# Computational Architecture – Development, Optimization and Design. Advances in Architectural Geometry 2014

Computational Architecture, Development, Optimization and Design

This is the poster I presented at the AAG14, trying to sum up my Master’s Thesis project.

### Introduction

Modern architecture brought innovative ways to approach the field. New materials, integrated workflows, complex structures, energetics, structural optimization and new shapes, specifically free forms. In such landscape in constant evolution, computational and design tools must keep up the pace as long with designer’s in-depth knowledge of them. The process bringing to construction an architectural idea is long and requires a great amount of variations. Those may be due to the client, the architect, the industry, the engineer and anyone who’s working on it. Not only traditional design methods are running obsolete, but the possibilities arising from having the whole project fully parametrized are becoming more and more compelling. Having the whole project parametrized makes easy to explore all possible solutions and eventually find the best one, using scripts appropriately. This paper presents the workflow necessary to design a free form roof from scratches and to optimize it. In order to do so, several problems arose which would have been very hard to deal with, without the automatisms inherent to a parametric conception.

The first problem of course, is to draw such a complex surface and the relative structure. As said traditional tools, such as CAD modeling, are not suited enough. Furthermore, it is well known that projects change hundreds of times during their conception and even their realization: updating it with CAD and subsequently with all the softwares involved (for structural analysis, energetics, fabrication, costs, etc…) is unfeasible.

Those architectures are highly expensive, thus every effort should be conducted to reduce costs. This practice is commonly referred to as optimization, since the aim is optimizing the use of material or the fabrication (both related to costs) to have best results with least expenses. Certain assemblies are expensive beyond imagination or not even possible to realize. In this case, the geometry itself must be modeled to make feasible what otherwise could be unfeasible.

For what concerns fabricability, the project presents a lot of different nodes, this again should catch the engineer’s attention and make him struggle to find a way to fabricate them in series to drive costs down.

For what concerns the project as a whole, the interoperability between softwares is crucial. As aforementioned, in the workflow of free form architectures a lot of softwares find their way in (e.g for structural analysis, energetics, construction phases, cost analysis, drawing). Models must pass seamlessly from one software to the other, without loss of information and within short time. Expanding the concept, the best would be found if the entire project, with all the information it carries, could flow from the first step to the last one without any external input required. The only way do so is by having the whole thing written in a parametric fashion, so that when an input is changed, the whole project updates as long with its various outputs.

### The project: between dream and reality

The Scuola Normale Superiore technical division asked to conceive and design a roofing for the Scuola‘s inner courtyard within a framework of accommodating the surrounding area. The courtyard is approximately 20x20m and the highest building is around 22m.
The building is a city landmark situated in the very city center, relevant both for its beauty and its being the pulsing heart of one of the most renowned university in Italy and in the world.

The client was very strict with its requirements. The new structure could not in any way relay on the ancient one and it should leave a certain gap between it and the surrounding roofs for natural air conditioning. Therefore, some kind of vertical structure should have been studied for the glass roofing to stand on, but that new structure should not prevent passers-by to enjoy the surrounding facades.

Designing complex structures requires iterating over various aspects, such as design, structural and geometry optimization and fabricability. Each aspect is related to each other, so the project evolves in every direction.

Several hypothesis have been studied from the architectural point of view and from the structural one. Free form architectures cannot be designed in sequential stages for a change in one aspect will most probably induce a change in one another. Those architectures must be designed in coordination with every professionals working on it and possibly their evolution must follow a loop (see figure on the left)

At last, the best solution was the one shown in the render. The shape resembles a Calla and the beam pattern is composed by intersecting logarithmic spirals. Here one can immediately spot a big difference with traditional gridshells used for roofing. Usually gridshells are found using a relaxation process, thus making the structure working mostly in pure compression. This case is different and was adopted because the client required the structure not to rely on the historical one. Because of this, beams work mainly in bending but by intertwining them the overall stiffness increases and resistance increase too, making the structure redundant.

### Form finding

As already said, traditionally free form roofing can be found by a relaxation procedure. Applying the same procedure in this case would result in a cone-like shape with approximately zero Gaussian curvature. A different approach could be finding the minimal surface joining some given curves, then superimposing those two solutions.

