Analysis of Bulky Crash Simulation Results: Deterministic and Stochastic Aspects - Simulation and Modeling Methodologies, Technologies and Application - page 216

In this paper we will concentrate on the stochastic aspects of simulation processes.
Industrial simulations, e.g. virtual crash tests, often possess a random component,
related to physical and numerical instabilities of the underlying simulation model and
uncertainties of its control parameters. Under these conditions the user is interested
not only in the mean value of an optimization criterion, e.g. crash intrusion, but also
in its scatter over simulations. In practice, it is required to fulfil optimization
objectives with a certain confidence, e.g. 6-sigma. This task belongs to the scope of
robustness or reliability analysis.
Often, the confidence intervals are so large that one has to reduce scatter before
optimization. There is a part of scatter deterministically related to the variation of
design variables, which can be found by means of sensitivity analysis. The other part
is purely stochastic. It can be triggered by microscopic variations of design variables
and - even if they are fixed - by the numerical process itself, e.g. by random
scheduling in multiprocessing simulation. These microscopic sources are then
amplified by inherent physical instabilities of the model related e.g. to buckling,
contact phenomena or material failure. Stochastic analysis allows to track the sources
of scatter, to reconstruct causal chains and to identify hidden parameters describing
chaotic behavior of the model. If uncontrolled, these parameters propagate scatter to
the optimization objectives. The challenge is to put them under control, at least
partially, e.g. by predeformation of buckling parts, adjustment of contact conditions,
placement of additional welding points etc. In this way the scatter of simulation can
be suppressed and optimization becomes more robust.
In Sec.2 we will overview the methods for metamodeling of bulky simulation
results; in Sec.3 we describe stochastic methods for reliability and causal analysis;
Sec.4 presents applications of these methods to real-life examples in automotive
design. The approaches presented in this paper have been implemented in software
tools DiffCrash [9-11] and DesParO [12-15] and have been subjects of international
patent applications (DPMA 10 2009 057295.3 and PCT/ EP2010/061439).

2 RBF Metamodel

Numerical simulations define a mapping y=f(x): R n R m from n-dimensional space of

simulation parameters to m-dimensional space of simulation results. In crash test
simulation the dimensionality of simulation parameters x is moderate (n~10-30),
while simulation results y are dynamical fields sampled on a large grid, typically
containing millions of nodes and hundreds of time steps, resulting in values of m~10 8 .
High computational complexity of crash test models restricts the number of
simulations available for analysis (typically Nexp<10 3 ) which is preferred to be as
small as possible.
Metamodeling with radial basis functions (RBF) is a representation of the form

→

f(x)= i=1.. Nexp c i Φ (|x-x i |), (1)

where () are special functions, depending only on the Euclidean distance between

the points x and x i . The coefficients c i can be obtained by solving a linear system

Φ