A Geometrical Method for Low-Dimensional Representations of Simulations

We propose a new data analysis approach for the efficient postprocessing 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 $\mathbb{R}^3$. 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, which 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, and 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.