Blake Courter

Differentiable Engineering

Executive summary

  • By including differentiable, parametric models in engineering processes, engineering software can better interoperate between human and artificial designers.
  • Existing CAD, CAM, and CAE tools can speak this language by adding differential interoperability to their APIs.
  • We provide a visual introduction to differential engineering using a cantilevered beam.
  • By examining the derivative of a rotation, we briefly unlock some deep math beauty and an application of Unit Gradient Fields (UGFs).
  • Differentiable engineering scales to product-level systems engineering.

✏️ Math advisory: this post assumes you’re okay with derivatives, the chain rule from basic calculus, and a little vector math. We will introduce intuitive visual tools to illustrate such concepts in design engineering. While I feel compelled to show the work, you can probably skim and glean the concepts from the illustrations.

👥 Lots of credit: These ideas came from discussions with many people, including:


Today, we practice three paradigms of computer-aided design (“CAD”), manufacturing (“CAM”), and engineering (“CAE”):

  • One-off design, where the focus is producing individual parts or products;
  • Parametric generative design, where the result is recipe to produce variants of similar parts or products; and
  • Computational generative design, where the final geometry is guided by simulation, often iteratively and with spatially-varying parameters.

As each of these generations has built on earlier technology, the emerging generation of engineering software powered by artificial intelligence and machine learning algorithms (“AI/ML”) is being trained on existing empirical, simulated, and textbook knowledge. However, while this new generation of tools promises ease-of-use, more accurate results, and orders of magnitude faster performance, it does not yet offer a meaningful shift in interaction paradigm. As these new tools become increasingly sophisticated, will new interaction paradigms emerge? Will we realize the sci-fi vision of product-level generative co-designers?

Let’s examine how AI and ML can blend with today’s optimization tech to expand engineers’ navigable design space. As generative design scales to the subsystem and product level, we’ll demonstrate how to delegate tasks to AI and ML without the meaning becoming hidden in a nonintuitive latent spaces, as with LLMs and generative art. We’ll focus on the role of a designer, human or automated, expressed in the language of optimization and machine learning: a differentiable approach to design engineering.

Abstracting the design engineer

Let’s propose a model for a design engineer, human or automated, which we’ll call “Mechanical Design Automation (MDA)”:

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Moat Map

Here’s just the map!

There was a time when I could keep track of all the engineering software companies. We had a few big CAD and CAE vendors, a handful of smaller companies defying VC pressure, and a CAM company for every manufacturing market. 3D printers were things that our resellers lugged around but didn’t really work. Life was simple. I could keep it all in my head.

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Parameterizing LatticeRobot

LatticeRobot Logo

Over on the LatticeRobot blog, a long post about our approach to LatticeRobot’s parameterization.

The LatticeRobot Unit Cell Parameter System

In addition, it’s worth mentioning that these parameterizations are build on top of Gradient Control Laboratories’ high level implicit scripting language, GCL Script (“GCLS”), provides a novel, high level API to implicits and powers LatticeRobot’s CodeRep output. Please be in touch if you’d like to learn more.

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Unit Gradient Fields: The Two-Body Field, Ξ

So far in the series, we’ve defined the basic idea that UGFs generalize SDFs and examined that when representing shapes, UGFs offer design freedom in the shapes’ normal cones. In most of the examples, we’ve shown that this freedom helps recapitulate the kinds of edge treatments we see in engineering software like rolling-ball blends and chamfers. In this post, we’ll take a look at the clearance and midsurface fields that apply to SDFs and the two-body field that applies to all UGFs.

Use the slider to change viewing modes:

The clearance field, \(\ugf{A} + \ugf{B}\,\), the midsurface field, \(\ugf{A} - \ugf{B}\,\), and the two-body field: \(\twobody{\ugf{A}}{\ugf{B}} \equiv \frac{\ugf{A} - \ugf{B}}{\ugf{A} + \ugf{B}}\). The clearance and midsurface fields are overlaid to demonstrate their orthogonality.

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Gradient Control Laboratories and LatticeRobot

With the UGF research entering a public phase, some other collaborations from the background are entering the foreground. In particular, some investigations with some friends have evolved into two new entities: an incubator, Gradient Control Laboratories and its first spinoff, LatticeRobot!

LatticeRobot Logo

Media coverage from CDFAM ‘23 LatticeRobot Launches a Home for Lattices, Metamaterials, and Textures

Develop3D: LatticeRobot announces community for advancing lattices in products

TCT: LatticeRobot launches engineering community for lattice research and knowledge share

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Unit Gradient Fields: The Promise of Unbreakable Geometry

Something seemed different about Sarah Frisken and Ron Perry, researchers from MERL who had ventured out to Boston’s Route 128 tech corridor to present their new modeling technology to the top geometry engineers at a leading CAD company. Frisken and Perry demonstrated that their adaptively sampled distance fields (“ADFs”) encoded geometry in a way where the offsets, Booleans, and rounded blends that confounded contemporaneous CAD systems like ours, would always succeed. They demonstrated organic texturing and lattices that would be unthinkable on state-of-the-art boundary representation (B-rep) solids. On the other hand, the modeling operations seemed limited to CSG operations, which had been superseded in mechanical CAD by the more expressive B-reps, and their only practical output was meshed geometry, considered inferior to B-reps. Would it be possible to combine the benefits of B-rep modeling with robust offsets, Booleans, and blends? It was 2001, I’d never seen anything like it, and I was hooked.

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Unit Gradient Fields: SDFs, UGFs, and their friends

Many readers of the last post requested a more formal definition of a UGF. Let’s look a bit more closely at the definition of an SDF and how it compares to UGFs and other useful fields in engineering applications. Some readers may find the visual concepts more intuitive than the nuances, so let’s get a feel for the territory first by examining the field at the intersection of two planes:

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Unit Gradient Fields: What do we mean by "offset"?

For those of us who work in engineering and geometric modeling, “offset” is an everyday operation. We use it in 2D and 3D to produce curves and surfaces at constant distance from other curves and surfaces. With experience, we learn that offset can be failure-prone, especially with precise B-rep solids and meshes. Implicit modeling, in particular, the signed distance field (SDF) representation of shapes, offers robust offsetting, but again, with experience, we learn that the results aren’t always what we expect.

Take these three examples of an offset rectangle, created using three different “line joining” approaches that date back to the early days of 2D graphics and are built into your browser:

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