#### Minimal surfaces

A minimal surface has the least area respecting a set of boundaries (i.e. curves). Minimal surfaces have several interesting properties such as having zero mean curvature everywhere. Moreover, if we think such surface as being composed of an elastic tensile wrap, in every point the sum of stresses in any direction must be equal to zero or the applied external load if there is any.

This property is used to discretize the surface with a mesh where each edge is a spring with zero rest length since we want to find the minimal area. This procedure is not free from errors, the most evident one is that this way principal directions are somehow set by the initial mesh. The minimal surface found has been tweaked by adding an inverse gravity load so that it would assume an inverse catenary shape.

#### Tweaked surface

Because of the way beams entwine each other, intersection planes are always perpendicular to each other. Nodes are made of cast iron all of the same cross shape, then their arms are cut to meet the required length. Glass panels are clamped to spiders, all identical and fixed by a stem which is to be plugged into a hole on the upper face of the cross and welded.

The eam pattern is derived by intersecting logarithmic spirals projected onto the surface. But there it comes another problem. In fact, fabricability is the key issue for this kind of architecture, with node-torsion being one of its main enemies, therefore the beam pattern must be checked against this issue. It’s known that conical meshes have no node-torsion. Since the beam weaving is made of quads, it could be transformed in a mesh. Once the mesh was found, an optimization procedure was conducted to make it the most conical possible.

### Structural optimization

Structural parts of such complex free forms can hardly be dimensioned. In addition, given the high redundancy, an optimization procedure can produce interesting results and in structures with lots of beams a sharp decrease of costs. In this project structural optimization was approached in two ways:

• Loop optimization;
• Evolutionary multi-objective optimization (EMO).

The former is quite straightforward. A loop tries to optimize beam section’s thickness according to the ratio between load and resistance. The loop runs until a certain threshold is achieved, in this case there were two:

• The whole structure is verified;
• Displacement and deformations consistent with Eurocodes.

Even though this method not necessarily leads to the best solution possible, EMO does. EMO searches the whole solution domain therefore it is part of the so-called global optimization algorithms.

#### Loop optimization

This is how the iterative optimization works: 1) The structure is analyzed and for each beam the Utilization Factor UF=load/resistance is calculated; 2) Each thickness is updated according to the equation plotted; 3) Thickness can’t be smaller than 4mm for because of local buckling; 4) Go to 1.

The loop optimization works as follows:

1. Initially the algorithm analyzes the structure and calculates the ratio between carrying load and resistance (later on referred to as Utilization Factor, UF) for each beam.
2. According to the UF the section thickness is updated using the following expression, plotted in the above figure $f\left(UF\right)=t\left(1+a\left(UF-\overline{UF}\right)\right)$ , where  a=1.2 is a parameter to speed up convergence and $\overline{UF}=0.9$ is the target Utilization Factor, less than 1.0 to guarantee a certain degree of safety. Basically, when a section is overstressed the algorithm increases its thickness and viceversa. Please note that this process is done using real (float) numbers even if thickness should be an integer, because float numbers produce a better numerical stability than integers.

All beams are RHS 250×100, thickness varies from 4-12.5 mm and relative quantities are shown in figure.

1. The structure is analyzed again using the updated thicknesses and UFs are recalculated. The loop ends when all $UF\leq0.9$ and stiffness requirements are met.
2. Once convergence is reached, thicknesses were reduced using the following mapping: $[4,32]\rightarrow[4,12.5]$. This is because the above expression does not attempt to reduce overall mass, so this had to be done after. A few beams were no more verified and had to be updated manually. Then, thicknesses that were defined by real numbers had to be rounded up to the nearest integer, actually to the first integer included in a given set of industrial rectangular hollow sections

### Panelization

Free forms are not just structure but rather aesthetic. Those shapes need to be covered by panels. This procedure is known as panelization. Glass is possibly one of the most difficult material to use for panelization. There are concurrent necessities in this practice:

• Resemblance with initial shape (position and derivative continuity);
• Cost increases with panel curvature; \item Since each panel is different it requires a specific mold, which are really expensive.

