Optimization Tools for the Design of Structures

Abstract

Inspired by the seminal paper by Michell (1904), engineers have investigated several tools for the optimization of structural shapes and systems to be employed during the design process. Structural optimization attracts increasing interest in the building industry, especially in the design of high-rise buildings. By selectively distributing the structural members in the building, the efficiency of the resulting design can be optimized; often aesthetically pleasant form can be achieved. Although Michell trusses represent a valuable starting point in defining optimal structural systems, analytical solutions have been derived only for relatively simple load conditions and geometries. Recently, along with the increased computing power, efficient numerical methods have been developed to generate optimal structural solutions for various problems. This paper presents recent developments in optimization methods which have been used for the design of novel structures. Concepts of growth, genetics, and emergence are introduced as context into the fundamental concepts presented. Growth patterns in nature can service as guidelines of how an optimal structure might be conceived. Genetics provide an understanding of how the blueprint of life, DNA, can be used to develop optimal solutions using genetic algorithms. Emergence theory suggests that interactions among parts serve to create something that is not realized until individual parts work in concurrence to comprise the whole. This concept is displayed through topology optimization. Several high-rise structures are described as examples of how these concepts can be applied to the development of structural systems.

Full text

Optimization Tools for the Design of Structures
M. Sarkisian1, E. Long2, D. Shook3, C. S. Doo4
1SE, Director, Skidmore, Owings & Merrill LLP, San Francisco, CA
2SE, Associate Director, Skidmore, Owings & Merrill LLP, San Francisco, CA
3PE, Skidmore, Owings & Merrill LLP, San Francisco, CA
4PhD, SE, Skidmore, Owings & Merrill LLP, San Francisco, CA
ABSTRACT
Inspired by the seminal paper by Michell (1904), engineers have investigated several tools
for the optimization of structural shapes and systems to be employed during the design process.
Structural optimization attracts increasing interest in the building industry, especially in the design
of high-rise buildings. By selectively distributing the structural members in the building, the
efficiency of the resulting design can be optimized; often aesthetically pleasant form can be
achieved.
Although Michell trusses represent a valuable starting point in defining optimal structural
systems, analytical solutions have been derived only for relatively simple load conditions and
geometries. Recently, along with the increased computing power, efficient numerical methods have
been developed to generate optimal structural solutions for various problems. This paper presents
recent developments in optimization methods which have been used for the design of novel
structures. Concepts of growth, genetics, and emergence are introduced as context into the
fundamental concepts presented. Growth patterns in nature can service as guidelines of how an
optimal structure might be conceived. Genetics provide an understanding of how the blueprint of
life, DNA, can be used to develop optimal solutions using genetic algorithms. Emergence theory
suggests that interactions among parts serve to create something that is not realized until individual
parts work in concurrence to comprise the whole. This concept is displayed through topology
optimization. Several high-rise structures are described as examples of how these concepts can be
applied to the development of structural systems.
INTRODUCTION
Natural forms are not created by a single force or event, nor by the simple
coexistence of many parts; it is the interaction of each part to its immediate
surroundings that initiates processes that over time produce coherent forms.
Michael Weinstock, “The Architecture of Emergence”
Observation of the natural environment reveals underlying patterns and intrinsic
relationships which transcend individual species and have elemental relevance to the built
environment. Explorations of the golden ratio, fractal geometry, and biomimicry have profiled
these phenomenon, but have not fully understood the depth of the processes from which they are
yielded. The study of genetics has provided a fundamental understanding to the composition of life
and how life changes over time. Nature’s intrinsic rules and relationships govern how elementary
components combine to create complex organisms and systems. These rules and relationships often

