The modeling tools are often used in conjunction with numerical design of experiments (DOE) to reduce modeling effort and to gain more insight into design parameter interactions. Often the optimal solution isn’t the numerical optimal, but the optimum within an insensitive region based on manufacturing tolerances and noise variables. My personal philosophy is to use numerical modeling as digital experiments, often run in conjunction with lab experiments. The experiments confirm the models and the models inform the experiments.

Note that due to the proprietary nature of the work done within GEHC, only high-level details can be shared.

Types of CFD Modeling at GEHC

CFD modeling spans the range from simple, such as pneumatic pressure drops or temperature limited electronics cooling problems, to the complex, such as precise temperature control problems or compressible flow studies through valves. Additionally, time depend transient studies are often needed in modalities such as respiratory and anesthesia care (ARC).

Below is a summary of simulation uses within GHEC:

• Concept Development: Map Design option for NPI’s
• Design Optimization: Hone selected concepts
• Design Implementation: Optimize system trade offs
• Root Cause Analysis: Understand field failure/Test results

Modeling Examples From GEHC

Electronics Cooling Example – FloEFD w/ Electronics Cooling Module

This is a typical electronics cooling problem. In this case, a new system on chip module was replacing an obsolete processor, and a new heat sink cooling solution that could fit within the existing space needed to be developed. To solve this problem, the SOLIDWORKS CFD package with the electronics cooling module was utilized to design the heat sink blower solution.

Figure 1. Example of a Typical Electronics Cooling Problem.

Figure 2. Example results.

This example demonstrates use of 2R chip models, PWB tool and flow and surface plots post processing of the temperature and flow results.

Valve Characterization Example

This effort consisted of mapping the performance of a high-speed fluid pulse width modulation (PWM) controlled injection valve as a function of inlet pressure, temperature and fluid. The results were used to determine the correct valve geometry, PWM duty cycles and the controls plant model of the valve, (partial results shown in Figure 3). The model established the dynamic range capability versus inlet pressure and fluid type, and the impact of noise variables such as temperature, pressure fluctuations and valve opening time.

Figure 3. Some results from high speed CFD modeling.

Ultrasonic Flow Sensor Design

The initial request was to evaluate a design that was developed experimentally. Empirical development led to poor performance and lack of understanding of what was driving it. CFD simulations were used to identify issues with flow field uniformity and transient flow pulsations. Ultimately, a new design was proposed and quantified numerically and experimentally.

Figure 4. Overview of ultrasonic flow sensor simulation project.

Figure 5. Sample CFD results from Ultra Sonic flow sensor.

In Figures 5 and 6, it can be seen that the flow fields are non-uniform spatially (steady state), and temporally (transient) with large difference between the minimum and maximum flows (0.15lpm to 15lpm).

Figure 6. Overview of transient analysis of ultrasonic flow sensor.

A new design was proposed where the flow was brought in around the perimeter of the ultrasonic transducers. The passageways around the perimeter were gridded to simulate a “honeycomb” structure to eliminate flow pulsations. As can be seen in Figure 8, the flow fields are very uniform, which was confirmed experimentally.

Figure 7. Design developed through CFD modeling of Ultra sonic flow sensor.

Particle Study Capability

In this example, a CFD particle study was used to understand how injected liquid would be entrained by flowing gas and how both fluids would be heated in a mixing chamber.

Figure 8. An example of using particle study.

This model was used to gain qualitative flow interaction knowledge, which helped guide the inlet and mixing design.

Case Study Example of Design Space Mapping and Use of Numerical DOE

CT Detector Temperature Control System Development Example – VCT

In the early 2000’s, GE designed a revolutionary 64 slice CT system from the ground up, which presented significant temperature control design challenges. The heat generating from the A to D electronics needed to be packaged near the temperature sensitive photodiode and scintillator of the x-ray sensor. The available cooling air DT was limited, as the maximum electronic temperature was close the maximum T_air. Additionally, an altitude of 0-3500m had to be accommodated.

A CT scan cycle consists of rotating from rest to high speed in just a few seconds, which caused a simultaneous rise in T_air and a change in air velocity near the detector (convective boundary condition shift). For artifact-free imaging, the x-ray sensor needs to remain essentially constant throughout a scan cycle. To solve this problem an architecture was proposed, then CFD/electronics modeling was used to map the design space and drive the design details.

