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{ 3D_OILDRUM.PDE  

 

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   This example illustrates the use of FlexPDE in Eigenvalue problems, or

   Modal Analysis.

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   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 }  

    cell_limit = 3000   { keep problem small for demo }