orchestrate the growth of the higher-level system without any global oversight or guidance. This
process is known as emergence. In this organic process, the whole becomes more than the sum of
its parts.
The following sections, Growth-Genetics-Emergence, examples investigate how both
classical biomimicry and modern emergent design techniques can be used for the design of novel
and efficient structures. A focus on the tools used for these applications brings to light the
importance of understanding the science behind these natural phenomenons.
GROWTH
Naturalists have long studied the growth of
organisms of all scales to look for themes and patterns
which transcend individual species. These have been
well documented in several texts including the seminal
work by D'Arcy Wentworth Thompson, On Growth
and Form (1917). These principles can be applied to
design to improve not only the efficiency of the
structure, but to reveal aesthetically interesting built
environments.
Turgut Tower. A proposed concept design
scheme for the Turgut Tower in Istanbul, Turkey
brings fundamental principles of nature to the design
of structural systems (see Figure 1). In nature, a state
of equalized stress is often resolved at a node by three
stiffer elements which are 120° apart. This is also
easily inferred by a simple free body-diagram of a
node with three equal forces acting on it. This
principle is demonstrated in the in-fill of a dragon fly
wing. The lattice pattern of the dragon fly wing is an
expression of the forces which flow through the
membrane and how they collect to the main body of
the insect during flight.
The proposed design concept for Turgut Tower
is a perimeter structural frame composed of two layers.
The first is an external mega-frame which acts as the
perimeter gravity and lateral load resistance system. A
secondary frame is set back from the mega-frame and
has dual architectural and structural purposes.
Architecturally, the in-fill pattern provides sun shading
to help mitigate thermal effects on the interior
environment. Structurally, it acts as an in-fill
structural system, analogous to the in-fill of the dragon
fly wings; collecting loads and brining them to stiffer
elements. This application of biomimicry does not
explicitly use optimization, but gleans concepts of flow
from nature and applies it to structural systems.
Figure 1. Principles of the Dragon Fly
Wing applied to the Turgut Tower
Concept Scheme

The internal lattice acts in concurrence with the external mega-bracing and internal
reinforced concrete shear wall core as the primarily gravity and lateral force resistance systems.
The in-fill also brings a quality of redundancy and repetition which is often found in natures
designs. This redundancy allows for multiple load paths in the event one or more members are
damaged, for example, in a seismic event.
Transbay Transit Tower Competition. The proposed design for the Transbay Transit
Tower Competition in San Francisco, California is inspired by natural forms. These
mathematically-derived forms define systems that are safe, sustainable, cost-effective to construct,
and provide optimal performance in seismic events.
Figure 2. Natural Forms Which Exhibit Logarithmic Spirals
The logarithmic spiral—found in forms ranging all
the way from shells, seeds, and plants to spider webs,
hurricanes, and galaxies—is interpreted and applied to the
ultra-tall tower, scientifically mimicking natural force flows
of a cantilevered structure to its foundation (see Figure 2).
The spiral inherent in these natural forms traverses around a
fixed center and gradually recedes from the center. Engineer
Anthony Michell (1904) captured this behavior through his
research in the early 1900s by describing the radiating lines of
a pure cantilever, where force flow lines of equivalent
constant stress result in specific spacing and orientations from
the fixed support to the tip of the cantilever (Figure 3, Michell, 1904). The result is the most
efficient cantilever system with the least material. The Michell Truss diagram is mathematically
interpreted and overlaid on the tower form defining an optimal perimeter bracing configuration
(Figure 4).
Figure 3. Michell Truss
Figure 4. Application of Mitchell Truss to Transbay Tower Competition Entry

GENETICS
Recent discoveries in the field of
genetics have brought to light the complexity,
robustness, and beauty of how one of the
fundamental characteristic of life, DNA,
composes organisms. This is revealed through
the study of DNA which compose an organism
through florescent microscopy. DNA is
composed of two key components: genes and
switches. Genes are the functions, the process
which can create parts of an organism: a liver,
a bone, a cell. Switches tell which genes to
implement, when to act, and to what degree.
Florescent microscopy attaches a protein
which will glow with florescence under
ultraviolet light to a switch. Then, once the
gene has had sufficient time to express itself,
photographs are taken under ultraviolet light.
The results are beautiful and telling of what
genes are used by the identified switches.
Genes can be reused and combined with other
genes for different applications. Similar to a
piano player combining keys to yield a chord;
switches combine genes to produce life. See Figure 5 (Carroll Labs, 2011) for example of an insect
whose tissue has been colored according to what genes were used to produce it. This displays the
minimalism and yet complexity of the biological environment.
Genetic Algorithms. Genetic algorithms have been
used in a wide-range of applications for improved
performance in numerous trades such as the aerospace,
automobile, and medical industries. This simple, yet robust
algorithm facilitates multi-variable and multi-objective
searches in large, often poorly defined, search spaces. Early
investigations of evolutionary algorithms were conducted by
Holland (1975) and inspired from observations made by
Darwin (1859). GA is a heuristic optimization method which
utilizes trial-and-error of mass populations as a basis of
optimization. To demonstrate GA concepts, a simple truss
optimization problem is illustrated in the following text.
For GA optimization to begin, an initial population
must first be generated. A population is a group of
candidate-solutions. For the example truss problem, a
population would consist of a set of potential truss
configurations. Each truss would have a different member
configuration but the loading and boundary conditions would
be the same. This concept is illustrated in Figure 6.
Figure 5. Example of Florescent Microscopy
(Carroll Labs, 2011)
Figure 6. Example GA Operations

