# Analysis Convergence ## 01 Jun 2021 Analysis Convergence

When referring to convergence in FEA, it could mean one of two things:
Mesh Convergence (https://www.fea-solutions.co.uk/mesh-convergence/)
Analysis Convergence

Analysis convergence is the calculation of the equilibrium of externally applied forces to internally developed forces in a non-linear analysis (https://www.fea-solutions.co.uk/non-linear-behaviour/).

In a non-linear analysis, loads are gradually applied in increments (substeps) rather than in one step. At each increment, the software attempts to solve the analysis in iterations to get to a solution. For a solution to be reached, the structure must be in equilibrium. To see if a solution is in equilibrium, convergence checks are performed by the software using results called residuals. These residuals exist for the following parameters:
Forces
Displacements
Moments

The force residual is the applied force minus the internally developed force, and similarly for the other residuals. These residuals must be below a given tolerance limit for the solution to converge.

Once an iteration has converged and an equilibrium is reached, this increment is stopped, the load is increased and the next increment begins. Once this has been done for all increments in the analysis, the solution has fully converged.

If an iteration has not converged, the solution will bi-sect. This will reduce the size of the load increment, giving a greater chance of convergence. If the solution fails to find an equilibrium when it has reached a pre-set minimum increment size, the solution fails to converge and stops. One of the many reasons for such a non-convergence could be that the physical limits of the structure has been reached, e.g. the ultimate strength of the material.

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