PROVIDENCE, R.I. [Brown University] — With the rise of 3D printing and other advanced manufacturing methods, engineers can now build structures that were once impossible to fabricate. An emerging design strategy that takes full advantage of these new capabilities is topology optimization — a computer-driven technique that determines the most effective way to distribute material, leading to an optimized design.

Now, a research team including mathematicians from Brown University has developed a new approach that dramatically improves the speed and stability of topology optimization algorithms. The team, a collaboration between researchers at Brown, Lawrence Livermore National Laboratory and Simula Research Laboratory in Norway, detailed their work in two recently published papers in the SIAM Journal on Optimization and Structural and Multidisciplinary Optimization.

“Our method beats some existing methods by four or five times in terms of efficiency,” said Brendan Keith, an assistant professor of applied mathematics at Brown. “That’s a huge computational savings that could enable people to make designs more quickly and inexpensively, or to develop more complex designs with higher resolution.”

One way to think about topology optimization is that it’s a bit like painting in 3D, according to Boyan Lazarov, a research scientist at Lawrence Livermore National Lab.

“In the past, when we wanted to design something, we’d use simple geometric forms and then connect them together in some way,” Lazarov said. “But with topology optimization, we start with a blank canvas and we use a computer to place material on it such that we eventually get a structure that performs optimally with respect to certain criteria.”

A key aspect of the process is that it’s iterative. The optimizer repeatedly makes small updates to its material pattern, adding material in some places and taking it away in others, then testing the design’s physical properties with each iteration. The process repeats until the algorithm converges to a final design that maximizes structural properties using the least possible amount of material. 

The process is great for creating highly efficient structures, but the iteration is computationally expensive. It’s not unusual for an algorithm to run for a week or more to arrive at a final design even on high-performance computing clusters. The team’s new approach seeks to optimize the optimization algorithm itself — enabling it to reach a final design in significantly fewer iterations compared to traditional methods. 

The team calls the approach the SiMPL (Sigmoidal Mirror descent with a Projected Latent variable) method. 

It works by helping to alleviate a common problem with topology optimizers. Imagine a canvas divided into pixels. The amount of material placed in each pixel could be zero (no material), one (filled entirely with material) or somewhere in between. The problem is that traditional topology optimizers will often assign impossible values to certain pixels — values less than zero or more than one. Correcting these “impossible” solutions slows down the optimization process, leading to more iterations and more time waiting for the final design. 

The SiMPL method streamlines the process by eliminating these impossible solutions entirely. It does so by transforming the space between one and zero into a “latent” space between infinity and negative infinity. Each iteration can place or remove material in quantities that approach those infinities, but cannot reach them. The material values generated in that infinite space are then transformed back to the space between one and zero and incorporated in each iteration. 

Benchmark tests show that SiMPL requires up to 80% fewer iterations to arrive at an optimal design compared to traditional algorithms. That translates into vastly less computing time — potentially shrinking iteration time from days to hours in many cases. That could make topology optimization accessible for a broader range of industries or enable designs at much finer resolution than is currently feasible, the researchers say. 

The team has made of version of the algorithm freely available for engineers and other researchers to use. 

“While the mathematical theory behind this algorithm is quite complicated, it’s actually quite simple to incorporate into standard topology optimization methods with just a few lines of code,” said Dohyun Kim, a postdoctoral researcher at Brown and lead author of the studies describing the work. “We think this could be quite impactful in the engineering community.”

The research was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory (DE-AC52-07NA27344) and the LLNL-LDRD Program (22-ERD-009 and 25-ERD-030, LLNL-JRNL-871320, 22-ERD-009, 25-ERD-030. Keith and Kim received support from U.S. Department of Energy Office of Science Early Career Research Program (DE-SC0024335).