Singularities in Finite Element Models - Dealing With Red Spots

8/19/2019 Singularities in Finite Element Models - Dealing With Red Spots

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Singularities in Finite Element Models: Dealing

with Red Spots

Your finite element model will sometimes contain singularities — that is, points where some

aspect of the solution tends toward an infinite value. In this blog post, we will explore the

common causes of singularities, when and how to remove them, and how to interpret results when

singularities are present in your model. While most of this discussion is in terms of structural

mechanics, similar phenomena can also be found in many other physics fields.

The Problem

In my previous role as a structural analysis consultant, I sometimes came across the problem of

how to report ridiculously high stress peaks in a finite element model to a customer. Experienced

analysts know when stress peaks are an expected effect of modeling and can be safely ignored.

Though, when a requirement that “the stress must nowhere exceed 70% of the yield stress” has

been stated, this may still turn out to be an issue. Equally important is the fact that the small red

spots in the color plots cannot always be ignored. Thus, we must have appropriate techniques for

interpreting the model results.

The Sharp Corner: A Prototype of a Singularity

Sharp reentrant corners will cause a singularity in the derivatives of the dependent variables for all

elliptic partial differential equations. In structural mechanics, this means that the strains can

become unbounded since the degrees of freedom are the displacements. Unless limited by the

material model, the stresses will also be infinite in such a case.

Stresses are investigated in the majority of structural mechanics analyses. This is why singularities

present more of an issue in structural mechanics than in most other physics fields. In heat transfer

analyses, for instance, you are much more likely to be interested in the temperature than in the

local values of the heat flux, the area in which a singularity would become evident.

Let’s have a look at a prototype problem. This problem involves a 2 meters by 1 meter rectangular

plate, featuring a square cutout with a side of 0.2 meters, that is subjected to pure tension:


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The plate is constrained along the left edge and has a uniform load along the right edge.

With two different meshes around the hole, the default plots of the effective stress look completely

different. Since the peak stress is twice as high in the model with the finer mesh, most details in

the stress field are lost. This can of course be remedied by manually adjusting the range of the

plots, but it may hide important details at first glance.


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The same effective stress field in two plots. Both plots are automatically scaled by the mesh-

dependent peak stress.

In fact, the smaller the elements that are used in the corner, the higher the values of stress that will

be found. The results will not converge since the “true” solution tends toward an infinite value.


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Stress at the corner as a function of element size (logarithmic horizontal axis).

If we investigate the stress field close to the hole, we will find that the stress peak is very

localized. In the figure below, the stress is plotted along a vertical cut line drawn at a distance of

0.05 meters from the hole. At this distance, the stress is virtually unchanged, even though the peak stress at the corner varies by a factor of two.


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Stress variation along a cut line (represented in red). Five different mesh sizes are used.

In the real world, there are seldom perfectly sharp corners. Thus, you could argue that by using an

accurate geometry representation containing all fillets, it is possible to avoid singularities. While

true, this comes with a price tag. If very small geometrical details must be resolved by the mesh,

the model grows enormously in size (especially the case in 3D). Even when a perfect CAD

geometry is available, it is common practice to defeature the geometry to remove small details

that are not important within the scope of the analysis. Therefore, in many cases, we actually

deliberately introduce sharp corners at the preprocessing stage.

There are, however, some drawbacks to keeping the sharp corner:

If the material model is nonlinear, there may be numerical problems at the singularity. For

example, the strain rate predicted by a creep model is often proportional to a high power of

stress. The high stress at the singularity (a value determined only by the mesh) raised to a

power of five may result in strain rates so high that the time stepping is forced to be in the

order of milliseconds, when you actually want to study an event taking place over months. If

you still want to keep the sharp corner, the remedy here is to enclose the singularity in a small

elastic domain.

Adaptive meshing, error estimates, and the like can fail since the singularity will dominate

over the rest of the solution. Exclude the corner from any such procedures.


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When running an optimization where stresses are part of the problem formulation, the

singularity will lead to solutions that are optimal only in terms of reducing the amplitude of

the unphysical peak stress. In theMultistudy Optimization of a Bracket tutorial, the region

where the bracket is bolted is excluded from the search for a maximum stress.

As previously noted, the high stress peaks tend to obscure more interesting features in the

solution, both visually and psychologically.

Physically, if the corner is very sharp, the material will be damaged by the high strains. A brittle

material may crack; a ductile material may yield. While it may sound alarming, such damage will

only cause a local redistribution of the stresses in most cases. As seen from the perspective of the

surrounding structure, the effect is no more dramatic than that of somewhat changing the fillet

radius. High, very localized stresses will only be a true problem if the loading is cyclic, which

creates a risk for fatigue.

In a building, nobody is concerned that the holes for windows and doors are rectangular with

sharp corners. But, in an airliner, you will find that the windows are smoothly rounded since the

variation between the pressure in the cabin and the pressure outside will provide a cyclic stress

history.

http://www.comsol.com/multiphysics/material-fatiguehttp://www.comsol.com/model/multistudy-optimization-of-a-bracket-19761


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Left: A rectangular window featuring sharp corners. Image by Jose Mario Pires. Licensed under

CC BY-SA 4.0 via Wikimedia Commons. Right: A window with smoothly rounded corners. Image

by Orin Zebest. Licensed under CC BY-SA 2.0 viaWikimedia Commons.