The solution of this problem is not obvious and several papers have been written on that subject (see references at the bottom). Every architecture has its own story and requirements, thus requiring specific adaptation of general concepts such as those presented in the above mentioned papers. In this project panelization has followed two steps:

1. Reduction of overall curvature;
2. Reduction of the quantity of molds.

#### Curvature reduction

Even if shape is set, one can attempt to reduce average curvatures by slightly moving the surface control points, thus obtaining an optimized shape very close to the initial one. To do so, a principle of fictitious energy minimization was used:

$min(E_{tot})=A\sum_{i=0}^{n}(E_{disp})+ B\sum_{j=0}^{f}(E_{\chi})$

where $E_{disp}=(d-d_0)^{2}$ is counted for every node and $E_{\chi}=|\frac{\mathbf{b}\times\mathbf{d}}{|\mathbf{b}\times\mathbf{d}|}\cdot(\mathbf{c}-\mathbf{a})|$ is counted for every face, with $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$ being the points where the two diagonals pass. Weight coefficients $A=0.8$ and $B=0.2$ were used.

Minimization of the above expression can be done in several ways, in this project a physical spring particle system was used, Kangaroo a Grasshopper plug-in. The new surface was finally generated through those new points.

In blue are shown panel curvature before optimization, whilst in red they are shown after that. The overall distribution shifts toward the right, with low radii (r<12m) disappearing and high ones (r>30m) produce single curved or planar panels.

#### Mold rationalization

Curvature of the molds used with the number of panels relative to each one.

The reduction of the amount of molds is done as follows. First of all, for each panel the two main curvatures are calculated in the middle point. Then, the two lists of main curvature thus found are merged and divided in sub-lists, each one having the higher curvature value equal to 1.20 times the lowest one. To each sub-list the same mold curvature was assigned, equal to the average curvature in the set. At the end of this procedure, each panel was assigned to a group where every component had the same constant curvature. Then, panels having $\chi=0.033$ (i.e. with a curvature radius of 30m) where considered planar.

Mold distribution on the surface.

Because of the peculiar shape studied, the panelization was done using three types of mold:

• Planar;
• Cylindrical: single curved panels, zero Gaussian curvature;
• Toroidal: double anticlastic curved panels, negative Gaussian curvature;

In the following figures the types of molds used are shown and the number of panels they are used for.

### Output data

All the drawings, data, analysis results, are assembled and automatically added to report and graphical visualization. This is crucial to have a clear view on the project phase and to share it with all the professionals working on it. Imagine if every variation, every optimization result should have been drawn by hand or all results data put manually into reports. It would take time for engineers and architects that otherwise would be spent for more productive tasks.

### Conclusions

This paper showed the difficulties encountered when designing a free form architecture and the cutting-edge tools required to deal with it. It’s clear how traditional tools where intermediate steps are not linked to each other, cannot provide a sufficient help to the designer. Moreover without the full parametrization of the project it would be impossible to automatically optimize it by having a script screening thousands of different solutions. The procedure here described allowed to greatly reduce costs, especially for what concerns glass panels production, and it was made possible by the increasing use computer engineering is having in the architectural landscape.

# Dodo – Scientific tools for Grasshopper

Food4Rhino

### Dodo

Dodo is a plugin for Grasshopper where I’m collecting scientific tools useful for computational designers. You can find the plugin comprehensive of some examples here and the full manual here. If you have any questions, ideas to implement or insults, do not hesitate to contact me!

### Iso surfaces

An isosurface is the collection of points in space mapping to the same value. In a 2D domain the isosurface is an isoline and an example of them can be found in topographical maps where mountains and depression are represented as closed curves each one indicating a different height.
Isosurfaces are the equivalent of isocurves but are the result of a 3D domain where each point in space has a certain value. They connect all these points together thus forming one or multiple surfaces.

One of the main ways to produce these isosurfaces is using Marching Cubes or the theories that generated from this one. In this plugin Marching Tetrahedra have been used since they can manage better singular points.
Among the new implementations there are some making use of derivatives of the scalar field given, but unfortunately the grasshopper Field component does not calculate derivatives. Maybe in the future I will implement a new customized field component but for the time being I thought that using the already available tools is best.

