  We propose a new data analysis approach for the efficient post-processing of
bundles of finite element data from numerical simulations. The approach is
based on the mathematical principles of symmetry.
  We consider the case where simulations of an industrial product are contained
in the space of surface meshes embedded in R^3. Furthermore, we assume that
distance preserving transformations exist, albeit unknown, which map simulation
to simulation. In this setting, a discrete Laplace-Beltrami operator can be
constructed on the mesh, which is invariant to isometric transformations and
therefore valid for all simulations. The eigenfunctions of such an operator are
used as a common basis for all (isometric) simulations. One can use the
projection coefficients instead of the full simulations for further analysis.
To extend the idea of invariance, we employ a discrete Fokker-Planck operator,
that in the continuous limit converges to an operator invariant to a nonlinear
transformation, and use its eigendecomposition accordingly.
  The data analysis approach is applied to time-dependent datasets from
numerical car crash simulations. One observes that only a few spectral
coefficients are necessary to describe the data variability, and low
dimensional structures are obtained. The eigenvectors are seen to recover
different independent variation modes such as translation, rotation, or global
and local deformations. An effective analysis of the data from bundles of
numerical simulations is made possible, in particular an analysis for many
simulations in time.
