Content uploaded by Roberto Naboni

Author content

All content in this area was uploaded by Roberto Naboni on Mar 24, 2018

Content may be subject to copyright.

Handbook of Research on

Form and Morphogenesis

in Modern Architectural

Contexts

Domenico D’Uva

Politecnico di Milano, Italy

A volume in the Advances in Media,

Entertainment, and the Arts (AMEA) Book Series

Published in the United States of America by

IGI Global

Information Science Reference (an imprint of IGI Global)

701 E. Chocolate Avenue

Hershey PA, USA 17033

Tel: 717-533-8845

Fax: 717-533-8661

E-mail: cust@igi-global.com

Web site: http://www.igi-global.com

Copyright © 2018 by IGI Global. All rights reserved. No part of this publication may be reproduced, stored or distributed in

any form or by any means, electronic or mechanical, including photocopying, without written permission from the publisher.

Product or company names used in this set are for identification purposes only. Inclusion of the names of the products or

companies does not indicate a claim of ownership by IGI Global of the trademark or registered trademark.

Library of Congress Cataloging-in-Publication Data

British Cataloguing in Publication Data

A Cataloguing in Publication record for this book is available from the British Library.

All work contributed to this book is new, previously-unpublished material. The views expressed in this book are those of the

authors, but not necessarily of the publisher.

For electronic access to this publication, please contact: eresources@igi-global.com.

Names: D’Uva, Domenico, 1975- editor.

Title: Handbook of research on form and morphogenesis in modern architectural

contexts / Domenico D’Uva, editor.

Description: Hershey, PA : Information Science Reference (an imprint of IGI

Global), 2018. | Includes bibliographical references and index.

Identifiers: LCCN 2017026110| ISBN 9781522539933 (hardcover) | ISBN

9781522539940 (ebook)

Subjects: LCSH: Architectural design. | Space (Architecture)

Classification: LCC NA2750 .A615 2018 | DDC 729--dc23 LC record available at https://lccn.loc.gov/2017026110

This book is published in the IGI Global book series Advances in Media, Entertainment, and the Arts (AMEA) (ISSN:

2475-6814; eISSN: 2475-6830)

69

Copyright © 2018, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Chapter 4

DOI: 10.4018/978-1-5225-3993-3.ch004

ABSTRACT

This chapter illustrates the main approach for a generative use of structural optimization in architec-

ture. Structural optimization is very typical of sectors like mechanical, automotive engineering, while in

architecture it is a less used approach that however could give new possibilities to performative design.

Topology Optimization, one of its most developed sub-methods, is based on the idea of optimization of

material densities within a given design domain, along with least material used and wasted energy. In

the text is provided a description of TO methods and the principles of their utilization. The process of

topology optimization of microstructures of cellular materials is represented and illustrated, emphasiz-

ing the all-important criteria and parameters for structural design. A speciﬁc example is given of the

research at ACTLAB, ACB Dept, Politecnico di Milano, of performative design with lattice cellular solid

structures for architecture.

INTRODUCTION

The development of computational tools and design methods nowadays influence highly the development

of new forms of architectural design. Designers are accessing easily tools which allow to impregnate

their design decisions with an overlook on technical questions. Integrative design approaches are in a

process of consolidation and design thinking has moved gradually towards the idea of form-finding,

where performances are integrated not merely into an evaluative discourse, but as generative parameters.

Among the vast proliferation of available tools, there is a plethora of algorithmic-based solutions

for including structure-related enquiries in the design process. Among them, Topology Optimization

(TO) methods have emerged as an interesting approach to evaluate form and structure jointly, follow-

ing the example of aerospace and automotive industry, where the optimization of mechanical parts is

Architectural Morphogenesis

Through Topology Optimization

Roberto Naboni

Politecnico di Milano, Italy

Ingrid Paoletti

Politecnico di Milano, Italy

70

Architectural Morphogenesis Through Topology Optimization

performed in connection with an efficient material usage. In civil engineering, TO is a known approach

for the optimization of structural elements, such as beam, columns, or truss layout, with an emphasis

put on structural and material optimization (Amir and Bogomolny, 2011). In this chapter is discussed

how TO can be applied conveniently for the morphogenesis of a structural design, in an effort to look at

this technique not only under the perspective of the material optimization, but with an accent put on the

potential of creating shapes which can complement and inspire the generation of architecture.

BACKGROUND ON STRUCTURAL OPTIMIZATION

In structural engineering, a main goal is developing load-bearing systems which satisfy economically

the design performance objectives and safety constraints. Economical consideration is often the main

driver for developing a design process that enables the minimization of resource consumption. In doing

so, an important concept is the one of optimization, that refers to the selection of the best element from

some set of available alternatives (Radman, 2013). Optimality conditions of structural systems have been

introduced first in 1901 by Anthony Michell in his theoretical study The Limit of Economy of Material

in Frame-Structures” (Michell, 1904). What Michell claimed is a continuous displacement field with

equal and opposite principal strains, considered as limit strains of the material in compression and ten-

sion. If in a particular problem it is possible to design a structure all of whose members are in tension,

or alternatively compression, then the optimum design is achieved, since all the members of a truss are

laid along these principal strain lines. Also, the tension and compression members that meet at a node

must be orthogonal, since they lie along principal directions with unequal principal strains e and -e.

