Polynomial to represent a linear rectangle element

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Discussion Overview

The discussion revolves around the mathematical representation of a linear rectangle element using polynomial equations, specifically focusing on the function u(x,y) and its relation to the rectangle's corner points. Participants explore the derivation of these equations and their geometric implications, as well as comparisons to triangular elements and lines in analytic geometry.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents the polynomial equation u(x,y) = c1 + c2x + c3y + c4xy as a representation of a surface defined over a rectangle, questioning its derivation and the meaning of the corner values u1 to u4.
  • Another participant clarifies that u(x,y) does not represent the rectangle itself but rather a surface that intersects the rectangle at its corners, suggesting that the derivation can be found in introductory finite element analysis texts.
  • A participant introduces a related expression for a triangular element, f(x,y) = a + bx + cy, and inquires about the mathematical branch relevant to its derivation, indicating an interest in geometry.
  • Discussion includes the basic analytic geometry interpretation of the equation a + bx + cy = 0 as representing a line in two dimensions and z = f(x,y) as representing a plane in three dimensions.
  • One participant questions whether the line equation y = mx + c' can be derived from the earlier equation, leading to a discussion about the rearrangement of the equation and its implications for vertical lines.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the equations and their geometric representations. There is no consensus on the derivation methods or the implications of the equations presented.

Contextual Notes

Some participants reference foundational texts in finite element analysis for derivations, but specific assumptions and mathematical steps remain unresolved. The discussion also highlights the limitations of representing certain lines in the standard form.

bugatti79
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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
 

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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.
 
SteamKing said:
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
 
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:
Mark44 said:
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
 
bugatti79 said:
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.
bugatti79 said:
If we re-arrange the above equation we get

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

thanks
 

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