Civil Engineering Reference
In-Depth Information
features were identified in an automatic assembly process. Karamete et al. (2013) presented an
algorithm for discrete Booleans with applications to FE modelling of complex systems in which
triangulated surfaces are combined by means of Boolean operations derived from the intersec-
tion of the triangular elements. Foucault et al. (2013) applied the ADF to composite surfaces in
which the geometric features and the constraints of the objects were well respected. Cuillière
et al. (2013) discussed the MG and transformation for topology optimisation of objects with
heterogeneous geometry.
In this section, a simple data-verification scheme for discretised curved surfaces and
objects defined by the boundary surface modelling specifications will be presented. The
quality of individual elements, overall topological structures and geometrical correctness
in terms of intersections, close touches and sharp angles will all be examined and verified.
This will only provide a basic routine surface-verification process for MG and FE analysis;
however, much more in-house checking and correcting procedures can, and will definitely,
be implemented following the same concept of validation in automation adapted to specific
applications under various working environments.
8.1.1.1 Boundary surface of solid objects
The boundary surface of an object to be meshed can be conveniently represented by a collec-
tion of triangular facets. The following is a typical data structure for such a representation
in which the boundary surface
B
is given by
B
= {Δ
i
= (P
1
, P
2
, P
3
)
i
, i = 1, N
B
}
The spatial points
P
= {(x
i
, y
i
, z
i
), i = 1, N
P
}, where N
P
and N
B
are, respectively, the number
of points and the number of boundary triangular facets in
B
. This data structure is the sim-
plest possible, but is broad enough to include solid objects based on B-rep models in which
B
consists of all the triangles on the discretised boundary surfaces.
8.1.2 Preliminary checks and preparations
Simple routine checks are done on the boundary surface
B
to find out the limits of the co-
ordinates, the range of the node numbers, the number of active nodes, the spacing between
nodes, the quality of each boundary triangle, etc.
8.1.2.1 Limits of points
The limits of the co-ordinates x
min
, x
max
, y
min
, y
max
, z
min
and z
max
are calculated:
x
=
min,
x
x
=
max,
x
similarlyf
or yy z nd z
min
,
,
max
min
i
max
i
in
max
i
=
1
,
Np
i
=
1
,
Np
8.1.2.2 Normalisation of co-ordinates
To have a better idea about the size of the elements, the co-ordinates are mapped onto the
interval between 0 and 100 through shifting and the multiplication of a scaling factor λ:
x
i
↦ λ(x
i
− x
min
), y
i
↦ λ(y
i
− y
min
), z
i
↦ λ(z
i
− z
min
)
where λ is a scaling factor such that xi,
i
, y
i
, z
i
∈ [0,100], i = 1, N
P
. Of course, [0,100] is rather
arbitrary, and other ranges could have been chosen for convenience in particular applications.