The work of Michell was remarkable because he achieved these results without any prior work on

optimization theory, but largely based on intuition. However, it suffers some limitations: it is limited to

planar structures, distributed loads are not included, loads are applied only on the boundary, the man-

ner by which a truss approaches the infinitely refined limit is not addressed, and consequently the exact

relationship between the limit and the underlying discrete truss structures is unclear.

Wider access to computational work in 1990s justified the development of numerical procedures for

the TO of structures, aimed at finding the best layout, configuration and spatial distribution of materials

in the design domain of the continuum structure (Bendsøe and Kikuchi, 1988).

Figure 1. Loads in the members of the Michell cantilever due to a unit load at the tip

71

Architectural Morphogenesis Through Topology Optimization

Structural optimization is typically referring to three approaches focusing on sizing, shape and topol-

ogy optimization, where each one of them address different aspects of the structural design problem. In

a typical sizing problem, the goal is to find the optimal thickness distribution (e.g. the optimal member

areas in a truss structure). The optimal thickness distribution minimizes (or maximizes) a physical quan-

tity such as the external work, peak stress, deflection, etc. while equilibrium and other constraints on the

state and design variables are satisfied. The main feature of the sizing problem is that the domain of the

design model and state variables is known a priori and is fixed throughout the optimization process. On

the other hand, in a shape optimization problem the goal is to find the optimum shape of this domain,

that is, the shape problem is defined on a domain which is now the design variable. TO of solid struc-

tures involves the determination of features such as the number and location and shape of holes and the

connectivity of the domain (Bendsøe and Sigmund, 2003).

One of the most effective way to optimize a structure is by finding ideal topology configurations which

suggests a layout of structural members and eventually a specific material distribution given specific

design conditions. Topology is a major area of mathematics concerned with properties that are preserved

under continuous deformations of objects. TO in the context of structural problems is a mathematical

Figure 2. Michell’s optimality analysis of structural systems are based mostly on intuition, with no prior

work, but they are applied only on two dimensional structures and considering just loads applied on

the boundaries

72

Architectural Morphogenesis Through Topology Optimization

approach that optimizes material layout within a given design space, for a given set of loads and boundary

conditions such that the resulting layout meets a prescribed set of performance targets (Radman, 2013).

Moreover, the responses of structural systems to the external loading conditions are highly dependent

on the material they are built from. Therefore, besides the algorithmic studies of structural analysis, in

the last few decades, a special interest in composite materials and materials with tailored or improved

properties has emerged. Lightweight materials are being studied and developed in order to respond to

both mechanical and functional requirements: mechanical in terms of load carrying and structural ef-

ficiency and the functional ones in terms of improved thermal, optical and chemical properties.

Strategies for TO are not a mere tool for calculus, but an aid to generate structurally sound architecture.

The output of TO is strictly morphogenetic, it suggests actual forms for a structure in relation to a defined

spatial boundary and requires the needed integration of the structural problem with the architectural one.

Figure 3. Two examples of topology design for minimum compliance compared with optimal Michell

type structures (Michell, 1904). a) and b) design domains, c) and d) topology optimized solutions and

e) and f) corresponding Michell type optimal solutions (from Sigmund, 2000)

73

Architectural Morphogenesis Through Topology Optimization

SOLUTION METHODS IN TOPOLOGY OPTIMIZATION

TO has the purpose to find efficient structural layouts, usually within a predetermined region. In this

process, the designer is responsible for defining the so-called boundary conditions, such as the applied

external loads, the support conditions, the volume of the bounding box including the future structure and,

eventually, some additional design restrictions such as the location and size of prescribed holes or solid

areas. One main aspect of structural topology design is the determination of the optimal organisation of

a specific material in space most often according to structural compliance or stiffness, and sometimes

Von Mises stress values. In particular, an algorithmic procedure defines if any point in space should

be assigned with material or not, according to its mechanical utility. A typical representation of such

structural analysis is similar to a black-white raster representation with pixels or voxels given by the

finite element discretization.

This problem is defined in mathematical terms as follow: within the spatial domain Ω, the main goal

is seeking to determine the optimal subset Ω mat of material points. For a defined optimization problem,

this approach implies that the set of admissible stiffness tensors (Ead) consists of the tensors for which:

Eijkl = 1Ω mat E0ijkl, 1Ω mat = 1 if x ∈ Ω mat

⎨ 0 if x ∈ Ω \ Ω mat

∫Ω 1Ω mat dΩ = Vol (Ω mat) ≤ V

The presented inequality expresses a limit, V, on the amount of material at a specific disposal, so that

the minimum compliance design is for a limited (fixed) volume. The tensor E0ijkl is the stiffness tensor

Figure 4. Three categories of structural optimization. a) Sizing optimization of a truss structure, b) shape

optimization and d) topology optimization. The initial problems are shown at the left-hand side and the

optimal solutions arc shown at the right (Bendsøe and Sigmund, 2003)

74

Architectural Morphogenesis Through Topology Optimization

for the given material. This definition of Ead means that a distributed, discrete valued design problem

(a 0-1 problem) has been formulated. The design variable x indicates the presence (1) or absence (0)

of the element, similar to formulation for the pointwise material/no material also known as black/white

optimization.

The most commonly used approach to solve this problem is to replace the integer variables with

continuous variables and then introduce some form of penalty that steers the solution to discrete 0-1

values. The design problem for the fixed domain is then formulated as a sizing problem by modifying

the stiffness matrix so that it depends continuously on a function which is interpreted as a density of

material. This function then becomes the design variable.

