MMP.18:4.2 - Construct the interface relation

Express the exchange as a relation between the component descriptions. A deterministic map is appropriate when the supplied result determines the needed input. Use a broader relation or conditional law when several inputs remain possible.

For a conversion, translate the quantity and its reference point. For an aggregate, state the operation over the specified region or interval. For example, an amount over [t0,t1] is the integral of a rate over that interval; a rate at t0 alone determines that amount only under an additional evolution assumption.

Let x range over values expressed by the supplying model and y over those used by the receiving model. A relation H(x,y) states the compatible pairs. A.3.3.TR supplies composition with other relations, and MMP.10 expresses the joint possibilities. MATH.18 helps determine which operations and consequences the interpretation preserves. Equality of numerical values is one possible relation, not the default.

Derive the property the map must preserve. For amounts at locations, a linear map b=W*a preserves their total for every a when every column of W sums to one. For values of an intensive field, a map preserving a constant field instead has every row sum to one. If amounts must remain nonnegative, nonnegative weights are an additional sufficient condition. The needed property chooses the map; a convenient interpolation routine does not choose the property.

The two conditions solve different problems. A common field value should remain that value on another discretization. The total amount distributed among cells should remain the same total. When both requirements matter, construct a map with the appropriate geometry or integration weights rather than substituting one condition for the other.