Structural and Multidisciplinary Optimization 2024

Optimal design of frame structures with mixed categorical and continuous design variables

Using the Gumbel–Softmax method

Optimum design of a 258-bar bridge structure with 258 categorical and 445 continuous design variables.


In optimizing real-world structures, due to fabrication or budgetary restraints, the design variables may be restricted to a set of standard engineering choices. Such variables, commonly called categorical variables, are discrete and unordered in essence, precluding the utilization of gradient-based optimizers for the problems containing them. In this paper, incorporating the Gumbel–Softmax (GSM) method, we propose a new gradient-based optimizer for handling such variables in the optimal design of large-scale frame structures. The GSM method provides a means to draw differentiable samples from categorical distributions, thereby enabling sensitivity analysis for the variables generated from such distributions. The sensitivity information can greatly reduce the computational cost of traversing high-dimensional and discrete design spaces in comparison to employing gradient-free optimization methods. In addition, since the developed optimizer is gradient-based, it can naturally handle the simultaneous optimization of categorical and continuous design variables. Through three numerical case studies, different aspects of the proposed optimizer are studied and its advantages over population-based optimizers, specifically a genetic algorithm, are demonstrated.

Download publication
“This work presents a novel optimization approach using the Gumbel-Softmax method for optimal design of frame/truss structures with mixed categorical and continuous design variables.”

Mehran Ebrahimi

Principal Computational Physics Research Scientist

Other Resources

Related Publications



BeNTO: Beam Network Topology Optimization

We present an optimization framework that allows designers and…



A low order, torsion deformable spatial beam element based on the absolute nodal coordinate formulation and Bishop frame.

Heretofore, the Serret–Frenet frame has been the ubiquitous choice…



Configuration Design of Mechanical Assemblies using an Estimation of Distribution Algorithm and Constraint Programming

A configuration design problem in mechanical engineering involves…

Get in touch

Something pique your interest? Get in touch if you’d like to learn more about Autodesk Research, our projects, people, and potential collaboration opportunities.

Contact us