The requirement is that the optimized design results consist almost entirely of regions of material

(1) or no material (0). This means that intermediate values should be penalized in a manner similar to

other optimization approximations of 0-1 problems (Bendsøe and Sigmund, 2003).

Figure 5. A ‘1-0’ optimization problem; For each point in space should be decided if contains (1) or

not (0) the material

75

Architectural Morphogenesis Through Topology Optimization

Eijkl (x) = 𝜌(x)p E0ijkl, p > 1,

∫Ω 𝜌(x) dΩ ≤ V; 0 ≤ 𝜌(x) ≤ 1, x ∈ Ω

Here the “density” 𝜌(x) is the design function and E0ijkl represents the material properties of a given

material. The density interpolates between the material properties 0 and and E0ijkl:

Eijkl (𝜌 = 0) = 0,

Eijkl (𝜌 = 1) = E0ijkl

meaning that if a final design has density zero or one in all points, this design is a black-and-white design

for which the performance is evaluated with a correct physical model.

In cases where p > 1 so that intermediate densities are unfavourable in the sense that the stiffness

obtained is small compared to the cost (volume) of the material. In other words, by specifying a value

of p higher than one makes it “uneconomical” to have intermediate densities in the optimal design.

Thus, the penalization is achieved without the use of any explicit penalization scheme. We note that the

original “0-1” problem is defined on a fixed reference domain and this means that the optimal topology

problem takes on the form of a standard sizing problem on a fixed domain.

If a numerical scheme leads to black-and-white designs one can in essence choose to ignore the

physical relevance of intermediate steps which may include “grey”. However, the question of physical

relevance is often raised, especially as most computational schemes involving interpolations do give rise

to designs which are not completely clear of “grey”. Also, the physical realization of all feasible designs

plays a role when interpreting results from a premature termination of an optimization algorithm that,

has not converged fully to a 0-1 design (Bendsøe and Sigmund, 2003).

EXAMPLES OF APPLICATION OF TOPOLOGY

In the last years, architects started to have an interest in exploring TO as a morphogenetic process of

design, experimenting with it at different scales ranging from creating complex joinery systems for

complex structures to generating wide-span architecture. In general, this approach allows for solutions

beyond common structural typologies and are sustainable in terms of material usage.

Pioneering Projects of Arata Isozaki

A fundamental project in the use of TO for architectural design is from Arata Isozaki, in collabora-

tion with structural engineer Matsuro Sasaki for a project called Illa de Blanes at the seaside of Blanes

(Costa Brava, Spain) developed in the years 1998-2002 (Januszkiewicz 2013). This was one of the first

attempts to generate forms obtained from Topology Optimisation algorithms as an architectural form.

An enormous complex, covering 75.000 square meters, has a roof supported by a large structure gener-

ated by ESO algorithm for TO.

76

Architectural Morphogenesis Through Topology Optimization

The building was characterized by tree like, organic shaped columns, and a doubly curved roof cre-

ated with the application of TO. The project, never built, inspired the design proposal for the largest

train station in Italy - Santa Maria Novella in Florence. Japanese designers proposed a huge structure

generated by Topology Optimisation algorithms, namely 3D Extended Evolutionary Structure Optimisa-

tion (Januszkiewicz 2013). A 400 meters long and 42 meters wide flat roof, designed as a land strip for

lightweight aircrafts, elevated twenty meters above the ground, was supported by massive columns in a

few points. On the ground, those columns had only four main roots to grow from.

The vision of application of engineering tools, such as Topology Optimisation in architectural design,

finally was realized in 2008 for the Qatar National Convention Centre (QNCC) in Doha. This became

the occasion for Isosaki to implement his innovative vision of architecture driven by engineering com-

putational methods, in cooperation with Buro Happold. The design is mainly based on a supported 250

meter long and 110 m wide lobby roof, which is by now the largest structure ever created with tools for

TO (Zwierzycki 2013).

3D Printed Steel Joints

A team lead by Arup has developed a technique for realizing 3D printing structural steel joints in a project

revealed in 2014 in collaboration with WithinLab (engineering design software and consulting company),

CRDM/3D Systems (expert in additive manufacturing) and EOS, which was involved mainly in the

early stages of development. The pioneering proposal by Arup represents a solution for steel nodes in

lightweight tensile structures characterized by complex shapes and customized design. Generally shaped

elements were optimised by SIMP method and TO resulted in an organic form using less material while

the original functions as cable connectors are still ensured (Block et al. 2015). Compared to traditional

Figure 6. Qatar National Convention Centre (QNCC), 2008

77

Architectural Morphogenesis Through Topology Optimization

design, presented approach not only provide optimisation of material, but also caused an increase of

node stiffness. The developed process is based on the principle of additive laser sintering, employing

steel derivatives as the printing material. The structural nodes have been originally conceived in stain-

less steel and later produced with steel, compatible with the technology of the machine owned by the

partner CRDM. This material is about four times stronger than normal construction steel, which made

Arup eager to experiment with it and further explore its potentials. EOS, the additive manufacturing

expert involved in the project, reported that this technological solution guarantees a 40 percent reduction

of CO2 emissions over the whole lifecycle in respect to traditional casting processes. Furthermore, the

process of direct metal laser sintering (DMSL) satisfies many design requirements, reducing weight and

preserving geometrical freedom. Due to the nature of additive manufacturing techniques, the production

of waste materials is minimized and the weight of the final product reduced by 30%. In order to verify

and improve this method, testing prototypes were scaled down to 40 per cent of the original size, thus

being 14 centimeters high, without compromising the structural properties of the joints. Arup imagined

developing the technology in the application of large sculptures, as an intermediate test before using it

in buildings (Arup, 2014). An important consideration is that larger machines are currently being engi-

neered and special hybrid materials being developed, thus the building industry should soon be able to

answer to almost any specific demand of the designers and clients within the construction field.

