1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Green's function

  1. May 27, 2004 #1

    I'm trying to understand an article about thin plate splines. My understanding so far:

    Suppose you have a infinite thin metal plate and set of M points {(x_1,y_1),...,(x_M, y_M)}
    At each point in the set you want to constrain the metal plate to a certain height above or below the plane. So suppose f(x,y) represents the metal plate, then f(x_1,y_1) = z_1, ... ,
    f(x_M, y_M) = z_M. Now you want to know f(x,y) for all other points (x,y) in the domain.

    f(x,y) should be a solution to the biharmonic equation D^2 f = 0, because this equation governs the shape of a thin plate under an out-of-plane load. (D^2 is the biharmonic operator). Now in the article it is stated that U(r) = r^2 log r^2 is the fundamental solution (or Green's function) of D^2. ( r = sqrt( x^2 + y^2 ) )

    It turns out that f(x,y) can be expressed as a linear combination of M copies of U(r) centered on each of the M points from the set.

    so: f(x,y) = w1 * U( sqrt( (x - x_1)^2 + (y - y_1)^2 ) ) + ... + wM * U( sqrt( (x - x_M)^2 + (y - y_M)^2 ) )

    the weights w1 .. wM can be determined by solving a system of equations by constraining f(x_1,y_1) to z_1 ... f(x_M, y_M) to z_M.

    I understand that a linear combination of U(r)'s is still a solution to D^2 f = 0. What is don't get is how do you proceed from the Green's function to a solution of the biharmonic equation with constriants as described above? It's hard to find accessible explanations on the internet, so I hope someone could point me in the right direction. :smile:

  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted

Similar Discussions: Green's function
  1. Green's function (Replies: 1)