These details included the obvious, such as number of fans, heat sink geometry and ducting, to the non-obvious such as placement of the chips, underfilling of chips, circuit board copper layout and flex effective thermal conductivity, to name a few.

Pictures of the exterior and interior of the GE 64 slice scanner are given in Figure 9. In the right-most picture of Figure 9, the system is rotating at maximum speed, which translates to a linear velocity of ~35mph near the x-ray sensors. The transition from stationary to final rotation speed takes only a few seconds, resulting in a large transient shift in convective boundary condition.

Figure 9. GE VCT 64. (Images taken from public domain.)

System Architecture – Start of the Design Space Mapping

CFD modeling was initiated to both investigate feasibility and to drive the design decisions. Figure 10 shows a sketch of the system and lays out the noise and design variables along with design outputs. The approach from architectural concept, critical variable identification, modeling methodology and sample of numerical DOE results are shown in Figures 11-14.

Figure 10. Sketch of system air flow path.

To start, a simple network representation (figure 11) was developed and used to drive the numerical DOEs. A global local modeling approach was used, where flow resistances from local models (numerical wind tunnel studies) were used to obtain global model flow boundary conditions, which were in turn fed into local models. Detailed electronics cooling models were used to determine steady state and transient temperatures of key components. From the DOE studies, transfer functions were built and used to map the design space, significantly reducing the number of computational runs and providing key insight into variable interactions.  

Figure 11. Node Network representation.

Figure 12. Global-Local methodology.

Figure 13. Example of DOE results.

Once the design space and component interactions were mapped, detail optimization began.

Figure 14. Example of Numerical Optimization and experimental and CFD Data.

Summary

A combination of CFD (with electronics cooling module), conduction heat transfer modeling using global local modeling technique driven by numerical DOE were used to show and drive design feasibility. The modeling effort took approximately nine months and was confirmed experimentally with a ¼ system bench model, followed by a system prototype. Prototype to production consisted of controls algorithm development only.

To learn more about SOLIDWORKS simulation for product development in health care, check out the whitepaper Simulating for Better Health and the webinar Free Surface Flow Simulation at GE Healthcare.

Here are a few types of buckling.

• Column buckling (Will be discussed further below)
• Snap through buckling (Will also be discussed further below)
• Flexural-torsional buckling
• Lateral-torsional buckling

While there are several different types of failure with buckling at its core, overall buckling is a failure characterized by sudden movement of the geometry of the structure in response to a compressive load. If a column is loaded axially downward, then the column’s rapid deformation to the left or right is due to buckling.

The mechanism of column buckling is caused by the fact that loading something perfectly in line with its geometric center is an impossibility. Nothing is ever perfectly straight, nothing is ever perfectly loaded and no atomic structure is ever arranged without flaws. Our world just doesn’t work that perfectly—but that is all fine provided this behavior is understood and checked for safety.

Here are some rules to determine whether buckling needs to be considered:

1. Are compressive forces present? If not, forget about it—tensile forces don’t lead to buckling.
2. Is the component long, or does it contain slender features? A column that has a small cross-section is an example of this.

For number 1, compressive forces can lead to failure through crushing as well. That isn’t the same as a buckling failure. In buckling, failure typically happens before reaching the yield stress, which makes it the governing failure mode in those cases.

The Mathematics of Buckling

In the image below is the mathematical derivation, that will hopefully provide some insight into the mechanics of buckling. The mathematics start with this assumption that no matter how well-crafted, a structural member can never be manufactured completely straight.

As a result, a compressive load is never perfectly aligned with a straight axis, meaning that any axial load creates a moment in the structural member that causes it to start bending, which increases the misalignment of the load and the center of the beam leading to an increase in the moment of the beam. It’s basically a feedback loop.

When the mathematics of buckling were first being explored by Leonhard Euler, he didn’t understand how to calculate the stiffness of the beam, but he understood it was a constant, and so he used the variable C, in order to represent the stiffness of the beam’s cross-section. The image below shows the equation he came up with as well as the column to the right that he assumed represented any column.

Euler assumed that the beam had an infinitesimally small deformation at the middle, and that its deformation was the shape of a sine curve to make the math simpler. Below is an image of the breakdown of these mathematics.

Going through the equations above, equation 1 is from general beam theory. It says the moment at any point is related to the 2^nd derivative of the displacement function multiplied by the stiffness of the beam, C in this case, and acts in the opposite direction of the displacement.