Unikabeton Project

An interesting interdisciplinary project, led by the Aarhus School of Architecture was developed in 2007,

as investigation of TO for concrete structures. The project aimed at discussing an integral approach in

which the generative design was essentially supported by a fabrication setup of robotic CNC-milling of

EPS moulds. The project, which finally ended up with the construction of a full-scale prototype, unveiled

important findings. Firstly, TO proved to be fundamental in reducing material consumption up to 70% in

comparison with massive equivalent structures subjected to loading conditions and requirements from

Figure 7. Arup’s 3D printed structurally optimized steel joints

78

Architectural Morphogenesis Through Topology Optimization

normatives. Secondly, the emergent structural design proved a new tectonic language where the natural

load path is immediately visible. The result of the material optimization is indeed a complex shape which

would be difficult to manufacture with typical methods. With the use of robot CNC milling the project

investigated how to realize efficiently such structures, with high precision and ease of mould construc-

tion. To prove this approach, a concrete structure of 12 by 6 by 3.3 meter was conceived, designed and

built, in the form of an asymmetrical, doubly curved slab structures with a three columns support. The

shape resulting from the TO optimization was then remodelled and used as negative form to generate

the EPS moulds forms to be cut. The complex shape, required the use self-compacting concrete and the

use of steel reinforcements. This research is by now a fundamental milestone, in the advancement of TO.

New structural shapes are created, in a generative way which aims at material reduction. Fabrication is

here thought to support this design technique into construction.

Figure 8. The concrete prototype structure of Unikabeton Project

79

Architectural Morphogenesis Through Topology Optimization

TOPOLOGY OPTIMIZATION IN PERIODIC BASE CELLS

An interesting opportunity arising from the use of TO, is the one of optimizing the material organization

at material or micro scale. In essence, the problem of material organization is highly interscalar, and

different studies have focused on extending this to Periodic Base Cell (PBC) which represents a hetero-

geneous continuum structure, which is comprised of different constituent materials or phases (Bendsøe

and Sigmund, 2003). The topology of the PBC is what influences the properties of materials. Hence

the major challenge in the design of these materials would be the determination of the optimal spatial

distribution of the constituent materials within the PBC. In the simplest form, the periodic composite

materials consist of a 2D or 3D scaffold of matrix, in which the other phases are included. Therefore,

it is reasonable to apply the structural topology optimization methodologies for determination of the

spatial distribution of the phases (Radman, 2013).

Materials with repeating or periodic microstructures usually consist of one constituent phase and a

void phase (known as porous or cellular materials), or combinations of two or more different constituent

phases with or without the void phase (also named as “periodic composites”). The overall properties

of these type of materials are controlled by the spatial distribution of the constituent phases within the

PBC, as well as the properties of constituent phases. In comparison with traditional composites, peri-

odic composites demonstrate greater flexibility in terms of their capability to be tailored for prescribed

physical properties, by controlling the compositions and/or microstructural topology of the constituent

phases. They can also be easily tailored to have gradation in their functional properties, in the form of a

functionally graded material (FGM) through gradual changes in the microstructural topologies.

TOPOLOGY OPTIMIZATION OF LATTICE STRUCTURES

AND CELLULAR MATERIALS

Lightweight cellular materials might be characterised by advanced physical, mechanical and thermal

properties that extend far beyond those of solid materials. The physical characteristics of materials can

vary by changing the materials distribution within their microstructure. To make the best use of resources,

the spatial distribution of constituent phases within the microstructures can be defined by using topology

optimization techniques (Radman, 2013). These types of cellular solids are of interest in architecture

due to their high structural stiffness, high strength-to-weight ratio, low energy absorption, good thermal

conductivity and acoustic insulation.

Although the structural weight is not generally a functional property, it might happen to be one of

the important design factors. It is assumed that the material is composed of Periodic Base Cell (PBC),

which is the smallest repeating unit of material. The dimensions of the base cells are assumed to be

much less than the overall length scales of the material body, and at the same time much larger than the

atomic length scale (Radman, 2013). PBC are discretized into a finite elements model under periodic

boundary conditions. The Finite Element Analysis is performed to extract necessary information for

calculation of the effects of individual elements within the PBC, on the variation of homogenized (aver-

age) properties of material.

The stiffness of an elastic material can be described by the bulk modulus K or shear modulus G. In

the design setting the aim is to define cellular materials with the maximum effective bulk modulus or

shear modulus subject to a prescribed weight. Therefore the topology optimization problem is to find

80

Architectural Morphogenesis Through Topology Optimization

the appropriate distribution of the solid phase within the PBC, subject to a prescribed volume fraction

of the solid phase (Radman, 2013).

