From
SIAM News
, Volume 38, Number 4, May 2005
The Extensive Use of Splines at Boeing
By Thomas A. Grandine
Splines are used extensively at Boeing and throughout much of the industrial world. There are very few remaining areas in either
manufacturing or engineering in which these interesting functions have yet to play a role, and their use continues to grow at a rapid
rate.
A spline, for the purposes of this article, is defined as any function made up of one or more polynomial pieces joined together
to satisfy given (and possibly different) differentiability requirements. At Boeing, these functions are used not only to represent
geometric designs, by modeling curves and surfaces that describe the edges and faces of geometric solids, but also to model
engineering analysis and performance data. As an example of the latter, splines can be used to model airplane drag as a function
of mach number, the speed of the airplane with respect to the free stream airflow.
Ideal Modeling Technology
In these contexts, Boeing uses splines in four very different kinds of applications. The most familiar is computeraided design,
where splines are used to represent the geometric entities that form the basis of productdefinition data.
A related application, common across many industries, is computeraided manufacturing. Here, in addition to representing
geometric parts, splines can be used to represent machine tool cutter paths. They can also be used as engineering data modeling
functions. Compensation tables on machine tools, for example, can be represented this way; compensation tables themselves are
used to model positional corrections to a large variety of environmental or external conditions, such as temperature.
A third application class, engineering analysis and simulation, is especially important at a large engineering company like Boeing.
Examples range from sophisticated computational fluid dynamics simulations to simple linear analysis codes involving simplified
physical models. In many cases, splines are required to represent geometry or geometric boundary conditions. In others, splines
are called on to model material properties and thicknesses, atmospheric chemical composition and reactivity, and experimental
results, such as wind tunnel data. Some codes use them to calibrate other computations, while others use the Bsplines that form
a basis for any given spline function space as finite elements. In the finite element case, engineering codes create spline
approximations to the solution of ordinary differential equations,
partial differential equations, integral equations, and differential–
algebraic equations, an approach popularized by de Boor and
Swartz in 1973.
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 Fall '09
 The Boeing Company, splines, Carl R. de Boor

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