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Architectural Morphogenesis Through Topology Optimization


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This chapter illustrates the main approach for a generative use of structural optimization in architecture. 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 micro-structures of cellular materials is represented and illustrated, emphasizing the all-important criteria and parameters for structural design. A specific example is given from the research at ACTLAB, ACB Dept, Politecnico di Milano, of performative design with lattice cellular solid structures for architecture.
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Handbook of Research on
Form and Morphogenesis
in Modern Architectural
Domenico D’Uva
Politecnico di Milano, Italy
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Title: Handbook of research on form and morphogenesis in modern architectural
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Chapter 4
DOI: 10.4018/978-1-5225-3993-3.ch004
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.
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
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.
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
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
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)
Architectural Morphogenesis Through 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)
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
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
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).
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.
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
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
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
Architectural Morphogenesis Through Topology Optimization
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.
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
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).
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)
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)
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.
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)
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)
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
Architectural Morphogenesis Through Topology Optimization
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).
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
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.
Architectural Morphogenesis Through Topology Optimization
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
Architectural Morphogenesis Through Topology Optimization
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
Architectural Morphogenesis Through Topology Optimization
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
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
Architectural Morphogenesis Through Topology Optimization
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In this paper, a topology optimization approach is presented, where uncertain load and uncertain material parameters are considered. The concept of compliance minimization, i.e., stiffness maximization, is applied based on a plane stress finite element formulation. In order to take uncertain structural load parameters and uncertain material behavior into account, the topology optimization is embedded into a reliability-based design optimization approach. Uncertain structural parameters and design variables are quantified as random variables, intervals and probability boxes (p-boxes). This allows to consider aleatory and epistemic uncertainties by means of polymorphic uncertainty models within the topology optimization. Solving optimization problems with random variables, intervals and p-boxes leads to a high computational effort, because the objective functions and constraints have to evaluated millions of times. To speed up the optimization process, the finite element simulation of the topology optimization is replaced by artificial neural networks. This includes the topology dependent maximal stresses and displacements of the structure, which are used as constraints, and also the material density distribution inside the design domain. The reliability-based optimization of structural topologies approach is applied to a cantilever structure and a single span girder.
Full-text available
Today, concrete is (in weight) the most frequently-used material on the planet after water, with a quantity of one cubic metre per person per year. Every second, 126 tonnes of cement are poured across the world, amounting to some 3.4 billion tonnes per year, or the equivalent of just over 14,000 Empire State Buildings. Over the three years from 2011 to 2013, China consumed 6.6 billion tonnes of cement, in other words more than the 4.5 billion tonnes used by the USA during the entire 20th century (Sources: USGS, Cement Statistics 1900–2012; USGS, Mineral Industry of China, 1990–2013). If we add to these figures the fact that production of cement by the clinkerisation process involves firing at temperatures of around 1450–1500 °C, it is not difficult to understand why the cement industry is one of the most polluting industries. Just like the construction sector as a whole, this industry faces many challenges today. Amongst these challenges are optimising the cement production process in the face of the growing cost of more sophisticated “raw materials” and their proven or potential harmfulness, or rethinking the supply chains and the product lifecycle to reduce grey energy, and lastly developing new construction methods. This last problem, in which architects have the most influence, should be envisioned beyond the usual constraints associated with buildings’ regulations as a great opportunity for the architectural discipline, especially through the (re)examination of new or deprecated concrete and/or cement production methods. This search for new approaches is at the heart of the works we are presenting in this paper, oriented towards integrative computational and fabrication methods for the design and realization of three-dimensional concrete-based spatial lattices.
This book presents an introduction into the entire science of Continuum Mechanics in three parts. PART I: Continuum Mechanics introduces into the Foundations using tensors in Cartesian coordinate systems, classical theory of elasticity, and fluid mechanics. PART II: Mechanics of Materials has chapters on viscoelasticity, plasticity, principles of constitutive modelling, and thermodynamics. PART III presents Tensor Analysis and fundamental equations of Continuum Mechanics in curvilinear coordinates.
Digital materials are comprised of a small number of types of discrete physical building blocks, which assemble to form constructions that meet the versatility and scalability of digital computation and communication systems. This work seeks to demonstrate the applicability of a digital material approach in designing new cellular materials and methods for assembly of structures with static reconfigurability. The science of cellular solids has enabled the widespread use of lightweight materials to meet important engineering needs, such as passive energy absorption, but they are not in widespread use for structural applications, perhaps due to a large gap between the strength and stiffness to weight ratios of popular classical solids, and the performance of known lightweight cellular materials that are produced from the same constituent material. The engineering of fiber reinforced composite materials has enabled structures with large reductions in weight for given strength and stiffness targets, but at very high design and processing costs, and many challenges producing mechanical interfaces (joints). Digital materials promise scalable methods of producing functional things with reconfigurable sets of discrete and compatible parts, but the presence of many reversible connections raises questions about the performance of the end result. Digital Cellular Solids are cellular solids that exhibit improvements in relative stiffness and strength compared to relative density, over current practices for producing lightweight materials. This is accomplished by assembling lattice geometries that perform better than any that we know how to make with traditional methods. When implemented with fiber composites, the result is not only stiffer and stronger than any previously known ultra-light material, but it presents a new scalable and flexible workflow for applying fiber composites to engineering problems.
Preface. 0: Preliminary Results. 1: Framings. 2: Bodies and Motion. 3: Kinematics. 4: Cauchy Stress Tensor. 5: Examples on Stress Constitutive Relations. 6: Noll's Simple Material. 7: Internally Constrained Materials. 8: Material Classification from Symmetry. 9: Canonical Stress Functions for Isotropic Materials. 10: Classical Infinitesimal Theory of Elasticity. 11: Shear of an Isotropic Elastic Rectangular Block. 12: Torsion of an Isotropic Elastic Circular Cylinder. References. Quotation References. Name Index. Subject Index.
Chapters contained in this book include fundamental concepts of optimum design, mathematical programming methods for constrained optimization, function approximations, approximate reanalysis methods, dual mathematical programming methods for constrained optimization, a generalized optimality criteria method, and a tutorial and survey of multicriteria optimization in engineering. Also included are chapters on the compromise decision support problem and the adaptive linear programming algorithm, sensitivity analyses of discrete and distributed systems, the design sensitivity analysis of nonlinear structures, optimization by decomposition, mixed elements in shape sensitivity analysis of structures based on local criteria, and optimization of stiffened cylindrical shells subjected to destabilizing loads. Other chapters are on applications to fixed-wing aircraft and spacecraft, integrated optimum structural and control design, modeling concurrency in the design of composite structures, and tools for structural optimization. (No individual items are abstracted in this volume)
Frequency optimization is of great importance in the design of machines and structures subjected to dynamic loading. When the natural frequencies of considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this paper, a modified SIMP model is developed to effectively avoid the artificial modes. Based on this model, a new bi-directional evolutionary structural optimization (BESO) method combining with rigorous optimality criteria is developed for topology frequency optimization problems. Numerical results show that the proposed BESO method is efficient, and convergent solid-void or bi-material optimal solutions can be achieved for a variety of frequency optimization problems of continuum structures.