LATTICE MICROSTRUCTURED ARCHITECTURE

A specific type of cellular solids, the lattice microstructure, is here analyzed and developed as innovative

structural system for architectural applications. The potential of this system relies on its implicit resistance

and reduced use of material, combined with the possibility to adapt to a large variety of architectural

shapes. Lattice microstructures are considered both as a structure and as a material. They are composed

by an interconnected network of struts, pin-jointed or rigidly bonded at their connections. At one level,

they can be analyzed using classical methods of mechanics, as typical space frames. On the other side,

within a certain scale range, lattice can be considered as a material, with its own set of effective proper-

ties, allowing direct comparison with homogeneous materials. Mechanical properties of lattice materials

are governed, in part, by those of the material from which they are made, but most importantly by the

topology and relative density of the cellular structure.

Applications of lattice structures in construction are currently of interest of several research and design

groups, often in connection with the study of novel fabrication methods, involving AM and industrial

robotic arms. Some of these experimentations are based on polymer pultrusion in space to create wall

reinforcements (Hack et. al, 2015; Boyd, 2015); others use fibre-reinforced composites to produce modu-

lar struts, assembled by robots (Cheung, 2012); a third approach employs AM for producing sand mold

halves casted with Ultra High Performance Concrete (UHPC) for the realization of three-dimensional

spatial lattices (Morel and Schwartz, 2015).

Figure 9. Top: Array of designed base cells with maximum bulk modulus and various volume fractions

of solids (from left): 50% ; 40% ; 30% ; 20%. Below: array of the base cells with maximum shear modu-

lus with various prescribed volume of solid phase (from left): 45% ; 35% ; 25% (From Radman, 2013)

81

Architectural Morphogenesis Through Topology Optimization

Figure 10. 3D base cells, 2 x 2 x 2 cells and effective elasticity matrices of 3D cellular materials with

maximum shear modulus; volume fraction is (from top): 45% ; 35% ; 25% ; 15% (From Radman, 2013)

82

Architectural Morphogenesis Through Topology Optimization

TO is often used as an early-stage design tool to give the designer an insight into an efficient struc-

tural layout. However, in this work the output of this analysis is used to directly inform the generation

of a continuous lattice microstructure. The TO is fed with two-dimensional free-form shapes, which

represent a “draft” of building envelope configurations to be evaluated, along with a description of

specific boundary conditions such as loads, constraints and material properties. An algorithm based

on the Solid Isotropic Material Penalization (SIMP) iteratively computes stiffness values and allocates

material in a multi-phase process which tends to converge to 0/1 values. These are zones with lowest/

highest density of material, respectively represented with an interpolation of black/white color values

(Bendsøe & Sigmund, 2003). This representation is then converted into a Functionally Graded Lattice

structure, where mechanical behaviours provide the needed information to evolve a base polyhedron into

highly specific cells with locally optimized cell dimensions and orientation, struts diameter and section

as well as material characteristics.

A CASE STUDY ON LATTICE STRUCTURES FROM TO

An experimental skin system based on cellular solids is implemented through the combined use of ad-

vanced computational design tools and Additive Manufacturing (AM). In particular, the study focuses on

the investigation of open cellular solids, based on a lattice structured system, a model that has an efficient

way of structuring material (Gibson, 2005). Fundamentally, cellular lattice structures are composed of

an interconnected network of struts, pin-jointed or rigidly bonded at their connections (Ashby, 2005).

At one level, they can be analyzed using classical methods of mechanics, as space frames. On the other

side, within a certain scale range, lattice can be considered as a material, with its own set of effective

properties, allowing direct comparison with homogeneous materials.

Among the examples of lattice cellular solids, the hierarchical structure of bones is considered as

one of the most prominent examples of lightweight and structurally efficient natural systems. Bones are

made of a composite material that is about 95% hard calcium-based mineral (hydroxyapatite) marbled

with an elastic protein (collagen). The cortical bone makes up the exterior of the bone, while cancellous

bone is found in the interior. This has high material efficiency because of its constitutional microstructure

based on cells named trabeculae, that are formed through an iterative load-responsive process. Here, an

emergent latticework of fibers constitutes a cellular microstructure informed by its loading conditions,

which varies in porosity, and in orientation to align with the main stress trajectories to withstand both

tensile and compressive forces (Benyus, 2002) (Figure 11).

Interestingly, the process of bone remodelling is responsive to variable loading conditions which

an individual can encounter during our life. In particular, this process is subjected to the simultaneous

action of two cells - osteoblasts and osteoclasts, that are evaluating local strain values within the bone

trabecular structure and adding or removing material accordingly. High strain levels indicate that the

bone is weaker than expected and osteoblasts will compensate by adding material in order to reduce

strain. Analogously, excessively low strain levels show an unneeded over-mineralization, and the need for

osteoclasts to remove material. The balance between these two processes therefore provides a converg-

ing point where function and structure are optimized (Turner, 2012). This specific formation process

can be synthesized in an algorithm which constitutes the procedural base for the generation of the load-

responsive cellular envelope tackled in this paper (Figure 12)

83

Architectural Morphogenesis Through Topology Optimization

Figure 11. Section of a human femur bone showing degrees of porosity according to a load-responsive

material organization

Figure 12. Algorithmic interpretation of the bone remodeling process (adapted from J.S. Turner, 2012)