This is in fact recognized by many design standards, where high local stresses are allowed as long

as the loads are static. The local corner stresses will not in any way affect the load-bearingcapacity of the structure. Using this type of approach does rely on a systematic way of classifying

the stress fields. Such methods are, for example, described in the ASME Boiler & Pressure Vessel

Code.

For cyclic loads, on the other hand, it is important to obtain very accurate stress values. The

fatigue life depends strongly on the stress amplitude. In this case, an accurate representation of the

fillet is necessary, not only geometrically but also in terms of mesh resolution. If the model

becomes too large to handle, you can use submodeling , an approach that is described in detailin this blog post.

https://www.comsol.com/blogs/submodeling-how-analyze-local-effects-large-models-2/https://www.asme.org/shop/standards/new-releases/boiler-pressure-vessel-code-2013http://commons.wikimedia.org/wiki/File:Blue_void_out_the_porthole.jpg#/media/File:Blue_void_out_the_porthole.jpghttp://commons.wikimedia.org/wiki/File:Castelo_Novo_09,_window.jpg


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shell will cause a singularity.

Constraints

If we think of a constraint in terms of its capability to apply a reaction force, it is evident that the

same conclusions can be drawn as those for loads with respect to, for example, constraints appliedto a point. But, that is not all. Consider the seemingly symmetric problem below. Here, we have a

plate with a constant tensile load on one side and corresponding roller conditions on the other

side.

A square plate with one half of the vertical boundaries constrained and loaded.

When looking at the stress distribution, it is apparent that the end of the roller condition introduces

a singularity that the sudden change in the load does not. A general observation is that the end of a

constraint has an effect that is similar to that of a sharp corner.


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Horizontal stress distribution.

An infinitely stiff environment supporting the structure does not exist in reality. The analyst is

again left with a choice: Can I live with the little red spot, or do I need to pay more attention to

what is outside of my structure?

If the singularity caused by the boundary condition is not acceptable, you could consider the

following approaches:

Extend the model so that any singularity caused by the boundary condition is moved outside

of the area of interest.

Use a softer boundary condition by applying a Spring Foundation condition, for instance.

Use infinite elements, which offer a cheap method for extending the computational domain.

Learn more with this tutorial.

Situations similar to the one mentioned above are inevitable in many kinds of transitions. An

example of such a transition is connecting a rigid domain to a flexible domain.

Welds

The art of analyzing welds is so important and complex that it warrants its own blog post. Here,

we will only briefly touch on this subject.

https://www.comsol.com/model/flexible-and-smooth-strip-footing-on-stratum-of-clay-2203


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Welded structures often consist of thin plates, so it is natural to use shell models in this context.

Let’s have a look at the model below. In this example, a stress concentration is evident in the area

where the smaller plate is welded to the wide plate.

Stresses in a simple shell model of two plates welded together.

The geometry and loads are symmetric with respect to the center of the geometry. The mesh in

this model, however, is designed so that it is much finer at one end of the weld. A graph of the

stress along the weld line reveals a singularity in the stress field in both plates.


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A stress plot identifying a singularity.

For many welded structures — ship hulls, cargo cranes, and truck frames — dimensioning against

fatigue is important. Refining the modeling process by using a solid model is seldom the answer

here. The local geometry and quality of a weld is rarely well defined, unless it has been ground

and X-rayed. The local geometry will differ along the weld and between the corresponding welds

on two items that nominally should be identical.

When analyzing welds, the most common approach is to average the stress along the weld line or

along a parallel line a certain distance away. The cut lines in COMSOL Multiphysics are

particularly helpful here. The local coordinate systems also come in handy since stress

components parallel and normal to the weld need to be treated differently. These averaged stresses

are then compared with handbook values, which are available for a number of weld configurations

and weld qualities. To learn more, see Eurocode 3: Design of steel structures — Part 1-9: Fatigue.

Cracks

The worst conceivable geometrical singularity is the one caused by a crack. A crack can be seen

as a 180° re-entrant corner, so many aspects of the corner singularity are also applicable here.

When a crack is present in a finite element model, it is typically an area of focus within the study.

https://law.resource.org/pub/eur/ibr/en.1993.1.9.2005.pdf


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The stress field around a crack tip, with the deformation scaled.

The stress field around the crack tip is known from analytical solutions, at least for linear

elasticity and plasticity under some assumptions. Computing the stress field through finite element

analysis, however, can be difficult due to the singularity. Fortunately, it is usually not necessary to

study the details at the crack tip. When determining the stress intensity factor, for example, you

can use either the J-integral or energy release rate approach. These methods make use of global

quantities far from the crack tip, so that the details at the singularity become less important.

Tip: Looking to explore the use of the J-integral approach in further detail? Consult

the Single Edge Crack tutorial in our Application Gallery.

Conclusion

Singularities appear in many finite element models for a number of different reasons. As long as

you understand how to interpret the results and how to circumvent some of the consequences, the

presence of singularities should not be an issue in your modeling. In fact, many industrial-size

models require the intentional use of singularities. Keeping down model size and analysis timeoften necessitates simplification of geometrical details, loadings, and boundary conditions in a

way that introduces singularities.
http://www.comsol.com/model/single-edge-crack-988http://en.wikipedia.org/wiki/Strain_energy_release_ratehttp://en.wikipedia.org/wiki/J_integral

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Singularities in Finite Element Models - Dealing With Red Spots