Equation 2, w(x), represents the displacement function. Euler assumed the shape of a half sine curve with a maximum amplitude of an infinitely small misalignment, ẟ. From there, he calculated the first derivative and the second derivative of the displacement function, equations 3 and 4 above.

The image on the right is the free-body diagram Euler used to plug in values into equation 1, along with the equation 4. By calculating a value for M and plugging in the second derivative of the displacement function, Euler was left with equation 5. The sine functions on both sides of the equation cancel out, as do the ẟ’s, leaving Euler with equation 6, also known as the critical buckling load calculation.

This is the pure compressive load that a member can hold before it buckles. It should be noted that n was an integer which was removed because checking values with higher order sine functions would result in a displacement shape with more “waves.” It is clear that n=1 creates the lowest critical buckling load and as a result would be the worst-case scenario, so n can be removed from the equation entirely.

It should also be noted that decades after Euler’s work was done, it was discovered that the constant C was calculated as E*I, where E is the modulus of elasticity of a material, while I is the moment of inertia of the cross-section. Euler didn’t discover this portion because his forte was mathematics, while those who experimented to learn this were scientists.

Determining If Buckling is a Governing Failure Mode

With the mathematics of buckling explained, it’s important to understand when it governs as a failure mode. As mentioned, buckling only happens under compressive loads because tensile loads would “flatten” out the sine curve. Compressive loads instead exacerbate the issue. If high compressive forces exist a system can either fail via buckling, or it can fail due to crushing—so determining which failure mode governs is critical for engineers.

If P/A > σ then crushing will happen before buckling. In this case:

• P = Compressive Load Applied (the critical buckling load can be substituted here if the actual applied loads are unknown. This would tell you which would govern, if the critical buckling load/cross sectional area is above the yield stress, material failure is going to govern the design.)
• A = Cross-sectional area of the column
• σ = Ultimate compressive stress of the material or yield stress depending on whether it is a brittle material or ductile.

In general, buckling can be prevented by using a larger cross-section or stiffer material. Whatever can be done to increase the stiffness of the cross-section, E*I will help. Additionally, it can be seen in the critical load calculation that the buckling load is inversely proportional to the length of the structural member squared, so if required, reducing the length of the structural member or bracing the member can be used to increase the critical buckling load.

Tools for Determining Buckling Loads

There are many tools available for calculating the critical buckling load, including spreadsheets, tables and FEA software. Each has its own merits and benefits:

Spreadsheets are easy; just plug in data and go. They can be cumbersome to build, or can be purchased for a nominal cost. However, they usually aren’t customizable, so if your project is slightly different, you’re out of luck.

Tables are cheap and easy to use. The most common ones provide for determining different effective lengths that modify the calculation of the critical load. One such source is the Manual of Steel Construction that many civil engineers use, as well as mechanical engineers that work on parts of larger scale. Discussion of effective lengths will be provided later in this article.

Finally, software such as SOLIDWORKS Simulation professional can be used to run a buckling study to calculate the critical load. This can be particularly beneficial when the geometry involved is that of a system acting more as a larger system in a composite action, or if it has irregular cut outs that make the stiffness change along the length of the beam. That is something that Euler’s calculation didn’t account for specifically, as it assumes the moment of inertia is constant along the length of the beam, without the cross-section changing. If the cross-section does change, it requires a numerical approach to solve efficiently and accurately.

Spreadsheets can be created by a user. Tables can be found from sources like the Manual of Steel Construction, and as mentioned Finite Element Analysis software like SOLIDWORKS Simulation Professional can be used.

Learning How to Use Simulation for Buckling Analysis

For SOLIDWORKS Simulation Professional users, there is a tutorial built right-in. Open SOLIDWORKS, make sure the simulation tool add-in is turned on, then go to Help > Tutorials > Simulation Tutorials > Simulation Professionals then look for the buckling tutorial under the Frequency and Buckling tutorials section.

This tutorial will walk you through the basics of any buckling analysis:

1. Parts definition – Materials and mesh type. The goal of this step is to tell the software the properties and how to derive the stiffness.

2. Loads. This determines what the structure is attempting to withstand.

3. Fixtures. This tells the software how the structural system is restrained from moving.

4. Meshing. This describes the shape and stiffness of the structure. This step relates to the accuracy of the analysis. If the mesh is too coarse at this step, the model results will over-predict the size of the critical buckling load—a negative trait in this case.

