System of Equations Through Mortar Method

[nima_ikm]

Dear Professor Taylor,

Sorry to take your valuable time.

First of all, thank you for your kind correction of my input file about tied interface in my last post. 

Based on the following system of equations or in your book ' The Finite Element Method: Its Basis and Fundamentals, 7th edition' in Eq. 11.17 - page 367 .

***                              ***        ***   ***       ***     ***
*  K1        0          Q1    *        *  u1   *       *    f1   *
*                                   *        *         *       *          *
*  0          K2        Q2    *  x    *  u2   *  =   *    f2   *
*                                   *        *         *       *          *
*  Q1'      Q2'        0      *        *   l     *       *    0    *
***                              ***       ***   ***       ***     *** 

I have the following questions:(Already, I received correct results from Feap but I would like to know how Feap is doing so)

(a) How FEAP solve that system of equations? Is it based on the paper of Prof. Farhat for FETI method which means at first it solves for lambda then by substituting lambda in first and second equation, finding the first and second solution.

(b) Actually, I am using staggered approach in my input file (based on partition command). How FEAP solving mortar method for multi-filed problem, for example mechanical part and damage? Is FEAP solving that system of equation once for mechanical part including coupling part and then damage part also including coupling part (exactly above system of equation once for one field and next for another filed).

(c) If K2 is singular, how FEAP deal with rigid body mode? (is it based on paper of Prof. Farhat for FETI method)

I am really appreciated if I can receive your valuable comment for these questions. Once again, I am so grateful for your kind help.

Best regards

[Prof. R.L. Taylor]

In general feap requires that no singularity exist in the matrices.  In your example, if K2 is singular you should get warnings from the solver that a large number of significant digits were lost during the initial solution (provided you are using the standard profile solver).  However, once you have contact you may have a situation where the total matrix is not singular and a solution is possible.  In general we try to interlace the equations so that the dof's of K1 and K2 occur before the lambda's; then if the terms in Q1 and Q2 are present the solution can proceed without loss of digits. 

There is no guarantee that a failure might occur if any of your blocks have possible rigid body modes and not inertia terms are present.