84

Architectural Morphogenesis Through Topology Optimization

The logics of bone formation process are here developed into a computational workflow for the de-

sign, optimization and fabrication of a Cellular lattice-based envelope (Figure 13), an innovative system

for building skins. This methodology requires the description of custom algorithms to generate lattice

structures parametrized on the base of a continuous feedback loop from a Topology Optimization (TO)

and manage the additive process of materialization. In analogy with the bone remodelling process, it is

performed an iterative macroscale mechanical analysis with Finite Elements Methods to compute the

specific behaviour of free-form building envelopes. The outcome of this analysis is then directly translated

into a lattice microstructure which, in common with the bone trabecular structure, orients itself follow-

ing principal stress lines and varying material porosity, according to local stress values. In this process,

main input parameters are material properties and fabrication constraints of AM, overall geometry and

boundary conditions. Variations in any of these parameters generate different lattice structures, as this

research develops a global method for highly specific design, where morphological, material and per-

formative information is read, analyzed and modified iteratively.

The potential of this system relies on its implicit resistance and reduced use of material, combined

with the possibility to adapt to any architectural shape. Cellular lattice structures are composed by an

interconnected network of struts, pin-jointed or rigidly bonded at their connections (Ashby, 2005). At

one level, they can be analyzed using classical methods of mechanics, as typical space frames. On the

other side, within a certain scale range, lattice can be considered as a material, with its own set of ef-

fective properties, allowing direct comparison with homogeneous materials. Mechanical properties of

lattice materials are governed, in part, by those of the material from which they are made of, but most

importantly by the topology and relative density of the cellular structure. In this research, the design

and fabrication of cellular lattice based envelopes is developed starting from the analysis of the mate-

rial system of 3D printing with thermo polymers, implementing its peculiarities into specific cellular

geometries, experimenting different fabrication settings for large scale structures, and finally creating a

composite skin inspired by the tectonic of bones.

Figure 13. Overall workflow scheme

85

Architectural Morphogenesis Through Topology Optimization

MATERIAL SYSTEM OF ADDITIVE MANUFACTURED LATTICE STRUCTURES

Within the field of constructions, the shift from prototyping to direct manufacturing is mainly connected

to material improvement, which in comparison with product design is more complicated to achieve.

Material characteristics and behaviour, mechanical properties and dimensional requirements are key

elements in evaluating the use of AM for large scale applications (Naboni & Paoletti, 2015). Therefore,

the exploration of a material system should be held in order to understand the way it can be exploited,

with a rigorous multi-scalar analysis of the material coupled with the fabrication system that will be

used (Hensel, 2011). This process starts with analyzing the materialization process through fabrication

experiments and the observation of their geometrical and mechanical characteristics. As result, a set

of specific boundary conditions for the fabrication systems, involving machinic, software and material

interdependencies is defined.

In the frame of this research it is used a delta-robot, a typology of printer intrinsically agile that guar-

antees an ideal travel speed for the production of discontinuous geometries such as the lattice structures.

The employed material is High Performance PLA (Polylactic Acid), a polymer with discrete mechanical

properties which are leveraged by its superior printability. An extensive campaign of fabrication tests has

been conducted with it to define print settings in relation to geometric constraints, printing time, printing

resolution and mechanical resistance of the lattice microstructure. Among various aspects, an important

one emerged in the necessity of evaluating models to be printed according to geometry limitations in

overhanging angles, to avoid the need of support geometries with consequent inefficiency in the use of

material. The relation between the deviation angle from the vertical axis and the number and thickness

of shell elements is fundamentally driving the resolution and refinement of the production (Figure 15).

STUDY ON LATTICE CELL TYPOLOGIES

A critical phase in the development of a cellular solid structure is the definition of the base unit cell. In

nature this is direct expression of a material system, which accommodates the biological and mechanical

Figure 14. The scheme shows the generation of the lattice structure based on stiffness factor values

obtained from Topology Optimization

86

Architectural Morphogenesis Through Topology Optimization

needs of an organism. This implies that the above-mentioned geometry constraints of FDM are to be taken

first into account in this evaluation. A comparative multicriteria analysis of typical three-dimensional

cells have been conducted, with an evaluation of printability, relative density and visual permeability.

Eight typologies have been analyzed: orthogonal grid (A), star (B), tesseract (C), octahedron (D), cross

(E), octet (F), vintiles (G) and diamond (H) (Figure 16). Each specimen is bounded in a 10 000 mm3

cube and all the struts have a sectional diameter equal to 10 mm. An analysis of the geometry constraints

has been carried out, focusing on the evaluation of overhanging angles. Considering the XY plane as

the leaning plane, a critical threshold for printability is set at 65° angle deviation from the vertical axis.

Printing angles below this value guarantees production speed and quality, whereas larger angles can

be problematic, in particular with thicker layer heights, as emerged in the description of the material

system. From the cell analysis, the octahedron (D) and diamond (E) cells show optimal features for this

fabrication process.

In the analysis of relative density are highlighted large differences: on one hand, the Octahedron and

Diamond have the lowest relative density of 0.18 and 0.10. On the other hand, cells such as the Octet and

the Tesseract have the highest relative density over 0.50, meaning that more than half of the bounding

box is occupied by the cell struts, resulting in a stiffer but heavier structure. Finally, visual permeability is

measured in respect to the projection of the unit cells on a vertical plane using a 30° angle of view. This

analysis highlights again strong differences among the samples, being the dimensions of the projected

areas ranging from 4 900 mm2 to 12 100 mm2. Considering that the projected area of the bounding box

is 16 600 mm2, the octahedron with its area of 4 900 mm2 obstructs about 1/3 of the visual field, while

the octet cell blocks around ¾ of the view with a projected area of 12 100 mm2. Everything examined,

octahedral cells have proven to be ideal to guarantee a streamlined production while offering a degree

of freedom allowing variable mechanical and visual features.

