# Geometry of Structures

### The constrained geometry of structures:

Optimization methods for inverse form-finding design

###### Pierre Cuvilliers

Submitted to the Department of Architecture

in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Architecture: Building Technology

Massachusetts Institute of Technology

May 2020

This dissertation by Pierre Cuvilliers aims to improve form-finding workflows by giving more control on the obtained shapes to the designer. Traditional direct form-finding allows the designer to generate shapes for structures that need to verify a mechanical equilibrium when built; however, it produces shapes that are difficult to control.

This dissertation shows how the design of constrained structural systems is better solved by an inverse form-finding process, where the parameters and initial conditions of the direct form-finding process are automatically adjusted to match the design intent. By defining a general framework for the implementation of such workflows in a nested optimizer loop, the requirements on each component are articulated.

The inner optimizer is a specially selected direct form-finding solver, the outer optimizer is a general-purpose optimization routine. This is demonstrated with case studies of two structural systems: bending-active structures and funicular structures.

These two systems that can lead to efficient covering structures of long spans. For bending-active structures, the performance (speed, accuracy, reliability) of direct form-finding solvers is measured. Because the outer optimization loop in an inverse form-finding setup needs to rely on a robust forward simulation with minimal configuration, author finds that general-purpose optimizers like SLSQP and L-BFGS perform better than domain-specific algorithms like dynamic relaxation.

Using this insight, an inverse form-finding workflow is built and applied with a closest-fit optimization objective. In funicular structures, this dissertation first focuses on a closest-fit to target surface optimization, giving closed-form formulations of gradients and hessian of the problem. Finding closed-form expressions of these derivatives is a major blocking point in creating more versatile inverse form-finding workflows.

This process optimizer is then reimplemented in an Automatic Differentiation framework, to produce an inverse form-finding tool for funicular surfaces with modular design objectives. This is a novel way of implementing such tools, exposing how the design intent can be represented by more complex objects than a target surface.

Reproducing existing structures, and generating more efficient funicular shapes for them, the possibilities of the tool are demonstrated in exploring the design space and fine-tuned modifications, thanks to the fine control over the objectives representing the design intent.

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