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{ SURFACE_FIT.PDE

 

 This problem illustrates the use of FlexPDE in a data fitting

 application.

 

 THE NUMERICAL SOLUTION OF THE BIHARMONIC EQUATION WITH A DISCONTINUOUS

 LINEAR SOURCE TERM USING FlexPDE.

 

 STATEMENT OF THE PROBLEM:

 

 Find the solution U of the fourth order elliptic PDE

 

       (dxx + dyy)(dxx + dyy) (U) = -beta*(U - C)      in O,       (1)

 

 where in the usual FlexPDE notation, dxx indicates 2nd partial derivative

 with respect to x, and where O is a given connected domain.  Equation (1)

 arises from the minimization of the strain energy function of a thin plate

 which is constrained to nearly pass thru a given set of discrete set of

 points specified by C and beta.    Namely,  a given  set of n data values

 [C(i)] is assigned at locations [(x(i), y(i))], i=1,..n ,

 and the factor beta has its support only at the locations (x(i), y(i)).

 

 Along with equation (1), we must prescribe a set of boundary conditions

 involving U and its derivatives which must be satisfied everywhere on the

 domain boundary.

 

}

title " The Biharmonic Equation in Surface Fitting Designs and Visualization"

 

variables

   U

   V

definitions

   eps = .001

   beta0 = 1.e7

   beta = 0.0

   a = 1/sqrt(2.)

   two = 2.5

   b = two*a

   xbox = array (0, 1, -1, 0,  0, a, -a, a, -a,  two, -two, 0, 0, b, -b, b, -b )

   ybox = array (0, 0,  0, 1, -1, a, -a, -a, a,  0, 0, two, -two, b, -b, -b, b )

 

 

monitors

  contour(U)

  contour(C)

  contour(C-U)   as "Error C - U"

plots

  contour (U) as "Potential"

  surface(U) as "Potential"

  surface(C)   as " Expected Surface"

  contour(beta)

  surface(beta)

  surface(U-C)

end