A Proposal For Digitally Prepared Designs Of 3D Buildings Replacing Restricted Boltzmann Machines On Generating Layouts
By: Aswin Vijayakumar
Tags: Space Syntax, 3d Modeling, Statistics, Building, Design

The above presentation of dropping the building structure starts when you plan the building layout for incident solar light. That is when the designer considers the layout of the building with restrictions to access to each bounded region via doors and windows. I can demonstrate how elementary the design process is, by which, considering two points acting as a source and sink interacts with the system producing a descriptive representation of the room and consequently the building.

Parameters

A client must be able to search through the design space using three factors:

(1) Light requirement for the building (2) Circulation requirement (3) Movement requirement

A Metric Table exists which maps from approximate input room sizes to these requirements

Metric Table

Next, is a Design Chart that maps from individual room types to aspects of a room type

Design Chart

A Load Distribution Map that specifies how much load each room of a given building should take

The coloured regions are representing loads taken by the rooms

The design proposal stays on with the Adjacency Matrix as:

Adjacency Matrix

Here, 1 indicates the rooms are adjacent to each other whereas 0 indicates they are not adjacent rooms.

Design Philosophy

The design philosophy considers designing the structures using a load distribution map taking into consideration the aspect ratio and the energy of the building and by approximately modelling the movement of people within the building. They are divided into light considerations, space requirements and application of proximity models for defining the structures within the building.

A Lagrangian Map

The space within a room is defined using a Lagrangian Map with proximity considerations of:

- Not too close to the Windows and Doorway - Not too far from the Windows and Doorway

The windows and doorway are used to construct the functions of not too far distances and not too close distances are coded. The programming code will look something like this:

not_too_far_distances = exp(distances)

not_too_close_distances = exp(-1 * distances)

Next, is to find two Lagrangian points that maximise and minimise the Lagrangian values. Such two points are considered to be source and sink which interacts with the System. The idea is shown here.

data_h = distr_affinity

# initialising the minimal data_h values to zero

data_h.affinity -= np.min(distr_affinity.affinity)

data_f = distr_distance

result = pd.DataFrame(columns=result_cols)

epsilon = np.arange(0.05, 0.2, 0.05)

for eps in epsilon:

value_df = data_h.iloc\

[data_f[(np.abs(data_f['d0'] - data_f['d1']) < eps) & \

(np.abs(data_f['d1'] - data_f['d2']) < eps)].index, 3]

if(len(value_df) != 0):

val = value_df.astype('float32').idxmin()

result = result.append(dict(zip(result_cols,

[value_df[val], eps, val])), ignore_index=True)

epsilon_chosen = 0.05

eps_df = result[(result['epsilon'] <= epsilon_chosen)]

beta = np.mean(np.square(_distances[eps_df.row_index.values.astype(int).tolist(), :]))

data_l = data_f.loc[:, 'distance'].values + beta * data_h.loc[:, 'affinity'].values primal_value = data_l[data_l == np.max(data_l)]

dual_value = data_l[data_l == np.min(data_l)]

What I’ve done above is optimize the Lagrangian to come up with a formulated beta factor. The result is a regression of not_too_close_distances and not_too_far_distances. Finally we find the primal and dual points that maximize and minimize the Lagrangian values. These two points act as a Source-Sink system for a single room. Each room connects to the adjacent room using the Access Door through which the Source-Sink Systems communicate. It can be shown using formulation that a Lagrangian Map is appropriate for such cases. The formulation for such a source-sink system is given below:

In a hypothetical Source-Sink System:

The first equation is the state space equation of the input to the Source, A being the State Matrix and B being the Input Matrix

The second equation is the state space equation of the input to the Sink, E being the State Matrix when connecting to the Sink and F being the Feedback / Transmission Matrix to the Input W

The third equation is the state space equation of the output from the Sink, C being the Output Matrix and D being the Feedback / Transmission Matrix to the Input W

Taking the difference of 2nd equation and 1st equation, and squaring them on both sides, we get a similar equation to:

Input u or w is flow vector and x is state vector

Residual Sum of Squares (RSS) taken on the Left hand-side matches with the Square term of the value where x represents the state vector and u/w represents the flow vector.

In a Lagrangian Map, since the Residual Sum of Squares (RSS) can be interpreted as:

- Exposures of exposure of a Variable

We can determine our state by taking a Lagrangian on our state space model.

The Lagrangian is applied to our space for the window, access door and another small window as shown below:

Points representing the Primal and Dual Variables. The large distance from the Doorway and Window is exploited by the magenta dot and the window nearby the aqua dot acts as one of the main factors of circulation.

The Proximity Chart

We can describe that the RSS (Residual Sum of Squares) is an opportunity for another room to get connected to this Lagrangian Room. Balancing the RSS values observed is our initial motive.

Using our Design Chart, we construct a Principal Components Analysis Matrix that constitutes to our Design considerations for developing a Proximity Chart.

Derivation of Proximity Chart

The Factor Analysis Algorithm

The Factor Analysis algorithm models the posterior from the prior and it maximises the Likelihood of the variable. Using Factor Analysis, it is effective to model the light requirements from every window and access area such as an access door. Hence, we can form a search space using Lagrangian and Factor Analysis, which are then aggregated to denote the Metric Table values.

The Factor Analysis algorithm maximises the KL Divergence between the posterior and the prior, so what we represent our light requirements as will be effectively translated to Ephemeric data of the location. As far as we record our client preferences for sunlight or light for a typical room, we know our system is modelled appropriately.

Similarly the Lagrangian Map takes in the input data from a stochastic distribution regarding flow across the window and it is easily represented by our space and proximity requirements, or in turn termed as circulation for the room. The Lagrangian Map as well models the movement of people, it is just a matter of changing the functions and the state space equation to model the space and proximity values for movement.

Conclusion

We should strictly avoid stochastic solutions as the solution here is predominantly deterministic in nature. As a matter of fact, modelling improves the output of the layout generated and we can restrain from model-based iterations for generating the layout.

I have seen in some papers, they use a Restricted Boltzmann Machine (RBM) algorithm to model the accessibility, floors and various other parameters. Such descriptions require a lot of stewardship data as opposed to configuration data.