We had a lot of great questions during our Coreform Flex webinar on October 10, 2024. In this post, we’ve included those questions as well as our responses - edited and paraphrased from the live responses. Enjoy!
Question 1: For those 2 examples that you showed: Can you roughly compare the solve time for equivalent accuracy, using traditional FEA methods?
Answer, by @gvernon
That is a good question to ask. We don’t have access to any experimental data, nor simulation results from other tools, to perform verification of the accuracy. I can say that the customer who originally provided the direct ink write example needed to use various strategies such as an explicit dynamics solution approach, with mass-scaling and a long timescale to obtain a quasistatic solution – which led to an expensive solution time. Additionally, they had implemented a different model-setup approach, using discontinuous meshes and tied-constraints between the threads – which leads to various modeling inaccuracies.
So, I think what you’ll find with the approaches that we tend to take, on the problems that we’re currently able to solve, that we tend to be within an order of magnitude in terms of solve time to achieve a similar level of accuracy. That’s because what we’re doing really is just the finite element method that’s just been improved upon in a few key ways to support this meshing free workflow. And though the generalizations that improve upon and enable this workflow use basis functions that are more accurate per degree of freedom than traditional FEA, they are more expensive per operation and we’re still working on more efficient implementations within our young solver.
Question 2: In biomedical engineering, we need to show that the results, especially stresses, are mesh independent. And so we usually run mesh sensitivity analysis, and refine the mesh. Do you have ways of refining the mesh in Coreform Flex?
Answer, by @gvernon
As of today, we don’t support local adaptivity (e.g., local refinement) in Coreform Flex, so the approach that we take today is: perform multiple simulations at different mesh densities and observe the convergence. But the ability to do local refinement is a core feature of spline meshes in general, so that’s something that we want to do. Note that our technical team has a lot of experience developing local refinement in splines (see list of sample sources below) and our “U-spline” technology supports local refinement. So stay tuned.
- Isogeometric spline forests
- Bézier projection: A unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis
- Local refinement of analysis-suitable T-splines
- Analysis-suitable T-splines: Characterization, refineability, and approximation
- Truncated hierarchical Catmull–Clark subdivision with local refinement
- Hierarchical B-spline complexes of discrete differential forms
- Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis
Question 3: Can you capture complex, round features using this hex mesh method?
Answer, by @gvernon
Yeah, we can. Here’s an image that more clearly shows this approach that we call “volumetric trimming” on a cube that has a spherical cutout in it – demonstrating that we’re able to capture, in our volumetric trimming approach, curved surfaces. Essentially, the trimming simply tells us how to integrate our basis functions (“shape functions”) over the physical domain. So yes, we can absolutely support complex curve surfaces with our approach.
Question 4: Are the stiffness, matrix and load vectors available through a petsc4py interface in Python?
Answer, by @gvernon
In practice, no. I mean, you could probably find a way to hack it to do that. We are developing an interoperability library with some SBIR funding (SBIR is a U.S. Federal funding program for that allows small businesses to do research that is applicable to the US Government and its departments). We have been working on developing this interoperability library in support of the open source MOOSE solver from Idaho National Labs. But that’s not something that’s released today or is otherwise available. But if you’re interested in that, we can talk more about how to enable that for you.
Question 5: Is the trim process parallelized? Or is it just the linear solver that’s parallelized.
Answer, by @gvernon
Yes, the trim process is parallelized and the solver is parallelized, both via MPI.
Question 6: How much error do we see compared to traditional, finite elements?
Answer, by @gvernon
What we’ve done is take traditional finite elements and replaced its low-order traditional basis functions with high-order smooth splines – you can use the same theoretical proofs of the accuracy for traditional FEM to verify the accuracy of our method. Of course, theory and implementation are often orthogonal, so we also publish a verification manual, which I encourage you to review, at docs.coreform.com. Note that this is a subsample of our internal verification documentation, and we’ve not spent a lot of time “prettifying” it.
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Cantilever beam with end shear load
- Note that we’re able to converge to the exact analytic solution using a cubic basis (the through thickness shear-stress is a quadratic function, so a cubic basis is required to obtain the exact solution).
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Composite cantilever beam with end shear load
- Similar to before, but now with two different materials (one 10x stiffer than the other). Note that, once again, we converge to the exact analytic solutions for the through-thickness normal and shear stresses when using a cubic basis – even when the two trimmed meshes don’t align with each other.
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Infinite plate with a hole
- Observe we match the optimal convergence rates, to the analytic solution, as predicted by FEM theory
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Stresses in long pressurized pipe with butt-welded supporting trunnion pipes
- This is a non-trivial geometry where NAFEMS performed an analysis using traditional FEM methods. If you review their results you’ll see that we converge to the same value as their reference solution.
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Scordelis-Lo roof
- This is a classical example for shell elements, however we solve this problem using (trimmed) solid elements. Note that, even with elements that are 8x larger than the thickness of the roof, we can obtain highly accurate approximations of the radial, shear, and membrane stresses when using a quartic (degree-4) basis – and results for cubics are pretty good too.