This is an Isosurface produced by superposing the field generated by a cylinder and one generated by random points in space.

Resources: Marching Tetrahedra of Paul Bourke.

### Non-Linear Optimization

Non-linear optimization differs from linear optimization for the function it tries to minimize and/or the constraints it is subject to are non linear, which is the case for almost several engineering application. NL-opt makes use of gradient free algorithms to achieve this and in order to do that it needs to calculate an approximate value of the gradient for a given point. Once this is done, it tries to move to a neighbour point according to the interpretation of the gradient given by the specific engine criteria.

This plugin uses the .NET implementations of the famous NLOpt library that you can find here, whilst the .NET implementation is here.
Due to the difficult interoperability of the dll, some engines are missings and non linear inequality constraints are harder to fulfill.

Optimization trend using Dodo’s NL-Optimization

### Artificial Neural Network

Neural Network schema

Artificial Neural Networks can date back to the second half of 1900 and take their name by their similarity with neuron interconnections in brain. ANN are composed by a series of layer each containing a number of neurons, each of which connects to its peers in the layer before and after, as shown in the following picture. The example in the picture shows an ANN mapping data from a 3D domain to a 2D on by making use of a hidden layer. Hidden layers are those which are not directly connected with the input data nor with the output. An ANN can have any number of neurons and hidden layer which can bring to completely different results as well as increased computational time. Neurons’ connections are drawn as arrow pointing in the direction in which the data flows. Usually neurons perform very basic operations and they are connected through value functions which are the targets ANNs optimize in order to fit the input data to the expected results.
The field of application of these ANN is data fitting, prediction and classification and their implementation is treated below.

These mentioned are not the only applications though as ANN can be used with unsupervised learning as Self-Organizing Map (SOM)

The library on which Dodo relies for Neural Networks can be found here.

### ANN – Supervised Training

This type of learning algorithms use sample inputs matched desired output values during the learning phase. The goal of this method is to shape to ANN so to provide a close fit output when given an input.

#### Delta Rule

Delta Rule dividing the sampling space in two parts

Delta Rule learning is one of the two threshold finders in Dodo’s ANN along with Perceptrons Rule.
In the example below the 8 vertices of a cube are assigned to two groups: group 0 above and group 1 below. The trained ANN finds the decision boundary depicted as a grey plane which splits the decision space in two. In the picture on the right a grid of 11x11x11 is sampled and only the points having a predicted class of 1 are shown which demonstrates they are respectful of the boundary surface.

#### Perceptron

Perceptron dividing the sampling space

In this example the NN is fed with a number of points in the 2D space, being part of three distinct groups which leads to a 3D solution space identified. To represent the group to which they are part of, a 3D vector can be used having for each dimension either 0 or 1. Moreover, in order to have a better graphical understanding, the vectors are multiplied by 255 so to transform them into colors.

#### Aproximation

Supervised training can be used to make an ANN be able to predict values of a series. In the example shown below 8 couples of number were used and plotted in teal. In red it is shown the approximating curve generated by the ANN trained using back-propagation learning. The two curves neatly superpose on the first part and diverge a bit on ending, but this behaviour can change widely playing with the learning coefficients.

#### Prediction

For prediction the procedure is the same as before but we feed the ANN with one single series of number but making it use sub-series of 5 numbers to predict the 6th and use the difference between the latter and the expected  value to rate the learning. In the example below 16 values from a cosine have been used and in the picture underneath one can see how well the ANN predicts the following 34 values without progressive increase of the error.

### ANN – Unsupervised Training

The ANN is given sample inputs without expected results and it will organize itself in order to find patterns in this series. Dodo implements two algorithms for this category: SOM and Elastic Network.

#### Elastic Network

Elastic Network used to solve the travelling salesman problem

Unsupervised training and elastic network can be used to find a solution to the travelling salesman problem. The elastic network is initially brought to the center of the data set, then slowly tries to replicate the values (position) basically working as a dumped spring system. The resulting weights are the Euclidean distances between the data points and the neurons’ output can be used how well the NN fit to the data samples. The neurons’ output can then be used to see how similar another dataset is to the first one and it is my understanding that this patter recognition strategies have been successfully used to recognize cancer cells from pictures.

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