Unformatted text preview: subspaces of HE and H01, respectively
however, this may not be the case because of errors introduced in approximating the
essential boundary conditions and/or the domain . These e ects will also have to be
appraised (cf. Section 7.3). Choosing a basis for S N , we write U and V in the form of
(3.3.1).
The variational problem is written as a sum of contributions over the elements and
the element sti ness and mass matrices and load vectors are generated. For the model
problem (3.1.1) this would involve solving
N
X
e=1 Ae(V U ) ; (V f )e; < V >e] = 0 8V 2 S0N (3.3.2a) 18 MultiDimensional Variational Principles
5 s 6 n
θ u=α 4
pu n+ γu = β 7
3
8 1
2 U y
7
x
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1
2
3
4
5
6
K=7
8 8
4
K e , le 1
0
11
00
11
00
1
0
11
00
11
00
1
0
11
00
11
00
1
0
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00
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00 1
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l=
8 1
0
1
0
1
0
1
0
1
0 Figure 3.3.1: Twodimensional domain having boundary @ = @ E @ N with unit
normal n discretized by triangular nite elements. Schematic representation of the assembly of the element sti ness matrix Ke and element load vector le into the global
sti ness matrix K and load vector l.
where Ae(V U ) = ZZ
e (VxpUx + Vy pUy + V qU )dxdy (3.3.2b) 3.3. Overview of the Finite Element Method
(V f )e = ZZ 19 V fdxdy (3.3.2c) e < V >e= Z V ds (3.3.2d) @ e \@ ~ N is the domain occupied by element e, and N is the number of elements in the mesh.
The boundary integral (3.3.2d) is zero unless a portion of @ e coincides with the boundary
of the nite element domain @ ~ .
Galerkin formulations for selfadjoint problems such as (3.1.6) lead to minimum problems in the sense of Theorem 3.1.1. Thus, the nite element solution is the best solution
in S N in the sense of minimizing the strain energy of the error A(u ; U u ; U ). The
N
strain energy of the error is orthogonal to all functions V in SE as illustrated in Figure
3.3.2 for threevectors.
e 1
0u
1
0
1
0
H1
E 1
0
1
0U
1
0 SN
E N
1
Figure 3.3.2: Subspace SE of HE illustrating the \best" approximation property of the
solution of Galerkin's method. 4. Assemble the global sti ness and mass matrices and load vector. The element
sti ness and mass matrices and load vectors that result from evaluating (3.3.2bd) are
added directly into global sti ness and mass matrices and a load vector. As depicted
in Figure 3.3.1, the indices assigned to unknowns associated with mesh entities (vertices
as shown) determine the correct positions of the elemental matrices and vectors in the
global sti ness and mass matrices and load vector. 20 MultiDimensional Variational Principles 5. Solve the algebraic system. For linear problems, the assembly of (3.3.2) gives rise
to a system of the form dT (K + M)c ; l] = 0 (3.3.3a) where K and M are the global sti ness and mass matrices, l is the global load vector, cT = c1 c2 ::: cN ]T (3.3.3b) dT = d1 d2 ::: dN ]T : (3.3.3c) and
Since (3.3.3a) must be satis ed for all choices of d, we must have
(K + M)c = l: (3.3.4) For the model problem (3.1.1), K + M will be sparse and positive de nite. With proper
treatment of the boundary conditions, it will also be symmetric (cf. Chapter 5).
Each step in the nite element solution will be examined in greater detai...
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 Spring '14
 JosephE.Flaherty
 Hilbert space, Boundary conditions, Galerkin, essential boundary conditions, MultiDimensional Variational Principles

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