BGM is a shape optimization method whose driving force is the
magnitude of surface stresses on structural components. This method
is based on the observation that biological structures, such as tree
trunks and animal bones, actually add layers of biological material to
the surface areas affected by activation stress. In the 1990s, Mattheck
proposed extending this concept to CAE and structural part design
by adding material to surfaces with high stresses and/or removing it
from surfaces where stresses were low.
A volumetric growth ε based
on the actual von Mises stress σvon Mises and a reference activation
stress σref was proposed:
The optimization behavior (add/remove material) is controlled
by setting the reference stress value σref. The method proposed in
this study is based on a more versatile and evolved version of the
Mattheck model and prescribes displacing the node in the direction
normal to the surface:
where σnode is the stress evaluated at each node, σth is a threshold value
for the stress defined by the analyst, and σmax and σmin respectively
are the maximum and minimum stress values in the current set. d is
the maximum offset between the nodes on which the maximum and
the minimum stresses are evaluated; this parameter is defined by
the user to control the displacement of the nodes whilst limiting the
possible distortion of the mesh. Nodes on the surface to be optimized
can either be moved inward, if the stress on node is less than the
threshold value, or outward, if the evaluated stress is greater.
The proposed BGM implementation can be guided by a variety of
stress/strain definitions (such as maximum principal stress, maximum
plastic strain, accumulated equivalent plastic strain, etc.), so that the
method can be tailored to the specific design requirements.
Propagation to the full computational mesh of the BGM field
calculated on the surfaces can be achieved using radial basis
functions (RBF) mesh morphing. RBFs were introduced in the early
1960s to interpolate scattered multidimensional data and to allow
interpolation of a scalar field anywhere in the defined space from
known values at discrete points called source points.
Since the data to be interpolated at x comprises scattered scalar
values at N source points Xsi, the interpolating function s(x) can be
obtained as
by summing the interactions at the probe location x with all the
source points that have been computed by weighting all the radial
interactions (calculated as Euclidean distances between each source
point) with γi coefficients, and the probe location transformed by the
radial function φ(r). The N unknown γi are calculated by imposing the
passage of the s(x) function through all the (known) source points.
In mesh morphing, the source points are the surface mesh nodes
over which the displacement is controlled, whilst the entire set of
nodes within the mesh is updated by receiving the interpolated
deformation field. Individual RBFs interpolate the three components
of the displacement. The above mesh morphing technique and BGM
can be successfully coupled in an optimization approach consisting
of the following steps:
- The baseline geometry is discretized into finite elements; load
and constraints are applied, and the FEM solution is evaluated;/
-
- From the FEM solution, the nodal stress on the surfaces to
be optimized are retrieved, σth and d are set by the user and
Snode displacement along the surface normal for each node is
evaluated for each selected node;
- The evaluated displacements are used to configure the RBF
problem by imposing them as values to be interpolated (values
on source points), the user can optionally set additional source
point values to complete the morphing configuration (i.e.,
points to maintain fixed);
- The FEM model mesh is morphed, and the FEM solution is
evaluated again;
- The stress values on the surface to be optimized are analyzed: if
the new stress levels can be further optimized, the procedure is
iterated from step 2; otherwise, the geometry can be considered
optimized.
In the methodology described above, the stress analyst must set
two BGM parameters: the threshold stress σth and the maximum
displacement d. The first parameter represents the stress level value
on which the optimization procedure will try to converge; the second
parameter represents the maximum displacement allowed within a
single iteration of optimization: the lower its value, the greater the
number of iterations that will be required to reach the optimum, and
the lower the risk of mesh distortions that could invalidate the FEM
model will be.