## Integration of numerical and geometrical modelling

The numerical outcomes of the LP optimisation comprise quantities of resources to be used. Many of these resources are expressed in numbers, surface areas or densities. An LP model is expressed in linear equations (equalities and inequalities), which implies that multiplication or division of two endogenous variables is impossible. For instance, the model cannot calculate the surface area of a spatial entity and simultaneously determine its length and breadth. The LP model does not provide a spatial plan. It leaves open where the various functions in a residential area or in a building will be located. In the LP model for a building, surface areas for various rooms and functions are known, but not their shape and physical location. The model does not say anything about being located at the outside of the building or compliance with rules for escape routes. The LP model can only achieve that certain boundary conditions are satisfied which allow an acceptable spatial arrangement to be made within the numerical outcomes of the model.

The LP model, moreover, does not take into account the requirement that in a layout the various elements have to fit like the pieces of a jigsaw puzzle without overlaps, ugly discontinuities in shape, or useless corners. Requirements related to light and sight cannot be accounted for either.

The conclusion is that, in addition to numerical models, we need geometrical models to generate spatial designs and plans. The numerical models are used to calculate optimum solutions. The geometrical models serve to translate the resulting quantities into spatial plans using classical sequential heuristic methods.

In Open Design, both methods - on the one hand numerical / integral / optimising, and on the other geometrical / sequential / heuristic - are applied in an integrated way. The integration is achieved in an iterative process (Fig. 3.7).

Usually, we have available at the beginning:

1. a bill of requirements, which in Open Design is always regarded as preliminary, and legislation;

2. a sketch of how the building or residential area looks, also regarded as preliminary, and often some reference designs.

A numerical (LP) model is built on the basis of the bill of requirements, legislation and physical constraints. The LP model provides a current solution space and an optimum solution point. In parallel, a geometrical model, consisting of 2D and 3D computer drawings is made on the basis of the preliminary sketch and reference designs. The geometrical model provides a list of relevant geometry related parameters - surface areas and volumes - and the current values of these parameters.

Figure 3.7 Integration of numerical and geometrical modelling through expert feedback

We can then compare these values from the geometrical model with the associated values from the LP model. Initially, they will fit poorly. To get a better fit, expert knowledge can be used to establish desirable changes in:

1. Current values of parameters in the LP model;

2. Features of the computer drawings.

This process is repeated until a satisfactory fit is achieved, upon which we can proceed to detailed design.

Instead of using expert knowledge to change the current values of the constraints concerned in the LP model, we can also do so by conducting a sensitivity analysis (as described in Section 7.1) on the constraints (Fig. 3.8). Conversely, if we wish to correct the geometrical model without calling on expert knowledge, we have to proceed as indicated in Figure 3.9.

In that case the feedback loop consists of two parts:

1. Extend the numerical model with geometry related parameters (which can be derived from the );

2. Conduct a sensitivity analysis (Section 7.1) on these parameters to establish the required changes in the features of the computer drawings.

Proceed to detailed design

Use sensitivity analysis to establish changes in current values of constraints in the numerical model

Figure 3.8 Integration of numerical and geometrical modelling through sensitivity analysis of constraints

The integration of the numerical LP model and the geometrical model, as described here, is absolutely essential to ensure that the model concerned has sufficient reality value, i.e. the extent to which the model reflects reality. Using a numerical model exclusively may give results which can turn out to be completely unrealistic when it comes to translating them into shapes and physical locations. Conversely, nice drawings - made by pencil or computer drawings - can turn out to be completely unrealistic because they violate physical and financial constraints.

Such loss of reality value can be avoided by continuously checking that the results of one model are meaningful in the other and vice versa. Hence, the necessity of integrating numerical and geometrical modelling.

Proceed to detailed design

Figure 3.8 Integration of numerical and geometrical modelling through sensitivity analysis of constraints

Proceed to detailed design

Extend numerical model with geometry related parameters

 Use sensitivity analysis on these parameters to establish changes in features of computer drawings

Figure 3.9 Integration of numerical and geometrical modelling through sensitivity analysis of geometry related parameters

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