Eigenvalue Solutions in the Instability Problems

[firat_dal]

Dear Prof. Taylor, Prof. Govindjee, and FEAP community,

When I read an article entitled "Size effects in nonlinear periodic materials exhibiting reversible pattern transformations" written by Ameen et al. (2018, published in Mechanics of Materials), I tried to reproduce the results given in the article using FEAP v8.5.

This is a 2-D plane strain problem exhibiting instability-induced pattern transformation. It is a heterogeneous medium with a rectangular domain and periodically located circular holes. The file (the nodal coordinates and element connectivity information are stored in the file 4761quad.str) and the input card (mesh.ameen and Irun) are at the attachments.

The nodes on the bottom and top edges are constrained in both directions. The nodes on the left and right edges are LINKed to impose periodic boundary conditions. Hyperelastic (Modified Mooney-Rivlin) model is employed (default FEAP model). I just apply the vertical (compressive) displacement on the top edge. When I solved the problem with the given boundary conditions, I got the same mechanical (nominal stress-strain) response and the same deformed geometry presented in the article. However, when I checked the (minimum) eigenvalue obtained by (IDEN+SUBS,,n commands), the eigenvalues decreased with the deformation (which is expected) until the critical (in-)stability point which is the point that pattern transformation begins. Once it reaches the critical stability point, eigenvalues increase instead of decreasing to negative values. I expected that if instability is the case, the stiffness matrix of the system loses its positive definiteness. Then, I try to solve the 2-D buckling problem (The input card is given as Ibvp). I have prepared a column whose bottom edge is fixed in each direction, while the top edge is restrained in the x-direction. The compressive force is applied eccentrically since a symmetric mesh has been prepared. The same constitutive model is employed with the same material parameters (Modified Mooney-Rivlin model). Using IDEN and SUBS,,n commands, I obtained the minimum eigenvalue of the system. The same thing happens in this case, too. Once minimum eigenvalues have been examined, a local minimum is observed at the beginning of the buckling. I do not know why we have such results.
Does it have to be transforming from positive to negative?
If so, why does FEAP give only positive values in those cases?
If so, is it possible to get negative eigenvalues and corresponding eigenmodes?
If so, where should I look for and change to adopt negative eigenvalues and corresponding eigenmode solutions?
If not, what is the reason? I try to understand what I missed up to now.

Thank you very much in advance for your help and advice.
Sincerely.

Firat

[Prof. S. Govindjee]

Have a look at this example:  http://feap.berkeley.edu/wiki/index.php?title=Branch_Switch

You if you toss in some extra IDENtity/SUBSpace pairs you will see the behavior of the eigenvalues (just make sure that you only look at the eigenvalues
for converged states). 

In short when you have converged onto an unstable branch you will see negative eigenvalues.  But when you are converged on a stable branch you will see positive values.

[firat_dal]

Dear Prof. Govindjee and FEAP Community,

Thank you very much for your prompt and detailed reply. I sincerely apologize for my delayed response to this post.

From your explanation, I have learned that the branch-switch feature enables us to perturb the system in the direction of the most critical eigenvector. To achieve this, the command arcl,add,1,tau (where tau is the scaling factor) is used. I understand that the subroutine in program/pmacr3/paddv is invoked to add the scaled eigenvector to the displacement vector when arcl,add,1 is specified in the input file. This arc-length solution procedure effectively addresses instability problems, such as buckling in column-type structures.

I have tested the input card and successfully obtained negative eigenvalues for a buckling problem. However, when I applied this solution procedure to an instability-induced pattern transformation problem, I was unable to observe negative eigenvalues during the pattern transformation process. I suspect this might be due to the choice of scaling factor and plan to revisit and review the associated code for further investigation.

If I may, I would like to ask a follow-up question related to the same topic. Using the computational homogenization method, microscopic stability points can be detected by examining the macroscopic stress-strain response. By applying a macroscopic displacement gradient with the PERIodic,CAUChy input and employing the Hill-Mandel averaging approach via the HILL,TANG command, macroscopic stress values can be obtained. When plotting the macroscopic stress-strain curve, I observe a sharp change in macroscopic stiffness, indicating the onset of pattern transformation (i.e., the critical stability point).

