Numerical Methods for PDEs
MATH 610 - 600
Alan Demlow
Computer Assignment 2
Spring 2015
Computer Assignment 2
Exercise 1
100%
FEM for Two-Point Boundary Value Problems Using Linear Elements
Write a program for solving two-point boundary value problems for second order ordinary differential equations by Ritz-Galerkin method using quadratic finite elements.
Specifications
1. Use double precision (which is the default in MATLAB). If you are using MATLAB, you can
solve the corresponding tridiagonal linear system using the backslash operator. You should
however be sure to use sparse data structures in your implementation.
2. Use 10, 20, 40, 80 mesh intervals. In a table give the error in L2 , H 1 , L∞ and also make
plots of the error norms versus the mesh size.
3. The expected format of your computer assignment will be discussed in lab. Please be sure to
follow your TA’s instructions in this regard.
Computational examples - solve the following problems :
1. The deflection of a uniformly loaded, long rectangular plate under axial tension force and
fixed ends, for small deflections, is governed by the second order differential equation. Let S
represent the axial force and q the intensity of the uniform load. The deflection W along the
elemental length is given by :
W 00 (x) −
qx
S
W (x) = −
(l − x), 0 < x < l, W (0) = W (l) = 0,
D
2D
(1)
where l is the length of the plate, and D is the flexural rigidity of the plate. Take q =
200 lb/in2 , S = 100 lb/in., D = 8.8 107 lb in, and l = 50 in. The exact solution is given by :
2
ql4
a = Sl
D , b = 2D , t = x/l and
√
√
b
2
2
√ [sinh( at) + sinh( a(1 − t))] .
W (t) =
−t2 + t − +
a
a a sinh( a)
2. Consider the problem (1) when the r.h.s.
solution for Q =
2
ql
2D
qx
2D (l − x)
is replaced by the constant
q
2D .
The exact
is :
W (t) =
Q
a
1−
√
√
1
√ sinh( at) + sinh( a(1 − t)) .
sinh a
Investigate how using different quadrature rules in the computation of the stiffness matrix
and mass matrix affects
the quality of the
consider the folloR1
R 1 approximation. In particular,
1
1
)
+
f
(0.5 + 2√
)], and
wing three rules : 0 f (x) dx ≈ f (0.5) ; 0 f (x) dx ≈ 12 [f (0.5 − 2√
3
3
p
p
R1
4
5
f (x) dx ≈ 9 f (0.5) + 18 [f (0.5 − 0.5 3/5) + f (0.5 + 0.5 3/5)]. First use the third rule to
0
compute the mass matrix while trying the first and second to compute the stiffness matrix.
Then use the second rule to compute the stiffness matrix while trying the first, second, and
third to compute the mass matrix. Report how these variations affect the order of convergence
as measured in the three norms specified above.
3. Consider the problem (1) when the right end of the plate is elastically supported, i.e. instead
β
of the boundary condition W (l) = 0 now we have the condition W 0 (l) + D
W (l) = 0, where β
is the characteristic of the elastic support. Solve the problem in the following two cases : (a)
β = 0, i.e., the end is free ; (b) β = 2 1010 lb (you can also try the case of a very stiff elastic
support, e.g., β = 2 1013 lb). The exact solution for the case of a plate with a free end (i.e.,
β = 0) is :
√
√
√
2
1
b
2
√ [ a sinh( at) + 2 cosh( a(1 − t))] .
−t + t − +
W (t) =
a
a a cosh( a)
Compare the solution of the case β = 2 1010 lb with the solution of problem (??).