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Unformatted text preview: 5.5.17b) s being a coordinate on @ N . As discussed in Chapter 3, smooth solutions of (5.5.17)
satisfy (5.5.2a), the essential boundary conditions (5.5.2b), and the natural boundary
condition
pun = (x y ) 2 @ N : (5.5.18) The line integral (5.5.17b) is evaluated in the same manner as the area integrals and it
will alter the load vector f (cf. Problem 3 at the end of this section). Problems 1. Determine the number of degrees of freedom when a scalar nite element Galerkin
problem is solved using either Lagrange or hierarchical bases on a square region
having a uniform mesh of either 2n2 triangular or n2 square elements. Express
your answer in terms of p and n and compare it with the results of Tables 5.5.1
and 5.5.2.
2. Assume that K is invertible and show that the following algorithm provides a
solution of (5.5.16).
Solve KW = TT for W
Let Y = TW
Solve Ky = f for y
Solve Y = Ty; for
Solve Kc = f ; TT for c 34 Mesh Generation and Assembly
3. Calculate the e ect on the element load vector fe of a nontrivial Neumann condition
having the form (5.5.18).
4. Consider the solution of Laplace's equation uxx + uyy = 0 (x y) 2 on the unit square := f(x y)j0 < x y < 1g with Dirichlet boundary conditions u= (x y) 2 @ : As described in the beginning of this section, create a mesh by dividing the unit
square into n2 uniform square elements and then into 2n2 triangles by cutting each
square element in half along its positive sloping diagonal.
4.1. Using a Galerkin formulation with a piecewiselinear basis, develop the element
sti ness matrices for each of the two types of elements in the mesh.
4.2. Assemble the element sti ness matrices to form the global sti ness matrix.
4.3. Apply the Dirichlet boundary conditions and exhibit the nal linear algebraic
system for the nodal unknowns.
5. The task is is to solve a Dirichlet problem on a square using available nite element
software. The problem is
;uxx ; uyy + f (x y) = 0 (x y) 2 with u = 0 on the boundary of the unit square = f(x y)j0
f (x y) so that the exact solution of the problem is xy 1g. Select u(x y) = exy sin x sin 2 y:
The Galerkin form of this problem is to nd u 2 H01 satisfying
ZZ vxux + vy uy + vf ]dxdy = 0...
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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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