How Is a Nonlinear Equation Linearized in FEM Software?

[aamirmub]

Hi,

I am trying to understand an example from a FEM software manual. The manual mentions a nonlinear equation http://aamir-pc:2080/v6.9/books/exa/graphics/exa_eqn00137.gif and this equation is linearized to obtain http://aamir-pc:2080/v6.9/books/exa/graphics/exa_eqn00152.gif .[/URL] Can anyone please explain how this has been done?

 

[mathman]

You need to fix your messages. The two equations don't show up.

 

[aamirmub]

The nonlinear equation is Y= G^(-1) * X + a * X^3 where G and a are constants. The linearized equation is Y(i+1) = (G^(-1) + a * X(i)^2) * X(i+1) where i and i+1 are superscripts.

 

[HallsofIvy]

The first thing done is factor out an "X": Y= (G^-1+ aX²)X. The next thing done was convert to a recursive form by treating the separate "X"s as if they were different terms in a sequence: Yⁱ⁺¹= (G^-1+ aXⁱ²)Xⁱ⁺¹. Given a starting value, X¹, you could then calculate a sequence of "Y"s. If that sequence convertes, then Y= \lim_{i\to\infty}Y^i will satisfy that equation: \lim_{i\to \infty} Y^i= (G^{-1}+ a(\lim_{i\to\infty}X^i)^2)(\lim_{i\to\infty}X^{i+1}) and, since "Xⁱ" and "Xⁱ⁺¹" refer to the same sequence they both converge to the same limit, X.

 

[aamirmub]

Thank you for your reply. Is the linearization carried out using the first two terms of the taylor series in incremental form?