The problem "The eigenvalues cannot be found..." was finally solved. I realized that the springs were not well defined, thus leading to several rigid body modes in the model. The problem was solved by defining the springs to resist translation in the 1, 2 and 3- directions using SPRING2 in Abaqus.

1) In the interaction Module of Abaqus, Click on Special, ...Spring/Dashpots... Create.

2)Choose Connect two points...Continue

3) Select the points to be connected...done

4) In the "Edit Springs/Dashpots" dialogue box, Choose Specify Fixed Direction as the Axis.

5) Enter the degree of freedom of the points. (1-1 for translation in the 1-direction)

6)Enter the spring stiffness

7) Ok.

The steps explained above would have successfully defined the spring with translation stiffness in the 1-direction.

To take care of translation in the 2 and 3 directions, repeat steps 1-7 for the same set of points, and adjust the degree of freedom to 2-2 and 3-3 for the 2 and 3 directions respectively (in step 5)

NB:1,2,3 directions as used here, coincide with the x,y,z directions of the global axis.

A big thank you to all of you who replied to my question. I appreciate all your responses.

Hello Stanly

check the constraints (degree of freedom of model or contacts between parts of the model) that you defined.

make sure that the models don't have sliding, rotation or any movement.

Regards

Hi Amirhossein,

Thank you for your swift response. A colleague and I have checked the model in line with your recommendation but haven't found out the issue yet. If you do not mind, I could send the files for you to help me crosscheck the model at your convenient time.

Hi again Stanley

I send my email address to you. I would be happy if I can help you

I do not know the problem in any depth to give a definitive answer to the question, but it seems that the software you are using calculates eigenvalues on the basis of some iterative procedure. The message it produces suggests that this procedure does not converge or has a very slow rate of convergence. If so, you will have to precondition the problem. One way of doing so is multiplying the matrix to be diagonalised by its incomplete Cholesky factorisation. Explicitly, suppose

(1) A v = λ v

is the eigenvalue problem to be solved iteratively. The convergence of the precess depends on the condition number of the matrix A. Let now B be an approximate inverse of A. With

(2) C = BA

and

(3) w = B v,

the eigenvalue problem in Eq. (1) can be expressed as

(4) C w = λ w.

Since C is approximately equal to the unit matrix, so that its condition number is close to 1, its iterative solution has a relatively fast rate of convergence (there are other subtleties involved here that I do not go into). Once λ and w have been determined, the eigenvector v is formally obtained from

(5) v = B-1 w.

Use of the incomplete Cholesky factorisation of A provides a B as a product of an upper- and a lower-triangular matrix. This enables one to by-pass the explicit calculation of B-1, as the vector v can be obtained from Eq. (3) by two processes of forward and backward substitutions (this is how the method of Gaussian elimination works). For the relevant details, consult any good text on numerical analysis, such as [1] and [2].

____

[1] WH Press et al., Numerical Recipes, 2nd edition (Cambridge University Press, 2001).

[2] GH Golub, and CF van Loan, Matrix Computations, 4th edition (The Johns Hopkins University Press, Baltimore, 2013).

Hi Behnam,

Thank you very much for your response. I am looking into it.

Stanley Iwuoha : My pleasure.