Confusion with categorising solvers

[jd90]

Hello everyone,

I have been recently getting stuck into FEA for my work, and I am having some real difficulty classifying or categorizing the breakdown of solvers and what fits in where.
I'm hoping that someone can clarify a few things or set me straight.

So for all of our problems our choices are either an implicit or explicit analysis. Then the decision is focusing around what type of steps we will use for the model, for simplicity I am going to stick to general static and dynamic.

General Static is when the problem at hand is not dependent on time, the geometric changes aren't too drastic and inertial, damping, and other dynamic effects aren't taken into account.
The general equation we are aiming to solve is P=Ku by a direct solver inverting our stiffness matrix, Gaussian elimination is used for this when the system is linear, when non-linearity occurs does the solver switch to a Newton method?



On the other hand dynamic analysis (implicit) is solving the equation Ma+Cv+Ku=F, hence we now have mass and damping matrices to deal with. In these cases Newtons method is used rather than the Gaussian elimination as to cope with the non-linear terms that arise?

I have read that the reason Newtons method is used is to ensure the method is converging (unlike in explicit) but it must therefore be used in conjunction with something, is that correct?




I do apologize if this is all over the place but I am trying to make sense of it all.
Any help would be greatly appreciated.

J

 

[DrBwts]

The explicit solver is better for dealing with dynamic systems NOT the implicit solver.

Also I think you are confusing what Gaussian elimination & Newton's method do.

Gaussian elimination is used to solve any matrix system whether you are using explicit or implicit.

Newton's method is an iterative approach to solving non-linear equations & during iterations will use Gaussian elimination to solve systems of equations. This is used in the implicit solver.

You are more likely to get convergence problems with implicit solvers due to Newton's method when dealing with contacts, high speed dynamics, buckling etc, for further info re. Abaqus explicit v implicit

Hope this helps.

 

[jd90]

Thanks for your response the information is very helpful. Just to ask, when the solver is working through an implicit dynamic non linear problem are you saying it is using gaussian elimination? I thought that could only be used for a set of linear equations

 

[DrBwts]

Newton's method makes use of linear approximations of the systems of equations which, are stored in matrices, which are then in turn solved using Gaussian elimination.

Newton's method

 

[jd90]

DrBwts your comments are very insightful but I am unfortunately left confused, I am new to this software and these solver methods so please bear with me.

So the system of non-linear equations in my dynamic implicit analysis are being approximated by Newtons Method then the outcome from this is then being stored in matrices and solved via Gaussian elimination? I have exclusively thought of the Gaussian elimination method as a direct solution and Newtons method as an iterative solution, but it seems here there is a combination of Gaussian elimination and Newtons method, but this is still an iterative solution.

 

[DrBwts]

So the system of non-linear equations in my dynamic implicit analysis are being approximated by Newtons Method then the outcome from this is then being stored in matrices and solved via Gaussian elimination?

Newtons's method makes successive approximations (i.e iterations) to a solution of a system of equations until an acceptable solution is found.
Something like this happens on each iteration,
1) a linear approximation to the system of equations is formulated, lets call it K.
2) a solution to K is calculated using Gaussian elimination, as it is system of linear equations what better method?
3) a residual is then calculated
4) if this residual is within a given tolerance then Newtons method is exited, if not then go back to (1)

Newtons Method is the overall scheme for solving the equations (I have left a lot of the detail about initial guesses & the residual), Gaussian elimination is used in one of the steps.