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Proceedings of the:
3rd ASME/JSME Joint Fluids Engineering Conference
July 18-23, 1999, San Francisco, California
A B-SPLINE BASED COLLOCATION METHOD TO APPROXIMATE THE SOLUTIONS
TO THE EQUATIONS OF FLUID DYNAMICS
Richard W. Johnson and Mark D. Landon
Idaho National Engineering and Environmental Laboratory
Idaho Falls, ID 83415-3805
Keywords: B-spline, numerical method, collocation, CFD
ABSTRACT
The potential of a B-spline collocation method for
numerically solving the equations of fluid dynamics is
discussed. It is known that B-splines can resolve complex
curves with drastically fewer data than can their standard shape
function counterparts. This feature promises to allow much
faster numerical simulations of fluid flow than standard finite
volume/finite element methods without sacrificing accuracy. An
example channel flow problem is solved using the method.
INTRODUCTION
The amount of time required to obtain a numerical solution
is one of the most important criteria by which the usefulness of
computational fluid dynamics (CFD) is judged by its users.
These include basic researchers who use direct numerical
simulation (DNS) to compute the details of turbulence,
combustion or other physical phenomena; commercial code
developers; design engineers; and particularly designers who
employ optimization with analysis to optimize designs. An
important consideration for design and optimization engineers
is the overall design time required wherein a number of CFD
simulations are necessary to find the optimal design. At the
other end of the spectrum are those who are engaged in DNS of
fluid flow and want to push the envelope (increase the Reynolds
number) for which DNS calculations are possible. Of course,
the trade-off to speed is accuracy, and CFD users require that
numerical CFD solutions exhibit acceptable accuracy.
Solution speed is affected by the speed of the computer
used for the simulation, especially the level to which the code is
optimized for the specific hardware, including the level to
which the solution process has been parallelized when parallel
cpus are available. The speed is also a function of the efficiency
and stability of the numerical solution algorithm and the
efficiency of the numerical method used.
The present paper examines the potential for a new
approach to approximating solutions to the equations of fluid
dynamics for use in CFD. Although the method can technically
be classified as a finite element collocation method, it was
arrived at through a totally different thought process and is
more akin to the fields of Computer Aided Design (CAD) and
Computer-Aided Geometric Design (CAGD). In the fields of
CAD/CAGD, curves and surfaces are commonly represented
through the use of Bezier, B-spline, and NURBS (nonuniform
rational B-spline) curves and surfaces. The curves and surfaces
that are defined by these versatile and flexible advanced
geometric functions are typically used to represent the geometry
of solid objects. In fact, Bezier curves were originally created to
mathematically define arbitrary curves and surfaces that
automobile designers created for automobile shapes so that
numerically controlled machines could be used in the
manufacturing process.
The obvious connection between advanced geometric
functions that can represent arbitrary geometric shapes and CFD
is that the solution to the equations of fluid dynamics can be
thought of as curves in 1-D, surfaces in 2-D and volumes in 3-
D. We seek B-spline curves, surfaces or volumes that represent
the solutions to the fluid equations. Because n-degree B-spline
curves and surfaces are defined by very few data points and are
C"-i continuous in terms of their parametric derivatives, we
hope that we can reduce the amount of computational effort
required to find numerical solutions. Furthermore, we can
achieve any level of accuracy we choose by choosing the degree
of the curve and the number of splines that constitute a curve or
surface. Finally, we need not define a mesh for the solution
process in the usual sense. In fact, we will not integrate the fluid
equations; we can simply evaluate the B-spline functions to
obtain all of the terms that appear in the equations. This last
feature of the method is makes it a collocation method.
A finite element collocation method is characterized by
employing the Dirac delta function as the test function and then
integrating in the usual FEM manner, see Carey and Oden
(1983), and Lapidus and Pinder (1982). Using the Dirac delta
function for the test function has the effect of removing the
integral, leaving only the original differential equation. Because
there is no integration to perform on some finite element, the
differential equation must be evaluated on some network of
points; these are called collocation points. Of course, the idea of
simply evaluating curve or surface values and their derivatives
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