ME 422
FEM Homework #4
Distributed: December 20, 2012
Due: January 10, 2013
Problem 1: The Helmholtz equation is often used to find the vibration
modes of cavities, including exhaust systems. In 1-D the governing differential equation for the problem is
d2 p
+ k2 p = 0
2
dx
with k 2 a constant, and p the pressure in the cavity. Suppose that p(0) = 0
and p(L) = 0. What is the weak form for the problem? (Hint: the weighting
function would be called p since the unknown is p.)
Problem 2: Consider the three-node one-dimensional conduction element
shown below. Unlike ordinary isoparametric elements, this one has node
three at r = − 12 . What would the shape functions (N1 , N2 , N3 ) be for this
element?
1
r=-1/2
2
r=-1
3
r=+1
(next page)
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Problem 3: Suppose that we have a rod with a square cross-section, 2L×2L.
It has internal heat generation g and conductivity k. Natural convection
occurs off of the long edges, with convection coefficient h and ambient temperature Ta . The ends of the rod are insulated.
convection coefficient h
2L
insulated
ends
2L
conductivity k
internal heat generation g
insulated
ends
ambient temperature Ta
(a) The length of the rod in the longest dimension does not matter in this
analysis, because every cross-section is exactly the same. By balancing the
internal heat generation per unit time and the convection losses from the
surface, we can estimate the surface temperature of the rod. (This is much
the same as the process you used in homework 3.) What is (roughly) the surface temperature of the rod? (Your answer will be in terms of L, g, h, and Ta .)
(b) Use ANSYS to model the full three-dimensional rod with convection off
of the sides. Print a picture of the mesh and also a picture which shows a
view of the cross-section with temperature contours. Make sure that you use
enough elements. For this step you will need to put in some actual numbers,
so let us use
L
g
h
k
Ta
0.5m
4 × 103 W/m3
4W/m2 C
4W/mC
20C
Show that your ANSYS solution agrees reasonably well with your solution
from part (a).
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