Introduction:
In this post, we want to explain the negative eigenvalues error in Abaqus. In the Q&A part, we discussed this message briefly:
How to solve a negative eigenvalue error?
As mentioned, the basic reason for this warning message is stability. Generally, Abaqus warns such messages for the nonpositive definiteness of the system matrix. Mathematically, the appearance of a negative eigenvalue means that the system matrix is not positive definite.
As you know, in general, a finiteelement problem is written as:
F = K x
Where F, K, and x are the vectors of nodal load, stiffness matrix, and nodal displacement vector, respectively.
Then a positivedefinite system matrix K will be nonsingular and satisfy if:
For all nonzero x. Thus, when the system matrix is positive definite, any displacement the model experiences will produce positive strain energy.
Abaqus negative eigenvalues messages are generated during the solution process when the system matrix is being decomposed. Physically, negative eigenvalue messages are often associated with a loss of stiffness or solution uniqueness in the form of either material instability or the application of loading beyond a bifurcation point (possibly caused by a modeling error). During the iteration process, the stiffness matrix can then be assembled in a state which is far from equilibrium, which can cause the warnings to be issued.
Main reasons for negative eigenvalues errors:
In practice, the messages can be issued for a variety of reasons, some associated with the physics of the model and others associated with numerical issues. The Abaqus message itself shows some origins of this warning:
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In addition to the causes shown in the warning message, some situations in which negative eigenvalues messages can appear include:
1. Buckling analyses in which the prebuckling response is not stiff and linear elastic.
In this case, the negative eigenvalues often point to notreal modes. Remember that the formulation of the buckling problem is predicated on the response of the structure is stiff and linear elastic prior to buckling.
2. Unstable material response:
 A hyperelastic material becoming unstable at high values of strain.
 The onset of perfect plasticity.
 Cracking of concrete or other material failures that cause softening of the material response.
3. The use of anisotropic elasticity with shear moduli that are unrealistically very much lower than the direct moduli. In this case, illconditioning may occur, triggering negative eigenvalues during shearing deformation.
4. A nonpositivedefinite shell section stiffness is defined in a UGENS routine.
5. The use of a pretension node that is not controlled by using the *BOUNDARY option and lack of kinematic constraint of the components of the structure. In this case, the structure could fall apart due to the presence of rigid body modes. The warning messages that result may include one related to negative eigenvalues.
6. Some applications of hydrostatic fluid elements.
7. Rigid body modes due to errors in modeling (not enough BC, …)
Negative eigenvalue warnings will sometimes be accompanied by other warnings, addressing such things as excessive element distortion or magnitude of the current strain increment. In cases where the analysis will not converge, the resolution of the nonconvergence will often eliminate the negative eigenvalue warnings as well.
For analyses that do converge, carefully check the results if the warnings appear in converged iterations. A common cause of negative eigenvalue warnings is the assembly of the stiffness matrix about a nonequilibrium state. In these instances, the warnings will normally disappear with continued iteration, and, if there are no warnings in any iterations that have converged, warnings that appear in nonconverged iterations may safely be neglected. If the warnings appear in converged iterations, however, the solution must be checked to make sure it is physically realistic and acceptable. It may be the case that a solution satisfying the tolerance for convergence has been found for the model while it is in a nonequilibrium state. I hope you understand the negative eigenvalues (Abaqus negative eigenvalue) error origin in Abaqus and can solve your problem correctly. It would be useful to see Abaqus Documentation to understand how it would be hard to start an Abaqus simulation without any Abaqus tutorial.
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When does Abaqus give negative eigenvalues warning?
During the solution process, Abaqus checks to ensure the system is in equilibrium and stable. The basic reason for this warning message is stability. Mathematically, the appearance of a negative eigenvalue means that the system matrix is not positive definite. Therefore, if Abaqus gives this warning, it means the stiffness matrix has become unstable.
What causes to get negative eigenvalues errors?
In practice, the messages can be issued for a variety of reasons, some associated with the physics of the model and others associated with numerical issues. Physically, negative eigenvalue messages are often associated with a loss of stiffness or solution uniqueness in the form of either material instability or the application of loading beyond a bifurcation point (possibly caused by a modeling error).
What is the reason for negative eigenvalue warnings in convergent analyses?
For analyses that do converge, if the warnings appear in converged iterations, however, the solution must be checked to make sure it is physically realistic and acceptable. It may be the case that a solution satisfying the tolerance for convergence has been found for the model while it is in a nonequilibrium state.
What should be done to correct negative eigenvalues?
There are some recommendations to remove the instability warning. The user should reevaluate material models. Then, recheck to ensure realistic all boundary conditions and loading conditions. After that, find the areas that might be buckling or overly strained.
When does the instability of the solution occur due to the unstable behavior of the material?
The unstable material model response occurs in some situations, such as:
 A hyperelastic material becomes unstable at high values of strain.
 The onset of perfect plasticity.
 Cracking of concrete or other material failures that cause softening of the material response.
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