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The simplest problem of material modeling
involves a single material feature -- represented by
a set of points where material properties are known.
Global material functions (functions representing material
properties) are constructed and controlled by specifying
derivatives and the value at the points of the feature.
The modeling scheme also allows to control material
function with differential, integral, analytic and explicit
constraints. Our modeling scheme relies on the generalized
Taylor series expansion by powers of a distance field
of the feature and we showed that any material function
can be constructed with our approach [1]. Figure (1)
and (2) are examples of material functions constructed
with a single feature. For both of the distributions
the value of the material function attenuate with the
exponential function of distance from the 'S' curve
(4-th order B-splne curve). In Figure (1) a constant
value of 1 is specified at the feature but for (2) it
varies linearly from one end of the 'S' curve to the
other end and the gradient value of -4 is specified
to all points of the curve.
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| Figure 1:
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Figure 2:
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A more typical situation with heterogeneous material
modeling involves several material features with known
material characteristics. Such problem boils down to
construct a single material function with material functions
of all features, constructed with the above described
approach. We employ a distance function based
interpolation technique to construct the single
material function [1,2]. The underlying notion of the
interpolation method is that the influence of a feature
is more at points nearer to it than points away from
it. The material distribution function in Figure (3)
was constructed with two material features: the 'S'-shaped
curve and an annulus region. The function specified
at the 'S' curve was (1-1.5u), where u is the distance
function of the 'S' curve. The constant material function
2 was specified at the annulus feature. 
Figure 3: Material function in figure (4) was constructed
by specifying two material functions ( see Figure
(5) ) and the differential constraint, the Laplace's
equation. The two material features defined in the example
are the union of all vertical faces (circular hole and
four vertical faces of the cube), and spiral canal surface
through the interior of the solids.
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| Figure 4:
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Figure 5:
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The theoretical and computational approach
is inspired by our earlier work on meshfree engineering
analysis, using Rvachev's Function Method (RFM) [3,4].
In particular, the approach is theoretically complete
in the sense that it allows representation of all material
property functions.
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