However, if I attempt to solve the same problem using the computational homogenization method, I encounter an issue: at the critical stability point, I am unable to achieve negative eigenvalues. Based on my understanding of the literature, the minimum eigenvalue typically begins at a positive value and decreases with deformation. Upon reaching the critical instability point, the minimum eigenvalue should change its sign. Yet, in my simulations—whether using standard displacement-controlled analysis or computational homogenization analysis—the minimum eigenvalue exhibits a local minimum at the critical instability point without changing its sign.

Could you please indicate what adjustments or approaches I should consider to observe negative eigenvalues at this point? Alternatively, could you kindly help me understand why FEAP exhibits this behavior? When I apply the procedures outlined in the literature using FEAP, I can successfully observe values such as force, (macroscopic) stress, strain, and the evolving patterns after transformation. However, my inability to capture negative eigenvalues leads me to suspect that FEAP might be applying some kind of factorization/correction in the global stiffness matrix.

I would greatly appreciate your suggestions or comments to clarify this situation.

Thank you very much for your guidance and recommendations.

Sincerely,
Firat

[Prof. S. Govindjee]

To observe a negative eigenvalues, you will need to converge onto an unstable equilibrium solution.  It sounds like your system is always jumping onto the stable bifurcated solution.  Perhaps the geometry or loading is not perfect?  You could perhaps decrease your load increments and see the minimum eigenvalue get closer to zero and maybe you will get 'lucky' and also converge on an unstable equilibrium branch.

I would, however, encourage you to think deeply about why you even want to see negative eigenvalues.  Such solution branches are unstable and are unlikely to be observed in reality, so why chase them?

[firat_dal]

Dear Prof. Govindjee,

Thank you once again for your detailed and thoughtful response.

I have re-run the analyses as per your suggestions, but unfortunately, the results remain unchanged.

If I may, I would like to explain the motivation behind my focus on identifying negative eigenvalues in my computations.

In the article titled "Size effects in nonlinear periodic materials exhibiting reversible pattern transformations" by Ameen et al. (2018, Mechanics of Materials), the authors state:

   "Under compression, the microstructure exhibits local instabilities initiated by internal buckling. These instabilities trigger the pattern transformation, upon which the material behavior changes from an approximately linear initial response to a complex post-buckling response, depending on the geometry and type of boundary conditions applied. The instability is detected using pivot checking, see e.g. Wriggers (2008). When a negative pivot is encountered, the Newton solver is initialized towards the lowest mode by adding a linear perturbation to the current solution."

Following this, I referred to Peter Wriggers's book Nonlinear Finite Element Methods. Chapter 7 (Stability Problems) explicitly mentions that the first negative diagonal entry (often corresponding to a negative value of det KT​) signifies the onset of instability.

For a problem with the same boundary conditions (though employing a different element type—the authors used a quadratic triangular element, T2P1, while I employed a bilinear quadrilateral element, Q1P0), I anticipated encountering negative eigenvalue(s) or pivot(s) during the pattern transformation. However, despite my efforts, I only observe positive eigenvalues.

Given the explicit reference in these works that microscopic instability corresponds to the moment when the minimum eigenvalue of the global stiffness matrix changes sign from positive to negative, I have been trying to verify this behavior in my analysis. I aim to understand this phenomenon better.

I sincerely appreciate your guidance and recommendations.

Thank you for your time and consideration.

Sincerely,
Fırat

[Prof. S. Govindjee]

I tried to run the problem from the paper.  I am getting their deformation patterns but I am not seeing any bifurcations.  FEAP is always finding the stable branch of the solution.   I agree with the description given in the paper and would have expected to see bifurcations, but that seems not to be the case.

Attached are the contours of the horizontal displacement at a nominal strain of 0.1.

[Prof. S. Govindjee]

I have to take back my statement.  I had an error in my mesh which perturbed it.  If I use an unperturbed mesh then I see the flip of the eigenvalue and the solution remains on the unstable branch.  This occurs for a strain level of about 0.047, a bit larger than what is in the paper but I have a modestly coarse mesh.

Attached are the input files (Igeers is the main file, Igeers2 is a helper file) and a plot of the solution on the unstable branch at a strain of 0.1.
If you want to branch switch, you should use the ARCLength method that I pointed to in the FEAPWiki example

[firat_dal]

Dear Prof. Govindjee,

Thank you once again for your patience and for addressing my numerous questions. I apologize if I have imposed on you too much.

Sincerely,
Firat