2017 | Book

# Discrete Optimization in Architecture

## Extremely Modular Systems

Author: Machi Zawidzki

Publisher: Springer Singapore

Book Series : SpringerBriefs in Architectural Design and Technology

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This book is comprised of two parts, both of which explore modular systems: Pipe-Z (PZ) and Truss-Z (TZ), respectively. It presents several methods of creating PZ and TZ structures subjected to discrete optimization. The algorithms presented employ graph-theoretic and heuristic methods. The underlying idea of both systems is to create free-form structures using the minimal number of types of modular elements. PZ is more conceptual, as it forms single-branch mathematical knots with a single type of module. Conversely, TZ is a skeletal system for creating free-form pedestrian ramps and ramp networks among any number of terminals in space. In physical space, TZ uses two types of modules that are mirror reflections of each other. The optimization criteria discussed include: the minimal number of units, maximal adherence to the given guide paths, etc.

#### Pipe-Z

##### Chapter 1. Introduction
Abstract
This chapter introduces the concept of Extremely Modular System (EMS), and presents an example—Pipe-Z (PZ). Although PZ is a parametric design system comprised of only one module, it allows to create complex spatial single-branch structures, represented here by mathematical knots. The Pipe-Z module (PZM) is introduced and its parametrization is explained and illustrated. An algorithm for automated PZ structure generation based on alignment of PZMs along given spatial curves is introduced. The procedure is illustrated with trefoil, pentafoil and figure-eight knot. The problem of self-intersections and minimization of diversity of the twist angles are briefly discussed.
Machi Zawidzki
##### Chapter 2. Pipe-Z Optimization
Abstract
This chapter outlines the optimization methodology for a single-branch tubular structure created with Pipe-Z system. In this process the geometrical properties of the Pipe-Z module (PZM), and the shape of the entire structure comprised of PZM replicas are simultaneously optimized. The total number of modules to follow a given guide path is minimized under constraints. The algorithm is illustrated with planar simplification of the three-dimensional problem. The results produced by random search are presented and discussed. Additionally, the search domain has been sampled and visualized. Finally, as an example for three-dimensional shape optimization, $$6_{3}$$ knot is constructed with the minimal number of congruent modules.
Machi Zawidzki
##### Chapter 3. Pipe-Z Manipulatives
Abstract
This chapter presents a number of virtual and physical Pipe-Z (PZ) manipulatives in two closely related contexts:
• PZ as an non-intuitive construction system which requires certain “sensory augmentation”;
• PZ as a manipulator for haptic communication or teaching aid.
Machi Zawidzki
##### Chapter 4. Arm-Z
Abstract
This chapter presents Arm-Z—a concept of a kinematic system composed of congruent modules (PZM*s) and capable of complex movements. Three fundamental spatial movements of the modular arm are explained and illustrated: extension, translation, and flexure.
Machi Zawidzki
##### Chapter 5. Deployable Pipe-Z
Abstract
This chapter presents a concept of deployable Pipe-Z (dPZ). dPZ is a modular structural system taking advantage of the robustness of rigid-panel mechanisms. It supports creation of free-form connectors which are reconfigurable and deployable. The folding mechanisms of: the single foldable Pipe-Z module (fPZM), and entire dPZ are explained. Folding mechanism of dPZ is illustrated with asynchronous folding of a relatively complex spatial Unknot. A low-fidelity prototype of a six-module octagonal dPZ is presented; several folding schemes including concentric toric rings are demonstrated. “Outside-in” and “inside-out” deployment schemes are demonstrated and discussed in the context of packing. Low-fidelity prototype is presented.
Machi Zawidzki

#### Truss-Z

##### Chapter 6. Introduction
Abstract
Truss-Z (TZ) is another example of Extremely Modular Systems (EMS). TZs are comprised of a single Truss-Z module (TZM) subjected to affine transformations (mirror reflection, rotation, and their combinations). These transformations produce four variations of TZM which allow to create complex three-dimensional linkages. TZ is a self-supporting skeletal system for pedestrian traffic. It is intended as a universal and practical system for new installations, but most importantly, for retrofitting. TZs are particularly practical where the use of heavy equipment is limited, uneconomic or impossible. Moreover, TZ also supports automated generation of optimal three-dimensional connectors where the only required inputs are: the positions of the terminals and the geometrical informations of the obstacles. Firstly, the issue of the conflict between modularity and free-from in architectural and structural engineering is addressed. Secondly, the concept of TZ as a ramp system and TZM are described. Thirdly, preliminary static analysis of TZM is given, followed by the concept of foldable TZ system and presentation of various fabrication methods of TZ physical reduction scale models.
Machi Zawidzki
##### Chapter 7. Single-Branch Truss-Z (STZ)
Abstract
This chapter describes various methods of creating single-branch Truss-Z (STZ) structures. First, the alignment to the given path is described, followed by backtracking-based method illustrated with the Case Study I. Next, various evolutionary algorithms are implemented for optimization of STZ illustrated with the Case Study II.
Machi Zawidzki
##### Chapter 8. Multi-branch Truss-Z (MTZ)
Abstract
This chapter introduces the concept of multi-branch Truss-Z (MTZ). First, the creation of MTZ networks based on alignment to multiple guide paths is described. Next, the optimization method based on Evolution Strategy is described and illustrated with the Case Study III.
Machi Zawidzki
##### Backmatter
Title
Discrete Optimization in Architecture
Author
Machi Zawidzki