5. Run. Takes all the inputs and solves them.

6. Processing results, also known as post-processing. Every software is a little bit different, but in SOLIDWORKS Simulation, for example, the result provided is called a buckling load factor. It essentially tells you how much you would have to scale the loads by in order to cause buckling.

When looking at the results, buckling load factors are the factor of safety on the critical load. A buckling load factor of 3 means the applied load would have to be increased by a factor of 3 for buckling to happen. It is also possible to have negative buckling factors of safety. These mean that the system is in tension, so the load direction would have to be reversed and multiplied by that amount in order to see buckling occur.

When the buckling load factor is anywhere between 0 and 1, the design doesn’t work. In those cases, that means the software is predicting a buckling load failure.

Effective Lengths and Boundary Conditions

Like any analysis, results are dependent on your boundary conditions. The table below shows how important the boundary conditions of your simulation are. The values from these tables don’t need to be input into your simulation; the behavior is already included in the analysis.

The reason I’m including the table here is to illustrate that if you mismanage your boundary conditions, setting up your analysis with a fully fixed condition when in fact the true behavior is rotationally fixed-pinned, it’s possible to be off by a factor of 4! That’s not safe, so be certain your boundary conditions are correct in the simulation, or else use the worst-case scenario shown below.

The table below comes from the Manual of Steel Construction and is used to modify the hand calculations for different end mounting conditions, modifying the length in the critical buckling load calculation. The reasoning is that when Euler was doing his work, he was assuming both ends are pinned, so his deformation function representing half of a sine wave was correct.

However, other end conditions change this behavior. Both ends fixed “shortens” the sine function as seen in (a) in the table below, so the effective length of the column becomes half its normal length in the critical buckling load equation.

For other end conditions like (e) and (f) below, they only represent a quarter of a sine curve, as such they double the effective length in the critical buckling load calculation. The table shows you the numbers for other conditions as well.

Snap-Through Buckling

At this point, we’ve talked about column buckling and linear simulation methods that can be used to determine that behavior. However, there is also a more complex form of buckling called snap-through buckling. The most common example of this would be a Snapple cap. Taking that cap and pressing the “button” on it, it goes from bulging outward towards you to bulging away from you. That system has a “switch” and once it’s pushed to a neutral or flat position, just another small increase in force, and it “snaps through” and becomes resistant to the applied load in tension.

This is a non-linear phenomenon because it involves a significant change in the geometry as it happens. Linear analysis tools can’t predict this behavior; it must be solved iteratively, which is the realm of non-linear solvers.

Luckily, SOLIDWORKS also has a tutorial built into it for this purpose; the catch is that you need SOLIDWORKS Simulation Premium to have access to the non-linear analysis tools needed to analyze this. You can find this tutorial in SOLIDWORKS at Help > Tutorials > Simulation Tutorials Tab > Simulation Premium > “Snap through/Snap back Analysis of a cylindrical sheet”.

This is the model used in the tutorial. It is a thin sheet with a slight curvature to it. The tutorial utilizes symmetry to run a ¼ model, saving computational time. It will also walk you through the setup and running of this type of analysis.

Snap-through behaviors come from geometries that are thin sheets with curvature to begin with. Depending on the material used and the geometry, snap-through behavior may not damage a structure, as can be illustrated with the Snapple cap. It can be pressed repeatedly, however, if its purpose is to support a load and it snaps through, then this could be considered a failure.

I know far fewer rules of thumb for this sort of analysis, so it is generally only calculated using tables or FEA tools such as SOLIDWORKS Simulation Premium.

A couple questions to ask after running this sort of analysis:

1. Is there a material failure prior to this behavior? That can depend on your definition of failure. However, it could be that the yield is reached, or that ultimate failure is reached. If the yield stress is reached, the analysis needs to be re-run using a more complex material curve, like a bilinear model or a stress strain curve if linear behavior was assumed. If those were used and failure stress was reached prior to the snap-through behavior, then snap-through isn’t a governing failure mode.
2. What is the deformation that happens after the snap-through and is it acceptable to operation? If a component is attached to this, or if this behavior is represented as something like a mechanical switch, will it contact the piece it’s supposed to?

Buckling is one possible failure mode in a system that has compressive loading present. It needs to be understood and checked to make sure that designs are safe. By following the guidelines outlined above you can better analyze your designs and be more confident that they will operate as they are supposed to. While this article can’t possibly hope to cover every possible consideration for buckling, it has introduced some of the basic concepts and hopefully given you an understanding as a foundation.

