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Linearization procedure in nonlinear FEM

phunguyen's picture


As far as I know, there are two ways to derive the linearized equations (employed in the Newton-Raphson method) in nonlinear finite element methods. 

In the first approach, starting with the weak formulation, the FE approximations are substituted into the weak form, you then get the semi-discrete equations. After that, the linearization is applied on this discrete equation to get the tangent matrix. The book of Prof. Ted Belytschko followed this approach.

The second approach is to use linearized weak forms. So, you have a weak form, you do the linearization using Gateaux derivative (directional derivative) to obtain the so-called linearized weak form. Then, FE approximations are introduced into this liearized weak form, and the linearized discrete equations are obtained, together with a linear solver give you the increments of the state variables (displacement field in solid mechancis, for example).

 Personnaly, the first approach is easy and when performed in indicial notation, it is quite straightforward, at least for single field weak form, as described in Ted's book. So, I used this way to get the linearized equations for multi-field weak form, say three-field Hu-Washizu, and I was not successful. Doing the literature, examples are the work of Prof. Simo, the second approach was used intensively.

I am aware that the fact that I was not able to do this by using the first linearization approach is only due to my limited algebra ability.  

So, could any one of our experts share the experience, the pro and cons of the two approaches, how to master the second way. In my opinion, the second approach is more general than the first one and stands on itself in the theory of continuum mechanics regardless of finite elements. 

Many thanks.


jandrade's picture

Dear Vinh Puh,

Both procedures yield the same answer, so this is a quick way to check. As you noted, linearizing from the weak form may not be straight forward, but seems pretty general. We used this approach in a two-field problem. Here is the paper with the linearization procedure (sorry, don't know how to attach paper):





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