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The scatter associated with design variables can be treated by the same method, if
one puts data items containing variation of design variables as the first candidates for
bifurcation points. The corresponding
-modes will represent sensitivities of
simulation results to variation of parameters. The remaining scatter represents
indeterministic part of the dependence. The corresponding
Ψ
-modes are bifurcation
profiles and their g-coefficients are those hidden variables which govern purely
stochastic behavior of the model. One can either take hidden variables into account
when performing reliability analysis, or try to put them under control for reducing
scatter of the model.
Ψ
5
Examples
5.1
Audi B-Pillar Crash Test
The model shown on Fig.1 contains 10 thousand nodes, 45 timesteps, 101
simulations. Two parameters are varied representing thicknesses of two layers
composing a part of a B-pillar. The purpose is to find a Pareto-optimal combination of
parameters simultaneously minimizing the total mass of the part and crash intrusion in
the contact area. To solve this problem, we have applied the methods described in
Sec.2, namely RBF metamodeling of target criteria for multiobjective optimization
and PCA for compact representation of bulky data. Based on these methods, our
interactive optimization tool DesParO supports real-time interpolation of bulky data,
with response times in the range of milliseconds. As a result, the user can
interactively change parameter values and immediately see variations of complete
simulation result, even on an ordinary laptop computer.
In more details, Fig.1 shows the optimization problem loaded in the Metamodel
Explorer, where design variables (thicknesses1, 2) are presented at the left and design
objectives (intrusion and mass) at the right. First, the user imposes constraints on
design objectives, trying to minimize intrusion and mass simultaneously, as indicated
by red ovals on Fig.1 (upper part). As a result, “islands” of available solutions become
visible along the axes of design variables. Exploration of these islands by moving
corresponding sliders shows that there are two optimal configurations, related cross-
like, as indicated on Fig.1 (middle). For these configurations, both constraints on
mass and intrusion are satisfied, while they correspond to physically different
solutions, distinguished by an auxiliary velocity criterion. For every criterion also its
tolerance is shown corresponding to 1-sigma confidence limits, as indicated by
horizontal bars under the corresponding slider as well as +/- errors in the value box.
This indication allows satisfying constraints with 3-sigma (99.7%) confidence, as
shown on the images. The Geometry Viewer, shown at the bottom of Fig.1, allows to
inspect the optimal design in full details.
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