| This example shows a flexible design
combining geometric and analytic constraints in a single
representation. The problem of smoothing (or fairing)
stands as finding a surface which goes through the given
sets of points. For example, Figure
1 shows dimensionally heterogeneous regions over
which restrictions are given on the surface:
The fairing problem may also be considered
as an interpolation problem. It has no unique solution
in the sense that there are infinitely many functions
interpolating any given data. A suitable function can
be chosen by putting additional constraints on the interpolant.
Often such constraints appear as a requirement of minimization
of some quantity. For example, the well known Lagrange
interpolation minimizes the degree of the interpolating
polynomial. Many interpolation schemes use minimization
of potential energy of tension or bending as a means
for controlling the shape of the interpolant. Interpolating
cubic splines minimize potential energy of a bending
beam fixed at data points. In the case of functions
of two independent variables, minimization of potential
energy of membrane or a thin plate can be used. In all
such cases, the interpolation problem is a special case
of some boundary value problem.
On the other hand, mathematical formulation
of any boundary value problem consists of a differential
equation (or equivalent variational statement) constraining
the distribution of physical field inside domain and
boundary conditions specifying the interaction of the
field with the external environment. In this case, interpolation
of the prescribed boundary conditions describes the
behavior of the solution in the vicinity of the boundaries;
the behavior of the interpolant away from the boundaries
is quite arbitrary. Thus, the solution of a boundary
value problem can be viewed as an interpolating function
that extends the boundary conditions into the domain,
with differential equation playing the role of a constraining
or smoothing operator.
Depending on the particular application,
the designed surfaces are often chosen to minimize one
of several functionals:
-
Potential energy of tension of thin
membrane;
-
Potential energy of bending of thin
plate;
-
Energy of thin plate in tension
analogy, which is a basically a linear combination
of the two previous functionals.
Figure 2(a) shows
a function transfinitely
interpolating the given values. The function shown
in Figure 2(b) minimizes a potential
energy of tension of thin membrane. Functions presented
in Figures 2(c-f) minimize an energy
of thin plate in tension for different ratios between
tension and bending.
For more information see the reference
[1].
Figure 1: Dimensionally heterogeneous regions
Figure 2: (a) transfinite interpolant;
(b-f) results of fairing. Click
here to see the animation.
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