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Polynomial to represent a linear rectangle element

  1. Mar 25, 2013 #1
    Folks,

    I have attached a picture illustrating the labelling of the linear reactangle element which can be represented by the following equation

    ##u(x,y)=c_1+c_2 x +c_3 y +c_4 xy## (1)

    ##u_1=u(0,0)=c_1##
    ##u_2=u(a,0)=c_1+c_2a##
    ##u_3=u(a,b)=c_1+c_2a+c_3b+c_4ab##
    ##u_4=u(0,b)=c_1+c_3b##

    I dont really understand these equations. I mean, how is equation (1) derived to represent a rectangle and how is ##u_i## for ##i=1..4## derived?

    Thanks
     

    Attached Files:

  2. jcsd
  3. Mar 25, 2013 #2

    SteamKing

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    Staff Emeritus
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    The function u(x,y) does not represent the rectangle itself. u(x,y) represents a surface which is defined over the area bounded by the rectangle. The functional values u1 - u4 are the values of u(x,y) at the corner points of the rectangle. The surface produced by u(x,y) will be a plane passing through the points u1 - u4.

    The derivation of the element equations are given in most elementary intro to finite element analysis texts.
     
  4. Mar 26, 2013 #3
    OK. Taking a slight step back and looking at the triangular element case which can be desribed by the following expression

    ##f(x,y)=a+bx+cy##. What branch of mathematics are we looking at here, geometry? IM am interested to know how this simple equation was derived to represent a plane surface...

    Thanks
     
  5. Mar 26, 2013 #4

    Mark44

    Staff: Mentor

    The equation a + bx + cy = 0, which corresponds to f(x, y) = 0, represents a line in two dimensions (the x-y plane).

    The equation z = f(x, y) = a + bx + cy represents a plane in three dimensions. This is pretty basic analytic geometry.
     
    Last edited: Mar 26, 2013
  6. Mar 26, 2013 #5
    So can one determine the equation of a line ##y=mx+c'## from above equation? If we re-arrange the above equation we get

    ##y=-a/c -bx/c##....?

    thanks
     
  7. Mar 26, 2013 #6

    Mark44

    Staff: Mentor

    Usually, but not always. Equations that represent vertical lines can't be put in this form.
     
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