Disclaimer: When designing any structural component, make sure that you are qualified to do so and have proper credentials and do proper safety checks. There are numerous factors that this article cannot possibly account for.

Learn more with the whitepaper Design Through Analysis: Simulation-Driven Design Speeds System Level Design and Transition to Manufacturing.

About the Author

Brandon Donnelly is a Technical Account Specialist with GSC. He produces a podcast for engineers called, “Stories for Engineers” that can be found on Spotify, Apple Podcasts, Stitcher and iHeartRadio. He is always available for questions or conversations about SOLIDWORKS, Simulation, 3D Printing and general engineering talk.

How to Add Topology Optimization to SOLIDWORKS?

ParetoWorks is a SOLIDWORKS add-in that allows engineers to perform topology optimizations on their parts based on stiffness or strength. This software uses simulation and optimization algorithms to lightweight parts without sacrificing the structural integrity of the parts.

ParetoWorks can automate the lightweighting optimization of a part based on stiffness and strength. (All images courtesy of SOLIDWORKS and SciArt.)

The software imports the SOLIDWORKS geometry as an STL file. The program then runs a finite element analysis (FEA) solver and a design optimization engine to give engineers an idea of how to reduce the weight of a part. Though ParetoWorks operates within the SOLIDWORKS framework, it is also a cloud technology available through a browser.

“Design optimization lies at the heart of modern engineering,” said Krishnan Suresh, mechanical engineering professor and CTO of SciArt, makers of ParetoWorks. “It is critical in reducing cost, reducing material, reducing weight and increasing quality, and is a driving force behind innovation. [However], design optimization can be very tricky and difficult for humans to carry out manually.”

To set up the problem, the user needs to input the units of the system, any boundary conditions, the limiting variable for optimization (stiffness or strength) and the forces the part will experience. All of these inputs are made in a simplified field of entry.

“[ParetoWorks’] focus is to make sure these structural loads are done right,” said Praveen Yadav, director of engineering at SciArt, makers of ParetoWorks. “We have the capability of handling thermomechanical loading, transient loading, modal analysis and buckling analysis. We include all of these analyses in the optimization for multi-constrained, multi-material and multi-load optimizations.”

The optimizations can also take constraints into consideration when assessing the part. Some of these constraints include:

• Maximum volume fraction
• Stiffness and displacement constraints
• Manufacturing constraints like casting draw direction
• Assembly constraints like which surfaces to maintain

The Benefits of Subtractive Topology Optimization to Additive Topology Optimization

ParetoWorks isn’t the only topology optimization software out there. However, many of them are based on additive optimization of the part as opposed to subtractive optimizations.

Though additive can give engineers a good starting point for a part based on a design space and loads, it does take a lot of data to perform this optimization. Subtractive optimization, however, requires less data but it will need to start with a part before the optimization can be performed.

Instead of mapping the whole design space, ParetoWorks uses subtractive optimization to modify the geometry, saving data costs.

“[Additive optimizations] try to map out the entire design space, this requires the storage of a lot of data, which gets expensive as you get new designs and you compare to the existing stored variable,” said Yadav. “ParetoWorks, on the other hand, attempts to stay as close as possible to the optimality front. This reduces the requirement of storing data.”

“All we care about is where we are right now and where we are headed,” Yadav clarified. “Based on that, we can choose a suitable search direction and make small increments to update the geometry. Once the geometry is updated we also perform fixed-point filtration to make sure it is robust and stable in that region.”

Since the entire design space isn’t mapped out, engineers might wonder how ParetoWorks handles the constraints. When performing this optimization, if the part hits a constraint made by the engineer, the software will reassess the step size for subsequent iterations in the optimization. This will slow down the changes made to the part and allow for a more detailed localized search for the best design in that region.

For more on ParetoWorks, read this ENGINEERING.com article. To learn more about structural optimizations using simulation, follow this link.

About the Author

Shawn Wasserman (@ShawnWasserman) is the Internet of Things (IoT) and Simulation Editor at ENGINEERING.com. He is passionate about ensuring engineers make the right decisions when using computer-aided engineering (CAE) software and IoT development tools. Shawn has a Masters in Bio-Engineering from the University of Guelph and a BASc in Chemical Engineering from the University of Waterloo.