Constrained Form-Finding of Tension–Compression Structures using Automatic Differentiation

Abstract

This paper proposes a computational approach to form-find pin-jointed bar structures subjected to combinations of tension and compression forces. The generated equilibrium states can meet additional structural and geometrical constraints via gradient-based optimization. We achieve this by extending the combinatorial equilibrium modeling (CEM) framework in three important ways. First, we introduce a new topological object, the auxiliary trail, to expand the range of structures that can be form-found with the framework. Then, we leverage automatic differentiation (AD) to obtain an exact value of the gradient of the sequential and iterative calculations of the CEM form-finding algorithm, instead of a numerical approximation. Finally, we encapsulate our research developments into an open-source design tool written in Python that is usable across different CAD platforms and operating systems. After studying four different structures –a self-stressed tensegrity, a tree canopy, a curved bridge, and a spiral staircase– we demonstrate that our approach enables the solution of constrained form-finding problems on a diverse range of structures more efficiently than in previous work. • A computational form-finding method is extended to enhance the form generation of spatial structures in static equilibrium subjected to tension and compression forces. The output forms can satisfy combinations of force and geometric constraints via gradient-based optimization. • A new topological modeling object, the auxiliary trail, is presented which simplifies the representation of a structure as a valid input to the method herein extended. • Automatic differentiation (AD) is leveraged to obtain an exact value of the gradient of a sequential and iterative form-finding algorithm with marginal human intervention. The application of AD leads to significant computational performance gains, especially when solving large constrained form-finding problems. • A standalone, open-source design tool is developed which enables the formulation and the solution of constrained form-finding problems in plain and simple Python code. The design tool is readily usable on three different operating systems and on three distinct 3D modeling software environments.