With an initial population generated, candidate-solutions are evaluated. Their fitness, or
score, is determined by a fitness function. For this example a truss’s fitness is the sum of
normalized deflection and normalized weight. This GA is a minimization algorithm, thus the sum
of the normalized values is taken for the fitness. Both deflection and weight must each be
normalized to minimize bias in the fitness score. Analysis software can be used to quickly
determine the deflection and weight of each truss in the population. Increased weight and/or
deflection increase the fitness of a candidate-truss and therefore diminish its chances to be selected
by the GA for inclusion into future generations.
The initial population is the first parent population and is used to generate the first child
population. The child population is a new set of candidate-solutions which are derived from the
parent population. The child population is to be the same size as the parent population. Each
member of the child population is to be generated using GA-operators. Parameters to be optimized
by the GA are contained in a vector of values termed a chromosome. The first type of GA
operation is called ‘crossover’. A crossover operation takes two parents and combines
characteristics from each parent to form a child. The second type of GA operation is called
‘mutation’. A mutation operation takes one parent and
alters one or more characteristics of the parent to form
a child. Various derivatives of these operations exist
throughout literature. These concepts are illustrated in
Figure 6.
After the child population is generated, each
child is evaluated and fitness determined. Next, parent
and child populations are combined into a single pool
of candidate-solutions. The pooled set is ranked
according to each member’s fitness score. For the
truss example problem, the truss with lowest fitness
score is considered best and the truss with the highest
fitness score is considered worst. With the pooled
parent and child populations ranked, the top 50% are
elected to be the parent population for the next
generation. The remaining trusses are discarded.
The key to genetic algorithms is the same as nature: diversity. Diversity of the population
will ensure that search space is properly interrogated by the search algorithm. The power of genetic
algorithms is a broad search which is facilitated by population-based iterations. The final
population may reveal multiple minima (or maxima) in the search space. Optimization results
should be viewed in light of all generation cycles, not just the final answer. The story of how the
optimization routine obtained an optimal result can be just as useful as the result itself.
Parametric Structural Analysis. Parametric modeling has been developed for Rhinoceros
(2011) with a plug-in called Grasshopper (2011). This advanced method of code writing makes use
of pre-defined functions which are represented by “blocks” with inputs and outputs in a graphical
interface. The primary use of Grasshopper (2011) is geometry generation as it is part of a 3D
geometric modeling environment, but the use of custom scripting permits additional functions and
advanced analysis.
Figure 7. Example GA Results

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Designers investigated a diagrid of
varying slope for improved efficiency under
lateral loading. To conduct this study a
parametric model which could automatically
generate the complex geometry based on a set
of salient inputs, conduct structural analysis,
and perform optimization was required. The
previously mentioned Grasshopper (2011) and
Karamba (2011) tools were combined with a
genetic algorithm, Galapagos, to conduct
parametric structural optimization.
Optimization variables include slope,
or pitch, of the diagrid over the height of the
building; and the fitness is taken to be
maximum displacement of any node in the
analysis model. A population of 50 candidate
solutions was employed at each generation
cycle. Approximately 30 generation cycles
were required for convergence. Therefore,
over 1,500 diagrid patterns were generated and evaluated by the optimization routine. Crossover
and mutation operations were employed at each generation to various measures depending on the
desired diversity in the optimization. Although diversity is encouraged through all generations, the
range of solutions does reduce as generation cycles progress to allow for refinement of solutions.
As shown in Figure
11, parametric structural
optimization results reveal a
pattern which evolves over
the height of the tower. At
the base, members are
primarily vertical, while at
the top the members’
transition to 45°. This is
reflective of dominate
stresses in the extreme
fibers of a cantilever: axial
at the base and shear near
the free end. Therefore, the
optimal diagrid pattern is
found to be response to the
natural flow of forces at the
perimeter of the structure.
As opposed to traditional
moment frame system which attempt to resist loads through flexure, the identified diagrid resists
loads in the most efficient manner possible, axially.
As a means of comparison, a conventional shear wall core and moment frame system was
designed to meet the same architectural, seismic, and wind requirements. The resulting
Figure 10. Wuxi XSD Phase 7 Tower
Concept Scheme
GA Iterations Final
Figure 11. Representative Optimization Iterations and Final Elevation

