The finite element solution is an approximation to the true solution of a mathematical problem. From an analyst's standpoint, it is important to know the magnitude of error involved in the solution. The Mechanical APDL program offers a method for a posteriori estimation of the solution error due to mesh discretization. The method involves calculating the energy error within each finite element and expressing this error in terms of a global error energy norm.
The error energy within each finite element is calculated as
ei = 1/2 v {Δσ}T [D]-1 {Δσ}
dV |
where:
| ei = error energy in element i |
| {Δσ} = nodal stress error vector |
| [D] = stress-strain matrix |
The nodal stress error vector {Δσ} is the averaged nodal stresses minus the unaveraged nodal stresses.
By summing all element error energies e, the global energy error in the model, e, can be determined. This can be normalized against the total energy (u + e), where u is the strain energy, and expressed as a percent error in energy norm, E:
The percent error in energy norm E is a good overall global estimate of the discretization or mesh accuracy. Several VMD and VMC tests use this error norm to illustrate its behavior as a function of known displacement or stress error. It should be recognized that the correlation of the error energy norm to displacements or stress error is problem-dependent, and therefore this norm should only be viewed as a relative measure of accuracy.