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

Rafael Pastrana, Patrick Ole Ohlbrock, Thomas Oberbichler, Pierluigi D'Acunto, Stefana Parascho

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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 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 in 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.

Original languageEnglish
Article number103435
JournalCAD Computer Aided Design
StatePublished - Feb 2023


  • Automatic differentiation
  • Combinatorial equilibrium modeling
  • Design tool
  • Form-finding
  • Shape optimization
  • Structural design


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