Watch out — You can get different results depending on how you mesh the model.
Paul M. Kurowski
Design Generator Inc.
London, Ontario, Canada
pkurowski@rogers.com





Secondorder shell elements: Meshing the thin band with secondordershell elements produces precise curvilinear geometry. 



There was a time when the task of building a model for finiteelement analysis (FEA) was long and tedious. Those days are over. Modern CAD and FEA programs work together so closely that CAD geometry becomes a starting point for FEA. But what really happens after CAD geometry goes into an FEA program for meshing? And how does meshing affect FEA results? The answers call for examining the properties of finite elements and finiteelement meshes and seeing how they alter results. It's also useful to review how fundamental assumptions made in element formulation affect global behavior of the analyzed model.
WHAT GETS MESHED?
Geometry from CAD used for analysis must be meshable into correct and reasonably small finite elements. But the process of defining these elements often necessitates making changes to the CAD geometry. Typical modifications include defeaturing (removing unnecessary details), idealization (for example, representing thin wall with a surface), and geometry cleanup.
Having prepared meshable, but not yet meshed geometry, we define material properties, loads, supports and restraints, and select a type of analysis, such as structural, thermal, or some other. This procedure creates a mathematical model on which an FEA solver can work.
The mathematical model now must be split into finite elements through a process of discretization, more commonly called meshing. However, loads and supports are also discretized by applying them to nodes of the finiteelement mesh.
Continuous CAD geometry and discrete FEA geometry are only superficially similar. While CAD geometry has solids, surfaces, faces, lines, and vertexes, no such entities exist in FEA geometry. No part of CAD geometry becomes FEA geometry. FEA geometry only has nodes. Mathematical formulations define element behavior and element connectivity to nodes. Displaying a mesh as lines connecting nodes helps us visualize element locations and it makes FEA geometry look almost like a CAD model which adds to the confusion in distinguishing between the two.
What elements does meshing make? The meshing process splits continuous geometry into finite elements. The type of elements created in this process depends on the type of geometry meshed, the type of analysis, and sometimes on the preferences of theperson doing the modeling. For brevity, we'll discuss the two mostfrequently used elements: solids and shells.
It's important to clarify terminology here. For instance, "solid" has different meanings when referring to "solid geometry" and a "solid element." In a nutshell, solid CAD geometry is an FEA volume, and solid elements are created by meshing those volumes.
Automatic meshing of solid CAD geometry takes place most often with tetrahedral solid elements, or just tets. This is because present day automeshers can't reliably mesh complex geometries with anything but tets. Tetrahedral solid elements can either be first or secondorder elements. The user decides which to use. However, we will show that only secondorder elements should be used for an analysis of any importance.
Firstorder tetrahedral elements have four nodes, straight edges, and flat faces. Those edges and faces remain straight and flat after the element has experienced deformation under the applied load. Firstorder elements model the lineardisplacement field inside their volume, on faces, and along edges. The linear or firstorder displacement field gives these elements their name. Recall from mechanics of materials that strain is the first derivative of displacement. Therefore strain, and consequently stress, are both constant throughout firstorder tetrahedrons. The situation imposes a severe limitation on the capability of meshes constructed with firstorder elements to model stress distributions of any real complexity. To make matters worse, straight edges and flat faces of the elements do not map properly to curvilinear geometry.
Secondorder tetrahedral elements have 10 nodes and model the secondorder displacement field in their volume, along faces, and edges. The edges and faces before and after deformation can be curved. Therefore, these elements map precisely to curved surfaces. Secondorder tetrahedral elements model the secondorder (parabolic) distribution of displacements inside their volume, on faces, and along edges. Consequently, they model the linear distribution of strains and stresses. Though they are more computationally demanding than firstorder elements, secondordertetrahedral elements are used for most analyses. They represent a "happy middle" between numerical complexityand the ability to model reallife displacements and stresses.
What is the role played by shell elements? While solidtetrahedral elements can mesh volumes derived from solid CAD geometry, use shell elements to mesh surfaces and when analyzing thinwalled structures. Because surface geometry does not carry information about thickness, the user must provide it.
Similar to solid elements, shell elements also come as first and second order. There are analogical consequences to mapping their curvilinear geometry to model displacement and stress fields. Again, only secondorder shell elements should be used for analyses of importance.
Certain classes of shapes can be modeled using either solid or shell elements. The type of element used for modeling tetrahedral solids or shells sometimes depends on the objective of the analysis. The nature of the geometry often dictates what type of element should be used for meshing. For example, parts produced by casting lend themselves to meshing with solid elements, while sheetmetal structures are best meshed with shell elements.
How does the meshed model's behavior differ from the original? Suppose we want to model displacements in the bracket with a hole on the first page. Having settled on a certain mesh density, we mesh the part with secondorder tets. Will the finiteelement model return the same displacement field as in the continuous model? No, it will not. Secondorder tetrahedral elements can only return a displacement field which is piecewise parabolic. Of course, if elements are sufficiently small complex displacement fields can be well approximated by piecewise parabolic distribution of displacements. This is the essence of finiteelement approximation.
Now, let's make an important observation: By meshing a part with a certain type of element and using a certain size and shape, we impose additional constraints on the part. The meshed part must conform to applied loads and restraints. But being meshed, it must also conform to constraints imposed by meshing. In other words, deformation must satisfy loads and restraints and be piecewise parabolic. Because the meshed part has the additional constraints, the process of meshing makes it stiffer.
