{ 3D_OILDRUM.PDE }
{ *******************************************************************
This example illustrates the use of FlexPDE in Eigenvalue problems, or
Modal Analysis.
*******************************************************************
In this problem, we determine the four lowest-energy vibrational modes of
a circular cylinder, or "oil drum", clamped on the periphery.
What we see as results are the pressure distributions of the air inside the
drum.
The three-dimensional initial-boundary value problem associated with the
scalar wave equation for sound speed "c" can be written as
c^2*del2(u) - dtt(u) = 0,
with accompanying initial values and boundary conditions:
u = f(s,t) on some part S1 of the boundary
dn(u) + a*u = g(s,t) on the remainder S2 of the boundary.
If we assume that solutions have the form
u(x,y,z,t) = exp(i*w*t)*v(x,y,z)
(where "w" is a frequency) then the equation becomes
del2(v) + lambda*v = 0
with lambda = (w/c)^2, and with boundary conditions
v = 0 on S1
dn(v) + a*v = 0 on S2.
The values of lambda for which this system has a non-trivial solution
are known as the eigenvalues of the system, and the corresponding solutions
are known as the eigenfunctions or vibration modes of the system.
}
title "Vibrational modes of an Oil Drum"
coordinates cartesian3
select
modes=4 { Define the number of vibrational modes desired.
The appearance of this selector tells FlexPDE
to perform an eigenvalue calculation, and to
define the name LAMBDA to represent the eigenvalues }
ngrid=6 { reduced mesh density for demo }
nodelimit = 3000 { keep problem small for demo }
Variables
u
equations { the eigenvalue equation }
div(grad(u)) + lambda*u = 0
extrusion z = -1,1
boundaries
surface 1 value(u) = 0 { clamp the bottom and top faces }
surface 2 value(u) = 0
Region 1
start(0,-1)
value(u) = 0 { clamp the sides }
arc(center=0,0) angle 360
monitors { repeated for all modes }
contour(u) on x=0
contour(u) on y=0
contour(u) on z=1/2
plots { repeated for all modes }
contour(u) on x=0 surface(u) on x=0
contour(u) on y=0 surface(u) on y=0
contour(u) on z=1/2 surface(u) on z=1/2
end