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A reliable design via FEA techniques is highly dependent on the execution of an analysis which is accurate and represents the design precisely. The accuracy and efficiency of the two available methods of meshing of the design components (mapped and free meshing) for finite element analysis are addressed in this paper. This paper intends to clarify that the “shape” of the elements rather than the “pattern” is the distinguishing factor for accuracy of the mesh-quality-results. This paper concludes by comparison of the FEA analysis to the analytical theory that element integrity and results are not governed by the elemental aspect ratio alone. The mesh density has a greater effect. Four types of mesh are investigated and compared for results' accuracy. In specific, a low-density free mesh, a high-density free mesh, a mapped mesh with high aspect ratio, and a mapped mesh with low aspect ratio of a plate with center hole are examined and compared. A plate with center hole is selected for this purpose since its far-field structural behavior is predictable and is highly applicable in showing the differences in accuracy among the different meshing methods.

The ambiguity surrounding the selection of the type of the mesh that would produce highly accurate results has been there for every stress analyst. In this paper, an effort has been made to clear up this ambiguity in a very practical manner via examples and relative comparisons. In general, there are two distinct types of meshing available:

The free meshing can usually produce high aspect ratio type of mesh, where the ratio of the long side of the elements to the short side is large. This type of element geometry leads to inaccuracy and nonconvergence of the solution [

Thus, a lot of emphasis is put on structured mesh generation in finite element analysis. Here, in this paper four different possible meshes consisting of structured and unstructured mesh are examined. The platform in which this investigation is done is a plate with a center-hole. In specific, a 20 inch by 20 inch in square plate with thickness of 0.2 inches is selected. The plate contains a center hole located symmetrically at the center with a radius of 2 inches. Further, an applied tensile loading of 1000 lbf is applied to the plate in the horizontal

A plate with center hole model is utilized since its stress behavior around the center hole is predictable. There are many experimental investigations performed [

One can derive an expression representing the stress field in the plate shown in Figure

Plate model with center-hole.

A quarter symmetric model of a thin plate with center hole is built with 4-node shell elements in ANSYS. Four distinct meshes are generated to elaborate the shape dependency of the FEA results. The first model is built with free-mesh of shell elements that are of fairly equal sides (LD free mesh). Figure

(a) The free mesh and (b) high-density free mesh using membrane shell elements.

(a) The high aspect ratio mapped mesh and (b) low aspect ratio mapped mesh using membrane shell elements.

Symmetric properties are used in order to model only one-fourth of the full plate model. Symmetric boundary conditions are used at the left and bottom sides of the plate models. For applied loading, a 50 lb/in uniform line pressure is applied to the right side of the model (depicted by red arrows in the meshed figure). This 50 lb/in line loading is equivalent to a quasi-static 1000 lbf tensile load applied to the plate thickness, in the right side. The top side of the plate model is free from restraints.

Table

The mesh-type index for all models.

Mesh type | HD mesh | LD free mesh | LAR map mesh | HAR map mesh |
---|---|---|---|---|

Number of plate elements | 482 | 395 | 60 | 640 |

Average size (inch^{2}) |
0.08 | 0.25 | 3 | 0.5 |

Percentage of plate elements with high aspect ratio | 0 | 0 | 40 | 100 |

Percentage of plate elements with low aspect ratio | 100 | 100 | 60 | 0 |

The “HD mesh” is the high-density free mesh, “LD free mesh” is the low density free mesh, “LAR map mesh” is the low aspect ratio mapped mesh, and “HAR map mesh” is the high aspect ratio mapped mesh. This comparison table materializes the mesh represented in Figures

Figures

Displacement contour for the unstructured free-mesh model.

Displacement contour for the structured high aspect ratio mapped-mesh model.

Displacement contour for the structured low aspect ratio mapped-mesh model.

Displacement contour for the unstructured high-density free-mesh model.

The stress contours representing the stress field of the plate model are plotted in Figures

Stress contour for the unstructured free-mesh model.

Stress contour for the structured high aspect ratio mapped-mesh model.

Stress contour for the structured low aspect ratio mapped-mesh model.

Stress contour for the unstructured high density free-mesh model.

The results from the finite element analysis of the previous four mesh models are compared with the theoretical Von-Mises stress levels derived by using the expression in (

Figure

Von-Mises stress levels around the plate hole radius of different mesh configurations.

In comparison to the low aspect ratio (LAR) “map-mesh” results, the low aspect ratio mesh over estimates the Von-Mises behavior of the plate slightly. However, for the low aspect ratio model, the final maximum Von-Mises stress magnitude is closer to the theoretical value with only a 1.72% difference. All three meshes together are close to the theoretical Von-Mises stress values except at 30 degrees angle location where the dip in the stress nullity is not very well determined. The high-density mesh (HD free mesh) closely follows the low-density free mesh and the high aspect ratio mapped mesh with the exception that it estimates the theoretical stress deep closer than the other three models. This mesh over estimates the theoretical final maximum stress level by 9.73%. Yet it proves to be a sufficient mesh since it closely predicts the stress behavior as seen by Figure

Interestingly, from all of the four mesh models, it can be concluded that the proper aspect ratioed mapped mesh of the elements alone does not guaranty accurate results. An adequate mesh density is required to predict the stress behavior of a structure. Also, it could be concluded that low aspect ratio meshes are a good tool for determining margins of safety, since they may closely predict the maximum stress levels.

Most finite element analysis packages in some way provide meshing tools that generate well-behaved (moderate aspect ratio) elements. Yet none of them provide for an automatic mesh density measure. The analyst has to select and control the mesh density of the region of the interest being analyzed. This has to be done either by preliminary FEA trial runs detecting the stress concentration regions or predetermination of the specified region by theoretical means. In either way, also the degree mesh density refinement has to be dealt with that only comes with FEA experience of similar structures.

To elaborate on the far-field accuracy of the most accurate mesh (the high-density mesh-HD mesh), the plot of the normal stress versus the vertical distance from the hole to the horizontal edge of the plate in

The normal stress versus the vertical distance to the top.

Four-node rectangular element.

To explain the theoretical reasoning for having less accuracy for high aspect ratio elements, it is essential to understand the computational method for the finite element modeling. One can begin by understanding the isoparametric formulation of elements [

The relation (

Now the strain in terms of the displacement is

whereas for a plane stress condition, the stress vector

Four different mesh types consisting of a combination of structured and unstructured mesh where generated in the plate with central hole. High aspect ratio and low aspect ratio elements were generated for these models. The Von-Mises stresses and the plate displacements were developed and compared to each other. The results were in comparison to theoretical solutions. It was concluded that the proper aspect ratioed mapped mesh of the elements alone does not guarantee accurate results. An adequate mesh density is required to predict the stress behavior of a structure. A higher mesh density in the stress concentration region produces a high performance finite element analysis. The correct elemental aspect ratio helps the solution convergence but the mesh density increases the actual accuracy of the analysis. Further work will be carried on in concentration on the elemental formulation in FEA and its effects on the solution accuracy of the structured and unstructured meshing and produced in another paper. The subject is a universal subject for all available FEA packages since they use similar algorithms.