Figure 15. The scheme shows the geometrical discretization of inclined geometries with FDM. For a

printing configuration with 0.7 mm diameter of nozzle extrusion size, and 0.5 mm extrusion height, angles

larger than 60 degrees from the vertical axis require the use of multiple shell elements.

87

Architectural Morphogenesis Through Topology Optimization

FABRICATION EXPERIMENTS OF CELLULAR COMPONENTS

Cellular structures based on the selected octahedral cells have been subsequently tested in different scale

of fabrication, starting from samples inscribed in a cuboid with length 150 mm. At this scale, different

options of shell thickness and infill patterns have been tested in order to define convenient strength to

weight ratio. In order to evaluate stiffness, lightness and permeability at full scale according to variation in

porosity (relative density), larger samples of cellular structures have been manufactured inscribed within

a 500 mm wide cuboid component (Figure 18). Interestingly, same relative density can be reached with

different cells size, and very different visual perception. These test samples contributed to the definition

of an optimal fabrication resolution, with a measured tolerance of 0.1 mm using an extrusion height

of 0.5 mm and a nozzle diameter of 0.7 mm. This configuration proved to ensure the best compromise

between production precision and printing speed, with an average production time of fifteen hours.

Figure 16. The image shows eight different unit cells typologies for the Cellular Lattice Structure and

their observed characteristics; first column shows unit cell types: A - orthogonal grid, B - star, C - tes-

seract, D - octahedron, E - cross, F - octet, G - vintiles and H - diamond; second column shows the

relative density (ρ), printability (P) and light permeability (L); third column shows the repeated unit cell

in a skin system, highlighting in red elements that are not possible to be fabricated with FDM

88

Architectural Morphogenesis Through Topology Optimization

DESIGN AND FABRICATION OF LOAD-RESPONSIVE CELLULAR STRUCTURES

Findings on the fabrication experiments have been implemented in a larger mock-up, realized implement-

ing the overall design workflow outlined in the methodology section. Starting from the selection of a

free-form shape, this is evaluated under different external loading conditions, added to its self-weight,

obtaining different patterns of material distribution. In the case of this mock-up, it is chosen a configura-

tion emerging from the of dead and live vertical loads (1KN), out of different loading conditions. The

algorithmic workflow generates a grayscale representation of the desired stiffness values, which informs

the sizing of each single strut diameter (Figure 19). A portion of this envelope design is prototyped in

scale 1:1 to address construction aspects of the load bearing structure (Figure 20).

Figure 17. A small-scale sample of cellular structure based on octahedral cells produced with FDM

89

Architectural Morphogenesis Through Topology Optimization

CONCLUSION

This chapter illustrates how optimization algorithms for structural design can be used as an approach

to architectural morphogenesis, with structural efficiency being a design driver. In the text, Topology

Optimization is analyzed as an interesting technique to generate original design while creating efficient

material organization, and explained in its foundational theory and through the use of case studies in

architecture. Finally, a recent laboratory development from ACTLAB resulting in prototypical mock-ups

in the field is illustrated.

AM methods have undoubtedly introduced novel materialization processes, where logics of sustain-

ability and efficiency typical of mass-production are no longer applicable. Unprecedented control, pre-

cision and freedom of this manufacturing allow the conceptualization of unseen architectural systems.

Taking inspiration from the remodeling process of bones, a design methodology based on Topology

Optimization which adapts to different shapes and loading conditions is developed. This experimental

Figure 18. Prototypical components 500 mm wide with different relative density: from left - ρ = 0.04,

ρ = 0.05, ρ = (0.04 to 0.06)

Figure 19. Workflow of the lattice cellular structure generation from the initial shape definition

90

Architectural Morphogenesis Through Topology Optimization

approach challenges current design paradigms of lightweight architecture: complex shapes are neither

pre-optimized by shape, nor post-rationalized to meet manufacturing constraints. The system has been

successfully designed, prototyped and tested in a laboratory setup. This approach can be easily adapted

to the use of metal 3d printing, to offer a more robust material option at current time. However, the

rapid development of thermo polymers for 3d printing with increased chemical, mechanical and weather

resistance, offers interesting perspectives of application with FDM.

Together with TO a novel Architectural tectonic arise with performative design giving to designers

the chance to innovate while informing the whole process of construction. The methodology integrates

a performance assessment in the design phase and optimizes the mechanical behaviour through an ad-

vanced formal articulation.

This new awareness develops new profiles also in the architectural field, where the separation of

phases is often a consequence of tender requirements where different competences are quoted in separated

contracts. The new ability to compute with materials give to designers a new consciousness opening the

path also to experimental solutions that can increase the quality of building construction sector.

Figure 20. Full scale mock-up of a 3D printed Load-Responsive Lattice Structure

91

Architectural Morphogenesis Through Topology Optimization

REFERENCES

Amir, O., & Bogomolny, M. (2011). Topology optimization for conceptual design of reinforced concrete

structures. In Proceedings of the 9th World Congress on Structural and Multidisciplinary Optimization.

