Spatial form

3.3 Quadratic interpolation

The next logical spatial form to pick is that of a quadratic polynomial. Select three nodes on the line element, two at the ends and the third inside the element. In local space the third node is at the element center. Thus, the local unit coordinates are r₁ = 0, r2=12, and r₃ = 1. It is usually desirable to have x₃ also at the center of the element in global space. If we repeat the previous procedure using u(r) = c₁ + c₂r + c₃r², then the element interpolation functions are found to be
(3.17)H1(r)=1−3r+2r2H2(r)=4r−4r2H3(r)=−r+2r2_______________ΣHi(r)=1.
These quadratic functions are completely different from the linear functions. Note that these functions have a sum that is unity at any point, r, in the element. These three functions illustrate another common feature of all C⁰ Lagrangian interpolation functions. They are unity at one node and zero at all others: H_i (r_j) = δ_ij. In natural coordinates, on −1 ≤ n ≤ 1, they transform to
(3.18)H1(n)=n(n−1)2,H2(n)=1−n2,H3(n)=n(n+1)2.
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Introduction

Lóránt Tavasszy, Gerard de Jong, in Modelling Freight Transport, 2014

1.2.2 Inventory Networks

Inventory networks are a spatial form of organisation of inventories. Given the spatial patterns and volumes of trade, storage and (de)consolidation of flows may occur at intermediate locations that are in between places of production and consumption. The main purposes of these inventories is to keep logistics costs low by bundling inventories and transport flows and to maintain high service levels with proximity to markets. Whether or not such intermediate inventories are necessary or worthwhile depends on many factors, including the physical characteristics of goods that determine logistics costs structures (e.g. perishability) and the service requirements. The effect that these intermediate inventories have on flows is that the spatial patterns of trade are changed, i.e. new origins and destinations for transport are created. In addition, shipment sizes are determined. These decisions are usually taken by logistics managers of the firms that send or receive goods. Sometimes the role of a logistics manager is partly outsourced to one or more professional service firms (logistics service providers), in which case the decision on locations and volumes of intermediate stocks is taken jointly for several firms.
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