Polynomial to represent a linear rectangle element

[bugatti79]

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 don't 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

 

[SteamKing]

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.

 

[bugatti79]

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.

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

 

[Mark44]

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.

 

[bugatti79]

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

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

 

[Mark44]

So can one determine the equation of a line $y=mx+c'$ from above equation?

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

If we re-arrange the above equation we get

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

thanks