The amount of additional stiffness depends on the element and their size. Firstorder elements require that the displacement field be piecewise linear. This is more restrictive than in secondorder elements where the displacement field must be piecewise parabolic. Larger elements add more stiffness than small ones. However, the effect of added stiffness (call it artificial stiffness) always accompanies finiteelement models. The effect of artificial stiffness is small but demonstrable in most cases, even with a reasonably wellrefined mesh.
How does artificial stiffness affect results? Two examples from structural static analysis can show the effect of artificial stiffness introduced by meshing. We will then extend conclusions to modal and buckling analyses. All geometry, material, load, and support information is presented for readers who would like to reproduce results in their own FEA programs.
Consider a cantilever beam meshed with three different densities to see how the maximum deflection changes. To make this comparison fair, even the first mesh (the rough one) must be able to model bending stiffness.
The beam deflects further with finer meshing. This proves that the finiteelement model becomes "softer" with a rise in element density. The effect arises because artificial constraints become less imposing with mesh refinement. Notice, too, that the deflection differences between the three models are small. Thus even the first rough mesh can model beambending stiffness correctly. Indeed, one layer of secondorder elements produces linearstress distribution across the thickness and, therefore, can correctly model bending stiffness in a regular shape such as the beam where stresses change linearly from tensile to compressive. The difference in results between the first and second models is 0.07 mm, while between the second and third it is only 0.02 mm. This proves that displacement results converge, as they should, to the solution offered by a continuous mathematical model.
How do stress results change with mesh refinements? Select any location on the beam surface to examine stresses and you will see they increase with more refined mesh. This effect arises from two contributing factors. First is that stresses are calculated based on displacement results, so larger displacements lead to larger stresses. This contribution, however, is usually small. The second and more important factor is that smaller elements collect stress information closer to the beam surface where higher stresses reside. How easy this effect is to visualize depends on if and what stress averaging technique has been used to plot results. Finally, with mesh refinement you will notice highstress concentrations in all four corners of supported faces (Not illustrated. Do this one on your own). This arises because stresses in those corners are singular and diverge to infinity with mesh refinements.
A second example is that of a thin flat steel strip (2503 103 2 mm) with supports on both sides. The strip has a 5mmthick block attached to the middle of its span. A load displaces the top face down 3.37 mm. The objective is to find reaction forces on both ends of the strip. Notice that though the deflection is small (1.3% of the length), the problem requires nonlinear deformation analysis to account for membrane stiffness that develops during the deformation.
Suppose we now mesh and solve the model three times, with one, two, and four layers of elements across the thickness. We'll notice that the reaction forces drop.
Why does the reaction force drop with mesh refinement? As before, the strip "softens" as the mesh is refined. Therefore, lower reactions develop after application of the same displacement.
The effect of artificial constraints also shows in modal analysis. By simple reasoning we can see that a modal analysis will overestimate the frequencies of a continuous model. This is because the finiteelement model has the artificial stiffness mentioned earlier and stiffer models mean higher frequencies of vibration. Mesh refinement lets the frequencies converge from above to the exact solution.
HOW THE REACTIONS COMPARE  
Thickness  Horizontal, N  Vertical, N 
One element  1,937  102.5 
Two elements  1,935  102.3 
Four elements  1,934  102.1 
Horizontal is along the length on the strip and vertical is across the beam thickness. Prescribed displacements are applied in this vertical direction. 
STIFFENING WITH BUCKLING
The effect of artificial constraints on results of buckling analyses resembles that in modal analyses. The added stiffness results in an overly optimistic magnitude of buckling load which converges from above to the exact solution of the mathematical model. And that on top of the fact that buckling analysis overestimates buckling loads because it disregards details such as geometric and material imperfections and load offsets. We must be careful when interpreting results of a linear buckling analysis which produces nonconservative results. Often, calculating a buckling load factor will require a nonlinear buckling analysis.
We have demonstrated the stiffening effect of a mesh using correctly shaped elements and correct meshes that can properly model structural behavior (bending, for the example presented). The stiffening effect is more pronounced when the mesh contains many degenerated elements or when a mesh just can't correctly model the stiffness, or both.
GUIDELINES FOR BEAM COMPARISONS
A fair comparison of beam results requires that all meshes, even the coarsest, be able to model the physics of bending. One layer of secondorder elements can model bending stresses (and consequently bending stiffness) in a model of simple geometry such as the prismatic beam because those elements can model linearstress distribution. If we instead used firstorder elements, it would be impossible to model bending stiffness because firstorder elements can only model a constantstress distribution (across the beam thickness). Such an erroneous model would have produced gravely incorrect displacement results of 3.5 versus 13.65 mm.
It is important to note that one layer of secondorder elements can model bending stiffness, which predominates in our beam. But experts recommend using at least two layers of secondorder elements across members in bending for more complex shapes and when shear stiffness plays a bigger role in model behavior.
Why use nonlinear analyses for simple beams
Simple beams need a nonlineardisplacement analysis even though the magnitude of displacement is small. The reason is that small displacements still significantly change the stiffness of the strip. Initially, the flat strip can resist the load only with bending stiffness. But having deformed a bit, it acquires membrane stiffness. An entirely new mechanism of load support develops because of deformation.
The big trouble with nonlineardisplacement analysis is that it is commonly referred to as "large displacement analysis." This confusing terminology implies that nonlineardisplacement analysis is only necessary for analysis of large displacement. However, the displacements need not be large to require a nonlinear analysis. All that matters is whether or not those displacements significantly modify the stiffness of the analyzed structure.
Try them yourself
Readers who would like to experiment with the models in the article can download them at no cost in Parasolid format from designgenerator.com/MDmodels.