To summarize, Coreform Flex is based upon the same fundamentals as the finite element method and thus we have the same ability to achieve accurate results. When comparing accuracy per degree-of-freedom, Coreform Flex is more accurate than low-order finite elements – a result predicted by finite element theory.
Question 7: How does the software track convergence?
Answer, by @gvernon
There are a couple different definitions of convergence. What I’m guessing what you mean is convergence of the solution as an engineer might care: you care about converging the displacement or stress field… you want to know that you’re getting the correct solution so you can make the correct engineering decision.
As I mentioned earlier, what we do today is, effectively, a manual process. We, as the user, will define “probes” e.g., to probe stress at a given location or along a line, to probe a maximum stress value within a region, and then will either manually update our mesh density and rerun – or we’ll use Python as a wrapper around our tools and write routines to determine convergence and perform refinement if necessary. But we don’t have any of that kind of functionality internal to Coreform Flex. Again, that’s on our roadmap as we investigate implementing local refinement.
If your question is about convergence of the actual numerics of the problem, e.g., the convergence of the Newton nonlinear solver, this is just done the same as in the traditional FEM approach of having residual vectors and computing norms on them etc.
Question 8: Is it possible to have a trial to use Coreform Flex?
Answer, by @gvernon
We are welcoming early adopters using Coreform Flex. I think you’ve seen kind of where we’re at from a user interface standpoint. And the software maturity standpoint. This will continue to improve. And we’re adding features rapidly.
For anyone that’s interested, the procedure we have now is, you set up and have a chat with Greg and me, and we just determine what application you’re interested in. We run one problem for you to just verify that the maturity level and the workflow is suitable for you today. And then assuming that’s a check, then we go ahead and give you access to Coreform Flex. So if that’s interesting to you, we’d love to have that chat with you and you could reach out to me directly at matt@coreform.com. Or you can go on our website as well to request a trial there. And along those lines, if you’re interested in a couple of weeks we have a short course at that we’re hosting at the IGA Conference in Florida. We still have just a couple of spots left at this short course.
Question 9: Which contact, constraint, enforcement method do you use? And have you compared the results with traditional finite elements?
Answer, by @gvernon
Right now, we use penalty enforcement method for contact. But any of the methods that you would use in traditional FEA could be used with Coreform Flex. We’re currently working on additional improvements to our contact approach. We do have some internal verification documentation (see sneak-peek below) on contact convergence, we’re not ready to share the results yet, but they’re exciting!
Question 10: In those 2 examples that you showed you only included elastic properties of material. Can we also include plastic properties like stress strain data, and how accurate is Coreform Flex in failure prediction?
Answer, by @gvernon
Here is another problem, one of the Sandia fracture challenge problems that you can find in the open literature. What I’m showing here uses our isotropic J2 plasticity model, the plot is of the equivalent plastic strain (sometimes referred to as eqps or peeq), where purple corresponds to the strain at ultimate tensile failure. In this case the hardening curve was provided by a piecewise linear function defined, by the analyst, in the input file.
I will say that, currently, we don’t have a large list of implemented material models today in Coreform Flex. Our approach is to work with customers to identify which material models are necessary for them to have success on their problems, and then we implement those materials for them. So if there are material models that you use in your traditional tools that you’d like to see, you can reach out to us and talk to us about your application. And what I’ll say is that for this, for this example, we compared pretty well to the experimental data of this problem.
Question 11: What are the primary markets Coreform Flex?
Answer, by @matt
Coreform Flex is a general finite element solver that we see applications for across many industries. The most interest so far has been coming from automotive and defense companies. But we see applications across a breadth of industries for Coreform Flex.
Question 12: Does Coreform Flex support all element types commonly used in traditional FEA software? And is it capable of analyzing structures that contain a combination of shell, solid and beam elements?
Answer, by @gvernon
Coreform Flex today only supports solid elements. The flex representation method (FRM) that we’ve developed, and that underpins Coreform Flex, does support all of the different dimensionalities of elements with the same trimming approach. So element types like shells and beams are supported by the underlying technology, but we have not yet implemented those in Coreform Flex.
One of the reasons we’ve not implemented shells is that, as we showed in the aforementioned Scordelis-Lo roof problem in our verification manual (a classical problem in shell element verification), we can use solid elements with even less than one element through the thickness and still obtain accurate solutions with reasonable efficiency. And so, since our customers today who deal with shell or shell-like objects are finding success with this solid approach, we’ve not yet focused on implementing these element types.
Matt Sederberg: Thank you so much. We appreciate all of the questions. We’ve been able to get to most of them. There’s a couple of questions that we will go ahead and answer on our forum, and we’ll also include a transcript of this entire QA session on our website forum. So as more questions come in, if you think of one in the coming days, feel free to interact with us there.
And again, we’ll be sending out an email with this webinar recording with our contact information. If you’d like to talk with us more about trying out Coreform Flex yourself, then we’d love to engage with you more.