Bendsøe, M. P. (1989). Optimal shape design as a material distribution problem. Structural and Multi-

disciplinary Optimization, 1(4), 193–202. doi:10.1007/BF01650949

Bendsøe, M. P., & Kikuchi, N. (1988). Generating optimal topologies in structural design using a ho-

mogenization method. Computer Methods in Applied Mechanics and Engineering, 71(2), 197–224.

doi:10.1016/0045-7825(88)90086-2

Bendsøe, M. P., & Sigmund, O. (2003). Topology Optimization: Theory, Methods and Application.

Berlin: Springer.

Block, P. Knippers, J., Mitra, N.J. & Wang, W. (Eds.) (2015). Advances in Architectural Geometry 2014.

Springer International Publishing.

Boyd, P. (2015). Branch technology 3d prints building walls with world’s largest freeform 3D printer –

launches 3D printed home competition.

Burger, M., & Osher, S. J. (2005). A survey on level set methods for inverse problems and optimal de-

sign. European Journal of Applied Mathematics, 16(02), 263–301. doi:10.1017/S0956792505006182

Cheung, K. C. (2012). Digital Cellular Solids: reconfigurable composite materials [Doctoral thesis].

Massachusetts Institute of Technology.

Díaz, A., & Sigmund, O. (1995). Checkerboard patterns in layout optimization. Structural Optimization,

10(1), 40–45. doi:10.1007/BF01743693

Fanjoy, D. W., & Crossley, W. A. (2002). Topology Design of Planar Cross-Sections with a Genetic

Algorithm: Part 2--Bending, Torsion and Combined Loading Applications. Engineering Optimization,

34(1), 49–61. doi:10.1080/03052150210907

Hack, N., Lauer, W. V., Gramazio, F., & Kohler, M. (2015). Mesh Mould: Differentiation for enhanced

Performance. In Proceedings of the 11th International Symposium on Ferrocement and 3rd ICTRC

International Conference on Textile Reinforced Concrete. Aachen: Rilem.

Huang, X., Zuo, Z. H., & Xie, Y. M. (2010). Evolutionary topological optimization of vibrating con-

tinuum structures for natural frequencies. Computers & Structures, 88(5–6), 357–361. doi:10.1016/j.

compstruc.2009.11.011

Irgens, F. (2008). Continuum Mechanics. Springer Verlag.

Januszkiewicz, K. (2013). Evolutionary digital tools in designing nonlinear Shaping of concrete structures

in current architecture, Concrete structures in urban areas, Wroclaw, pp. 1-6

Kamat, M. P. (1993). Structural optimization: status and promise. American Institute of Aeronautics

and Astronautics, Inc. doi:10.2514/4.866234

92

Architectural Morphogenesis Through Topology Optimization

Michell, A. G. M. (1904). The limits of economy of material in frame structures. Philosophical Maga-

zine, 8(47), 589–597. doi:10.1080/14786440409463229

Morel, P., & Schwartz, T. (2015). Automated Casting Systems for Spatial Concrete Lattices. In M.R.

Thomsen, M. Tamke, C. Gengnagel et al. (Eds.), Modelling Behaviour - Design Modelling Symposium

(pp. 213-225). Switzerland: Springer International Publishing. doi:10.1007/978-3-319-24208-8_18

Radman, A. (2013). Bi-directional Evolutionary Structural Optimization (BESO) for Topology Optimiza-

tion of Material’s Microstructure [PhD thesis]. School of Civil, Environmental and Chemical Engineer-

ing, College of Science, Engineering and Health, RMIT University.

Sigmund, O. (1994). Design of material structures using topology optimization [PhD Thesis]. Technical

University of Denmark.

Sigmund, O. (2000). A new class of extremal composites. Journal of the Mechanics and Physics of

Solids, 48(2), 397–428. doi:10.1016/S0022-5096(99)00034-4

Sigmund, O., & Petersson, J. (1998). Numerical instabilities in topology optimization: A survey on

procedures dealing with checkerboards, mesh-dependencies and local minima. Structural and Multidis-

ciplinary Optimization, 16(1), 68–75. doi:10.1007/BF01214002

Smith, D. R., & Truesdell, C. (1993). An Introduction to Continuum Mechanics after Truesdell and Noll.

Springer Verlag. doi:10.1007/978-94-017-0713-8

Thomas, H. L., Zhou, M., & Schramm, U. (2002). Issues of commercial optimization software develop-

ment. Structural and Multidisciplinary Optimization, 23(2), 97–110. doi:10.1007/s00158-002-0170-x

Watanabe, R., & Kawasaki, A. (1990). Proceeding of the 1st international symposium of FGM, Sendai,

Japan.

Xie, Y. M., & Steven, G. P. (1993). A simple evolutionary procedure for structural optimization. Comp.

& Struct., 49(5), 885–896. doi:10.1016/0045-7949(93)90035-C

Zhu, J. H., Zhang, W. H., & Qiu, K. P. (2007). Bi-Directional Evolutionary Topology Optimization Using

Element Replaceable Method. Computational Mechanics, 40(1), 97–109. doi:10.1007/s00466-006-0087-0

Zienkiewicz, O. C., & Taylor, R. L. (2015). The finite element method for solid and structural mechan-

ics. Amsterdam, Boston: Elsevier Butterworth-Heinemann.