner as for the linear
scalar problems (3.1.1). As an example, consider a thin elastic sheet occupying a twodimensional region . As shown in Figure 3.1.2, the Cartesian components (u1 u2) of
the displacement vector vanish on the portion @ E of of the boundary @ and the components of the traction are prescribed as (S1 S2) on the remaining portion @ N of @ . The equations of equilibrium for such a problem are (cf., e.g., 6], Chapter 4) @ 11 + @ 12 = 0
@ 12 + @ 22 = 0
@x @y (3.1.9a)
(x y) 2 (3.1.9b) where ij , i j = 1 2, are the components of the two-dimensional symmetric stress tensor
(matrix). The stress components are related to the displacement components by Hooke's
11 22 12 E
= 1 ; 2 ( @u1 + @u2 )
@y (3.1.10a) E
= 1 ; 2 ( @u1 + @u2 )
@x @y (3.1.10b) = E ( @u1 + @u2 ) 2(1 + ) @y @x (3.1.10c) 3.1. Galerkin's Method and Extremal Principles 5 y
θ u1 = 0,
u2 = 0 Ω S2 S1 x Figure 3.1.2: Two-dimensional elastic sheet occupying the region . Displacement components (u1 u2) vanish on @ E and traction components (S1 S2) are prescribed on @ N .
where E and are constants called Young's modulus and Poisson's ratio, respectively.
The displacement and traction boundary conditions are u1(x y) = 0
n1 11 + n2 12 = S1 u2(x y) = 0
n1 12 + n2 22 = S2 (x y) 2 @ (3.1.11a) E (x y) 2 @ N (3.1.11b) where n = n1 n2]T = cos sin ]T is the unit outward normal vector to @ (Figure
Following the one-dimensional formulations, the Galerkin form of this problem is
obtained by multiplying (3.1.9a) and (3.1.9b) by test functions v1 and v2 , respectively,
integrated over , and using the divergence theorem. With u1 and u2 being components
of a displacement eld, the functions v1 and v2 are referred to as components of the
virtual displacement eld.
We use (3.1.9a) to illustrate the process thus, multiplying by v1 and integrating over
, we nd
v1 @x1 + @@y ]dxdy = 0:
The three stress components are dependent on the two displacement components and
are typically replaced by these using (3.1.10). Were this done, the variational principle 6 Multi-Dimensional Var...
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- Hilbert space, Boundary conditions, Galerkin, essential boundary conditions, Multi-Dimensional Variational Principles