New industrial design technology will gain speed boost

The rise of 3D printing and other advanced manufacturing methods has allowed engineers to build structures that were previously impossible to manufacture. A new design strategy that takes full advantage of these new features is topology optimization. This is a computer-driven technique that determines the most effective way to distribute materials and leads to optimized designs.

A research team, including mathematicians at Brown University, has now developed a new approach that dramatically improves the speed and stability of topology optimization algorithms. The team collaborated with researchers from Brown, Lawrence Livermore National Laboratory in Norway, and Simula Research Lab, where they detailed their research in two recently published papers in the SIAM Journal on Optimization and Structural and Interdisciplinary Optimization.

Brendan Keith, Brown’s assistant professor of applied mathematics, said: “This is a massive computational savings that allow people to make their designs faster and cheaper, or develop more complex designs at a higher resolution.”

One way to think about topology optimization is that it resembles paintings in 3D, according to Boyan Lazarov, a research scientist at the Lawrence Livermore National Laboratory.

“In the past, when you wanted to design something, you used simple geometric shapes and then you connected them in some way,” Lazarov said. “But with topology optimization, we start with a blank canvas and use a computer to place the material on it, and finally get the structure that is optimally executed with respect to a particular criteria.”

An important aspect of the process is that it is repetitive. Optimizer repeatedly updates its material patterns, adds materials to some locations, removes them elsewhere, and tests the physical properties of the design in each iteration. The process is repeated until the algorithm converges to a final design that maximizes structural properties using the final amount of material.

This process is great for creating highly efficient structures, but the iterations are computational. It is not uncommon for an algorithm to run for more than a week to reach the final design even in high-performance computing clusters. The team’s new approach is trying to optimize the optimization algorithm itself. This allows you to reach the final design with significantly fewer iterations compared to traditional methods.

The team calls the approach the Sigmoidal Mirror descent method with simple latent variables.

It works by helping to alleviate common issues with topology optimizers. Imagine a canvas split into pixels. The amount of material placed in each pixel can be zero (no material), one (completely filled with material), or somewhere in between. The problem is that traditional topology optimizers often assign impossible values ​​to a particular pixel. Fixing these “impossible” solutions slows down the optimization process and increases the time to wait more iterations and wait for the final design.

A simple method streamlines the process by completely eliminating these impossible solutions. It is done by transforming the space between one and zero into a “potential” space between infinite and negative infinite. Each iteration can place or remove material in an amount that approaches their infinity but cannot reach them. The material values ​​generated in that infinite space are returned to the space between one and zero and incorporated into each iteration.

Benchmark tests simply show that up to 80% less iteration is required to reach the optimal design compared to traditional algorithms. This often means reducing very few computing times, often from days to hours of iteration. This could allow topology optimization to be accessible to a wider range of industries and enable designs at a much better resolution than what is currently possible, researchers say.

The team is free to use versions of the algorithms that engineers and other researchers can use.

“The mathematical theory behind this algorithm is very complicated, but in practice it is very easy to incorporate into a method of optimizing a standard topology with several lines of code,” says Dohyun Kim, Brown’s postdoctoral researcher and lead author of the study. “I think this could have a huge impact in the engineering community.”