A framework for structural shape optimization based on automatic differentiation, the adjoint method and accelerated linear algebra

Abstract

Shape optimization is of great significance in structural engineering, as an efficient geometry leads to better performance of structures. However, the application of gradient-based shape optimization for form finding is limited, which is partly due to the difficulty and the complexity in gradient evaluation. In this work, an efficient framework based on automatic differentiation (AD), the adjoint method and accelerated linear algebra (XLA) is proposed to promote the implementation of gradient-based shape optimization for form finding. The framework is realized by the implementation of the high-performance computing (HPC) library JAX. We leverage AD and the adjoint method in the sensitivity analysis stage. The XLA feature is exploited by an efficient programming architecture that we proposed, which can boost gradient evaluation via just-in-time compilation of computer programs. The proposed framework also supports hardware acceleration such as GPUs. The framework is applied to the form finding of arches and different free-form gridshells: gridshell inspired by Mannheim Multihalle, four-point supported gridshell, and canopy-like structures. Two geometric descriptive methods are used: non-parametric and parametric description via Bézier surface. Unconstrained and constrained shape optimization problems are considered, where the former is solved by gradient descent and the latter is solved by sequential quadratic programming (SQP). Through these examples, the proposed framework is shown to be able to provide structural engineers with an efficient tool for form finding, enabling better design for the built environment.

Appendices

Appendix A: code example—derivative calculation using JAX for customized python objects

The code snippet (Code 3) illustrates how the JAX methods work by presenting a simple problem: calculating how the length of a 2D line changes with the nodal coordinates, i.e., the gradient of the length with respect to the nodal coordinates of the line. Firstly, a new class that represents a line in 2D world is defined and its attributes are the nodal coordinates. The line() class is registered to JAX by @register_pytree_node_class, tree_flatten and tree_unflatten. A function called L() is firstly defined, outputting the distance between two 2D points. Note that jax.numpy is used in the function L() for calculating distance instead of numpy, which is of great significance because we need to trace the operations so that the derivatives can be obtained through automatic differentiation. An external function called line_sens() is then defined for line() objects to calculate the derivatives of its length with respect to nodal coordinates. This original line_sens() works for a single line() object that represents one line. 100 lines with random coordinates are then created. We then implement jit to compile the line_sens() function into optimized machine codes. Lastly, we use vmap to vectorize the line_sens() function so that it can be applied to a single line() object that represents multiple lines.

Appendix B example codes in JAX-SSO