Machine Design

Converging on an accurate solution

Most FEA students quickly learn that finer meshes usually produce more accurate solutions.

A precision contour display indicates the effects the finite-element mesh has on accuracy for the square-plate problem. The highest values (in red) occur where the change in von Mises stress results is greatest between adjacent elements. These locations might require mesh refinement to obtain a more-continuous change in stress.

A steel plate (4 X 4 X 0.1 in.) with fixed boundary conditions on all sides carries a uniform pressure load of 100-psi normal to the element faces. A mesh convergence study is performed using an n X n mesh where n = 2, 4, 8, 16, and 32 plate elements.

Maximum displacement plotted against mesh density shows how density influences displacement. It converges as the mesh density increases.

A plot of maximum von Mises stress versus mesh density shows changes in stress results. Plotting stresses from the center of the plate shows they converge on a solution (about 22 ksi) as the density increases. (Maximum von Mises stresses are in the table.) Although stress results are a frequent goal, the same method can be used to perform mesh convergence studies for displacements, temperatures, and pressures.

Although the software typically cannot obtain exact solutions for models, it can find highly accurate approximations. Students also quickly learn that greater accuracy comes at the cost of longer compute times. The tradeoff begs the question: When is a mesh fine enough to accurately model a real-world event? The answer comes from a meshconvergence study.

"It's an empirical process that compares results of one meshed model to those derived with a denser mesh on the same model," says Bob Williams, product manager at Algor Inc., Pittsburgh ( "The best way to start is with the fewest, yet reasonable number of elements and run the simulation. Then remesh with a denser mesh, run the simulation again, compare results, and see if they are similar. If not, then the coarse mesh is not accurate. Increase the density and reanalyze the model."

Continue to increase the density globally, or use automatic and point-and-click mesh refinement options, and reanalyze the model until results satisfactorily converge. "Results eventually reach a point at which a finer mesh no longer yields an appreciable difference. This type of convergence study helps generate accurate solutions with meshes that are sufficiently dense and yet not overly demanding of computer resources," says Williams.


More than one way to study convergence

Algor's Bob Williams says any of the following methods will determine when results have converged satisfactorily and accurately:

Display precision contours. This is a color-coded graphical display of changes in results from one element to the next. This contour determines the mesh's effect on accuracy and supplies guidance for locations needing a finer mesh. A fine mesh on high-stress locations, rather than the whole model, trims analysis time.

Display unsmoothed result contours to see stepped changes between results for adjacent elements.

Display the model's residual forces and check reactions at supports to make sure they balance or at least meet expectations based on engineering judgment.

Inspect result values at the same location (the center for example) after each iteration.

Algor Inc., (412) 967-2700,

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