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# How to solve negative eigenvalue error?

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## What is the Abaqus Negative eigenvalues warning?

In Abaqus, the presence of negative eigenvalues can be a warning sign indicating potential issues in the analysis. Eigenvalues are used to determine the natural frequencies and modes of vibration of a structure. In a linear eigenvalue analysis, the system’s stiffness matrix is solved to obtain eigenvalues and corresponding eigenvectors.

Negative eigenvalues indicate that the system’s stiffness matrix is not positive definite, which means it has at least one negative eigenvalue. This can occur due to a few reasons:

1. Modeling errors: Negative eigenvalues can be a result of errors in the modeling process, such as incorrect boundary conditions, constraints, or material properties. These errors can lead to an unrealistic or unstable representation of the structure, causing negative eigenvalues to emerge.
2. Inadequate mesh: An inadequate mesh can lead to inaccurate results, including negative eigenvalues. Insufficient or distorted elements may fail to capture the behavior of the structure accurately, causing the computed stiffness matrix to be unstable and produce negative eigenvalues.
3. Structural instability: Negative eigenvalues can also indicate structural instability or buckling. Buckling occurs when a structure becomes unstable under compressive loads, causing it to suddenly deform or collapse. In such cases, negative eigenvalues can be an indication that the structure is in an unstable state.
4. If you encounter negative eigenvalue warnings in Abaqus, it is important to investigate the issue further. You can start by examining your model setup, including boundary conditions, material properties, and mesh quality. Ensure that your model accurately represents the physical behavior of the structure. Additionally, consider performing a nonlinear analysis to capture any potential instability or contact behavior that might be causing the negative eigenvalues.

## Some common modeling errors that can lead to negative eigenvalues

Several common modeling errors can lead to negative eigenvalues in a finite element analysis. Here are some of the typical mistakes to watch out for:

1. Incorrect boundary conditions: Applying incorrect or inconsistent boundary conditions can lead to negative eigenvalues. For example, fixing degrees of freedom that should be allowed to move or constraining degrees of freedom that should be free can result in an unrealistic representation of the structure’s behavior.
2. Material property errors: Incorrectly defining material properties, such as an incorrect Young’s modulus or Poisson’s ratio, can affect the stiffness matrix and lead to negative eigenvalues. Make sure to use accurate and appropriate material properties for your analysis.
3. Inadequate constraints: Incomplete or incorrect constraint definitions can cause negative eigenvalues. Ensure that you appropriately constrain the degrees of freedom that should be fixed, while leaving the necessary degrees of freedom free to deform.
4. Incorrect element types: Using inappropriate or incompatible element types for certain structural features can introduce errors. Ensure that you choose element types that are suitable for the specific behavior and geometry of your structure.
5. Mesh quality issues: Poor mesh quality, such as distorted elements, element aspect ratio problems, or inadequate element density, can lead to inaccurate results and negative eigenvalues. Ensure that your mesh is well-designed and captures the structural behavior effectively.
6. Modeling geometric nonlinearity as linear: If your structure exhibits significant geometric nonlinearity (large deformations or rotations), modeling it as linear can lead to unrealistic results, including negative eigenvalues. In such cases, consider using a nonlinear analysis approach to accurately capture the behavior of the structure.
7. Neglecting contact or interface behavior: If your structure involves contact or interface interactions, neglecting or improperly modeling these behaviors can lead to negative eigenvalues. Ensure that you appropriately define and model contact conditions between different parts or surfaces.

It’s crucial to carefully review your model setup, verify the boundary conditions, material properties, and element types, and ensure that the mesh is of good quality. Conducting validation studies and comparing your results with experimental data or analytical solutions can also help identify and rectify modeling errors that may lead to negative eigenvalues.

## How to deal with negative eigenvalues?

Developing a consistent practice of reviewing the message file for negative eigenvalues holds significant importance. In cases where negative eigenvalue warnings surface during converged iterations, it becomes crucial to scrutinize the solution thoroughly to ensure its accuracy.

To rectify negative eigenvalues, it is advisable to reassess the material models employed and verify the realism of the boundary conditions and loading conditions. When analyzing the outcomes of a model exhibiting negative eigenvalues, it is essential to focus on identifying regions that might be susceptible to buckling or excessive strain. Additionally, reevaluating the interplay between the loading and boundary conditions in those specific areas is recommended.

This answer is just a summary of the article:
Warning Messages Related to Negative Eigenvalues

First of all, you must know that it is just a warning and not an error. If the error and warning concepts in Abaqus is not clear for you, take a look at
What is the difference of error and warning in Abaqus?

The negative eigenvalue in context of buckling basically just means that the load to cause the buckle is of the opposite sign to the applied load. So if I applied a reference load of 10 MPa and the first eigenvalue was -0.5, then the load required to trigger the buckling mode is -5 MPa.

In other analyses, this message is related to instabilities in the model. Negative eigenvalues may be produced because of the elements, which goes under distorting conditions. Some of the reasons are:

* Material properties have not defined properly.

** Geometrical properties of the meshes are not applicable or not enough for converging the solutions.

*** Boundary conditions have not restrained enough, so the model goes to act as a mechanism. You must check your boundary conditions. Improper BC’s may be the reason of the instability in the model. Due to this instability, you can have this issue related to negative eigenvalues.

## Keep in mind:

1) Negative eigenvalues warnings that pop up during iterations that do not converge can generally be ignored.

2) If negative eigenvalues warnings appear during iterations that converge then the computed solution must be carefully evaluated.