Biology Reference
In-Depth Information
methods as described in Sec. 5; they are based on approximating the
smooth field functions of a continuous model by integrals that are being
discretized onto computational elements called particles. While the par-
ticles in discrete simulations represent real-world objects such as mole-
cules, atoms, animals, or cells, particles in continuous methods are
computational elements that collectively approximate a field quantity as
outlined in Sec. 5.1.
4.5. Representing Complex Geometries
in the Computer
Complex geometries and surfaces can be represented in the computer
using a variety of methods,
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which can be classified according to the
connectivity information they require. Triangulated surfaces
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are an
example of connectivity-based representations, as they require each tri-
angle to know which other triangles it is connected to. Establishing this
connectivity information on complex-shaped surfaces is computationally
expensive, so these representations are preferably used in conjunction
with numerical methods that operate on the same connectivity. This is
the case when using FE methods with triangular elements in simulations
involving triangulated surfaces,
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or FD methods in conjunction with
pixelated surface representations.
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An example of a complex triangulated
surface is shown in Fig. 1(a).
Connectivity-less surface representations include scattered point
clouds
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and implicit surface representations such as level sets.
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In level
set methods, the geometry is implicitly represented as an isosurface of a
higher-dimensional level function. Level sets are well suited to be used in
combination with particle methods because the level function can
directly be represented on the same set of computational particles.
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This allows treating arbitrarily complex geometries at constant computa-
tional cost, and simulating moving and deforming geometries with no
linear stability limit. The ER shown in Fig. 1(b) was, e.g. represented in
the computer as a level set in order to simulate diffusion processes on its
surface using particle methods.
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