conventional system was found to use approximately 20% more material than the optimized core
and diagrid. Thus, an efficient and aesthetically interesting structural system is conceived.
Al Sharq Tower. The Al Sharq Tower is to be located
in Dubai, United Arab Emirates (Figure 12). The plan form of
the structure is based on nine-adjoining cylinders. Since
traditional perimeter columns are not desired, the design teams
considered a cable-supported perimeter.
The 102-story residential tower has 39m x 39m floor
plan, a height of 365m, and therefore an aspect ratio nearing
10:1. The proposed structural system consists of reinforced
concrete systems with perimeter spiraling high-strength
galvanized steel cables. The lateral system is composed of
intersecting sets of parallel shear walls and perimeter high-
strength galvanized steel cables. A typical floor framing plan is
shown in Figure 13. The perimeter cable system consists of
approximately 70 kilometers (44 miles) of high-strength
galvanized steel cables. Initial cable profiles suggested a
helical formulation for each.
With the concepts of the genetic algorithm described,
its application to the optimization of cable filigree of the Al
Sharq Tower is now considered. GA operations are to optimize
three types of variables which include cable diameter, spacing
and pitch. At each generation the child population is formed by
executing crossovers and mutations. 25% of the child
population is created by crossovers, 25% by mutations, and 50%
are survivors from the parent generation. Of children produced by a crossover, 50% are single-
point crossover operations and 50% are shuffle crossover operations. Of children produced by a
mutation, 40% are blind mutations and 60% are Gaussian mutations. Since children are produced
by moderate and aggressive GA-operations, sufficient diversity should be maintained in the
population. This is done to ensure the search space is adequately interrogated. For this study a
fitness function which minimizes material and deflection is employed. Although these goals seem
competitive, Maxwell (1890) has shown they are complementary. Optimization terminates when
variations in the optimal member of the population has little variance from generation to generation.
Figure 12. Al Sharq Tower
Figure 13. Al-Sharq Tower Plan Figure 14. GA Variables Figure 15. GA Program

To implement GA for the optimization of Al Sharq cable filigree,
several tools are needed. Previously mentioned tools which are part of the
Grasshopper (2011) programming environment were not used in this study,
instead custom scripts and graphical user interfaces were generated. A
general purpose programming environment is needed for GA-operations,
interaction with finite element software, and collection of results. Visual
Basic .NET of Microsoft’s Visual Studio is well-suited for this task due to its
robust interaction with other software, relative ease of use. An illustration is
provided in Figure 15 of GA tools and their relationships.
Results of optimization reveal a pattern which transforms over the
height of the structure as shown in Figure 16. The pitch of the cables are
primarily vertical at the base and transition to 45° at the top. This profile is
reminiscent of the previously mentioned Wuxi SXD Phase 7 diagrid. For this
study the change in pitch over the height of the tower is determined to be
parabolic. It is noted that moment demands of cantilever beams are parabolic.
This trend has been observed in two independent studies and reveals a
potential optimal pattern for tall, slender towers. This could be an intrinsic
pattern of tall structures and could be traversed to other similar forms.
EMERGENCE
An entity or process is called emergent if some quality is not realized
when only considering sub-components independently. For example a few
ants are nothing remarkable on their own, but when they work together in a
colony they can accomplish extraordinary tasks. The success of the colony is
entirely dependent on the basic interactions between ants without any
supervising guidance. Instincts that have developed over millions of
generations guide the colony through survival, including such critical operations as food collection,
home building, and colony organization. As mentioned in the opening quote, emergence is found
in the interaction of each part to its surroundings which produces coherent form. Observing form
can be useful, but understanding the
relationships which produce form are far
more valuable as they can be transposed
other scenarios.
An intriguing example of
emergence is a work created by the
artist David Nash entitled Ash Dome
(Nash, 2007). Ash Dome is a set of 22
ash trees planted in a circular pattern
(see figure 17). The trees were grown
under typical conditions except that the
artist modified the trunk of each tree by
pruning it and tying it to the ground
every few years at various locations. This caused the trees to grow at a stepped incline. Here a
natural growth pattern was given an additional restriction that forced it to achieve a goal that was
not originally part of the plan of the maturing trees. The result was an intimate, exciting space
Figure 16. Final
Cable Profile
Figure 17. Ash Dome, David Nash

