Multidimensional Grids for Parametric Design and Analysis

We introduce the notion of $D$-grid representation, where $D$ is the grid dimension, for the purpose of parametric studies. This formalism is a methodological effort and contribution to the field of parametric design and analysis.

In such a study a system is analyzed regarding several variables named parameter. Each parameter can vary in a discrete or continuous range of scalar values. If not, the modeling of the system can easily be reinterpreted in that way. For instance a (x,y,z) point parameter can be understood as 3 scalar parameters.

Complex multivariable systems often lead to complex and costly analysis process. In practice, it is almost impossible to know the state of the system for every state of inputs ; nor analytically ; nor numerically. It might be a question of pure feasibility but it might also be a question of time or money.

Thus, it is common to analyze the system in a finite number of interesting states, where the choice of those states, and the corresponding values of the input parameters, comes down to the knowledge of the designer.

Depending on the sharpness of the grid, the designer would be interested in the results at grid nodes or, possibly, to interpolate results to predict any state of the system, thus reducing its margin of error.

We model the set of combinations of input values as a $D$-grid. A $D$-grid is a $D$-dimensional grid generated by $D$ finite and strictly ordered sets, where each grid dimension corresponds to a variable parameter of the study. The finite number of states that a (scalar) parameter can have, and where the system is to be evaluated, are stacked in ascending order into a set.

We found that the proper comprehension of this object, as detailed below, leads to a solid and generic methodology to organize and conduct a parametric study. It structures the way results can be browsed and interpolated over the parametric space.