Automatic Differentiation in Dynamic Topology Optimization

Recent advances in scientific programming, particularly with regards to topology optimization and machine learning, have necessitated computational methods that generate gradients for optimization. One method to do so is to utilize a tool called automatic differentiation, a mechanism to algorithmically calculate derivatives of functions and combine them to generate gradients of compositions of functions. Code bases such as Jax and PyTorch (which particularly focused on machine learning applications) have demonstrated the ability to scale automatic differentiation to large problems. This allows for rapid gradient calculations, leading to reduced development time as well as significantly higher complexity in the equations used to study a physical phenomenon. While these methods have been studied in the context of machine learning, these approaches have only been applied to mechanics in a handful of cases. This presents an opportunity to study a large variety of optimization problems, such as topology, material parameters, or initial-value problems using this automatic differentiation infrastructure. By taking advantage of the sequential structure of time dependent problems, we develop rapid algorithms for gradient calculations that can be utilized in a large variety of different contexts. We demonstrate the efficacy of these algorithms by studying various dynamics problems. One example is in designing the initial velocity profile of an elastic wave to generate a specific final state. This can be utilized to amplify or reduce particular wave characteristics in non-trivial ways. This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.