which is both unexpected and inviting. This example demonstrates that emergence can be natural,
man-made, or a combination of the two. Emergent processes can result in both functional and
aesthetic environments, as Ash Dome reveals.
Topology Optimization. The principle of emergence is observed to
free-form material optimization called topology optimization. Here,
homogonous materials are analyzed and materials are re-distributed
according to observed demands and boundary conditions. Topology
optimization, at a local level, moves material to areas which yield higher
structural efficiency. These local changes develop over several generations
and global structural systems being too emerge from the local optimization
processes. Thus, the local redistribution of materials results in global
structural schemes which are not readily apparent prior to optimization.
Gemdale. The Gemdale Tower Competition submission is planned
for Shenzhen, China (Figure 18). The 350 meter tall 71-story superstructure
consists of structural steel floor framing and composite metal deck slabs.
The lateral system for the building consists of a central shear wall core and
perimeter frame. The central core is composed of reinforced concrete walls
located around passenger elevators and service areas. The perimeter frame is
composed of an optimized filigree pattern of concrete filled steel tube
members.
Members are to occur at the perimeter or “skin” of the building and
act as part of the lateral and gravity systems. This location allows for greater
lateral system efficiency and the opportunity to use the structure as an
architectural expression for the built environment. To determine perimeter
member profiles topology optimization is employed (Optistruct, 2011).
Given a homogenous design space, topology optimization manipulates material densities to
determine an optimal structural system based on re-distribution of strain density. That is, topology
optimization moves material from regions of
low stress to regions of high stress over several
optimization cycles to increase stiffness and
minimize material. During re-distribution of
materials, algorithms identify potential discrete
member locations and then modify their size
and orientation for efficient transfer of loads.
Effectively, the algorithm takes a uniform
surface, and through local optimization of
materials, a global structural system emerges.
This emergent structural system is both
efficient and often aesthetically interesting.
First an initial form is determined that
meets architectural design intent with input
from the structural design team (Figure 19).
Before optimization, the form is specified to be
a uniform shell of constant material density and
thickness. Lateral loads and foundation
supports are specified in the analysis software.
Figure 18. Gemdale
Tower Competition
Figure 19. Form, Optimization, Interpretation

The topology optimization routine is defined by optimization goals and restraints. An optimization
goal is the maximization of global stiffness with a specified constraint to only use 50% of the initial
material which is a uniform, homogenous shell. Topology optimization is then conducted.
Results are displayed graphically (Figure 19) showing regions of high material density (red)
and low material density (blue). From this, regions of high density are studied (Figure 16). Finally
the structural and architectural design teams determine an interpreted structural system that meets
architectural design and structural design intents (Figure 19). Performance of the optimized
perimeter frame is compared to conventional construction of similar material quantities. Results
indicate that the optimized structural frame reduces building drift by 25% when compared to the
comparable moment frame. Through the employment of topology optimization in a free-form
manner an emergent process is utilized, and a unique solution is derived for this specific form. By
equally distributing the strain energy at a local level, a global structural system is realized which is
both efficient and aesthetically interesting.
CONCLUSIONS
Growth, genetics, and emergence are ideas which can guide designers towards efficient and
innovative built environments. Principles of growth are shown to be useful in the conceptualization
of structural systems which mimic patterns and forms in nature, but are limited in application. The
study of genetics and application of genetic algorithm optimization provides a robust and efficient
means of optimization through mimicking natural processes. Emergence theory, as applied through
topology optimization, is determined to be a robust investigation into the relationships of individual
elements in a system and how those relationships can be used to create unique and efficient
structural responses to form.
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